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Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone... 1IntroductionLet HHbe a real Hilbert space with inner product ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle and induced norm ‖⋅‖\Vert \cdot \Vert . In this paper, we consider the variational inequality problem (VIP) of finding a point p∈Cp\in Csuch that (1)⟨Ap,x−p⟩≥0,∀x∈C,\langle Ap,x-p\rangle \ge 0,\hspace{1em}\forall x\in C,where CCis a nonempty closed convex subset of HH, and A:H→HA:H\to His a nonlinear operator. We denote by VI(C,A)VI\left(C,A)the solution set of the VIP (1).Variational inequality theory, which was first introduced independently by Fichera [1] and Stampacchia [2], is a vital tool in mathematical analysis, and has a vast application across several fields of study, such as optimisation theory, engineering, physics, operator theory, economics, and many others (see [3,4, 5,6] and references therein). Over the years, several iterative methods have been formulated and adopted in solving VIP (1) (see [7,8,9, 10,11] and references therein). There are two common approaches to solving the VIP, namely, the regularised methods and the projection methods. These approaches usually require that the nonlinear operator AAin VIP (1) has certain monotonicity. In this study, we adopt the projection method and consider the case in which the associated nonlinear operator is pseudomonotone (see definition below) – a larger class than monotone mappings.Now, we review some nonlinear operators in nonlinear analysis.Definition 1.1A mapping A:H→HA:H\to His said to be (1)γ\gamma -strongly monotone on HHif there exists a constant γ>0\gamma \gt 0such that (2)⟨Ax−Ay,x−y⟩≥γ‖x−y‖2,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge \gamma \Vert x-y{\Vert }^{2},\hspace{1em}\forall x,y\in H.\hspace{1.4em}(2)γ\gamma -inverse strongly monotone on HHif there exists a constant γ>0\gamma \gt 0such that ⟨Ax−Ay,x−y⟩≥γ‖Ax−Ay‖2,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge \gamma \Vert Ax-Ay{\Vert }^{2},\hspace{1em}\forall x,y\in H.(3)Monotone on HH, if (3)⟨Ax−Ay,x−y⟩≥0,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge 0,\hspace{1em}\forall x,y\in H.\hspace{4.75em}(4)γ\gamma -strongly pseudomonotone on HH, if there exists a constant γ>0\gamma \gt 0such that (4)⟨Ay,x−y⟩≥0⇒⟨Ax,x−y⟩≥γ‖x−y‖2,∀x,y∈H.\langle Ay,x-y\rangle \ge 0\Rightarrow \langle Ax,x-y\rangle \ge \gamma \Vert x-y{\Vert }^{2},\hspace{1em}\forall x,y\in H.(5)Pseudomonotone on HH, if (5)⟨Ay,x−y⟩≥0⇒⟨Ax,x−y⟩≥0,∀x,y∈H.\langle Ay,x-y\rangle \ge 0\Rightarrow \langle Ax,x-y\rangle \ge 0,\hspace{1em}\forall x,y\in H.(6)Lipschitz-continuous on HH, if there exists a constant L>0L\gt 0such that (6)‖Ax−Ay‖≤L‖x−y‖,∀x,y∈H.\Vert Ax-Ay\Vert \le L\Vert x-y\Vert ,\hspace{1em}\forall x,y\in H.\hspace{1.55em}If L∈[0,1)L\in {[}0,1), then AAis said to be a contraction mapping.(7)Sequentially weakly continuous on HH, if for each sequence {xn}\left\{{x}_{n}\right\}, xn⇀ximpliesTxn⇀Tx,x∈H.{x}_{n}\rightharpoonup x\hspace{1em}{\rm{implies}}\hspace{1em}T{x}_{n}\rightharpoonup Tx,\hspace{1em}x\in H.From the above definitions, we observe that (1)⇒(3)⇒(5)\left(1)\Rightarrow \left(3)\Rightarrow \left(5)and (1)⇒(4)⇒(5)\left(1)\Rightarrow \left(4)\Rightarrow \left(5). However, the converses are not generally true. Moreover, if AAis γ\gamma - strongly monotone and LL- Lipschitz continuous, then AAis γL2\frac{\gamma }{{L}^{2}}- inverse strongly monotone (see [12,13]).The simplest known projection method for solving VIP is the gradient method (GM), which involves a single projection onto the feasible set CCper iteration. However, the algorithm only converges weakly under some strict conditions that the operator is either strongly monotone or inverse strongly monotone, but fails to converge if AAis monotone. The classical gradient projection algorithm proposed by Sibony [14] is given as follows: (7)xn+1=PC(xn−λAxn),{x}_{n+1}={P}_{C}\left({x}_{n}-\lambda A{x}_{n}),where AAis strongly monotone and LL-Lipschitz continuous, with step size λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right).Korpelevich [15] and Antipin [16] proposed the extragradient method (EGM) for solving VIP (1), thereby relaxing the conditions placed in (7). The initial algorithm proposed by Korpelevich was employed in solving saddle point problems, but was later extended to VIPs in both Euclidean space and infinite dimensional Hilbert spaces. The EGM method is given as follows: (8)x0∈Cyn=PC(xn−λAxn)xn+1=PC(xn−λAyn),\left\{\begin{array}{l}{x}_{0}\in C\\ {y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {x}_{n+1}={P}_{C}\left({x}_{n}-\lambda A{y}_{n}),\end{array}\right.where λ∈0,1L\lambda \in \left(0,\frac{1}{L}\right), AAis monotone and LL-Lipschitz continuous, and PC{P}_{C}denotes the metric projection from HHonto CC. If the set VI(C,A)VI\left(C,A)is nonempty, then the algorithm only converges weakly to an element in VI(C,A)VI\left(C,A).Over the years, EGM has been of interest to several researchers. Also, many results and variants have been developed from this method, using the assumptions of Lipschitz continuity, monotonicity, and pseudomonotonicity, see [17,18, 19,20] and references therein.Due to the extensive amount of time required in executing the EGM method, as a result of calculating two projections onto the closed convex set CCin each iteration, Censor et al. [8] proposed the subgradient extragradient method (SEGM) in which they replaced the second projection onto CCby a projection onto a half-space, thus, making computation easier and convergence rate faster. The SEGM is presented as follows: (9)yn=PC(xn−λAxn)Tn={w∈H:⟨xn−λAxn−yn,w−yn⟩≤0}xn+1=PTn(xn−λAyn),∀n≥0,\hspace{3.75em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {T}_{n}=\left\{w\in H:\langle {x}_{n}-\lambda A{x}_{n}-{y}_{n},w-{y}_{n}\rangle \le 0\right\}\\ {x}_{n+1}={P}_{{T}_{n}}\left({x}_{n}-\lambda A{y}_{n}),\hspace{1em}\forall n\ge 0,\end{array}\right.where λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right). The authors only obtained a weak convergence result for the proposed method. However, they later introduced a hybrid SEGM in [7] and obtained a strong convergence result. Likewise, Tseng [21], in the bid to improve on the EGM, proposed Tseng’s extragradient method (TEGM), which only requires one projection per iteration, as follows: (10)yn=PC(xn−λAxn)xn+1=yn+λ(Axn−Ayn),∀n≥0,\hspace{0.6em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {x}_{n+1}={y}_{n}+\lambda \left(A{x}_{n}-A{y}_{n}),\hspace{1em}\forall n\ge 0,\end{array}\right.where AAis monotone, LL-Lipschitz continuous, and λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right). The TEGM (10) converges to a weak solution of the VIP with the assumption that VI(C,A)VI\left(C,A)is nonempty. The TEGM is also known as the forward-backward method. Recently, some authors have carried out some interesting works on the TEGM (see [22,23] and references therein).In this work, we consider the inertial algorithm, which is a two-step iteration process and a technique for accelerating the speed of convergence of iterative schemes. The inertial extrapolation technique was derived by Polyak [24] from a dynamic system called the heavy ball with friction. Due to its efficiency, the inertial technique has become a centre of attraction and interest to many researchers in this field. Over the years, researchers have studied the inertial algorithm and applied it to solve different optimisation problems, see [25,26, 27,28] and references therein.Very recently, Tan and Qin [29] proposed the following Tseng’s extragradient algorithm for solving pseudomonotone VIP: (11)sn=xn+δn(xn−xn−1)yn=PC(sn−ψnAsn)zn=yn−ψn(Ayn−Asn)xn+1=αnf(zn)+(1−αn)zn,\left\{\begin{array}{l}{s}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1})\\ {y}_{n}={P}_{C}\left({s}_{n}-{\psi }_{n}A{s}_{n})\\ {z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{s}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({z}_{n})+\left(1-{\alpha }_{n}){z}_{n},\end{array}\right.\hspace{2.4em}δn=minεn‖xn−xn−1‖,δifxn≠xn−1δ,otherwise.{\delta }_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{\Vert {x}_{n}-{x}_{n-1}\Vert },\delta \right\}& {\rm{if}}\hspace{0.33em}{x}_{n}\ne {x}_{n-1}\\ \delta ,& {\rm{otherwise}}.\end{array}\right.\hspace{2.1em}ψn+1=minϕ‖sn−yn‖‖Asn−Ayn‖,ψnifAsn−Ayn≠0ψn,otherwise,{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\phi \Vert {s}_{n}-{y}_{n}\Vert }{\Vert A{s}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}& {\rm{if}}\hspace{0.33em}A{s}_{n}-A{y}_{n}\ne 0\\ {\psi }_{n},& {\rm{otherwise}},\end{array}\right.where ffis a contraction and AAis a pseudomonotone, Lipschitz continuous, and sequentially weakly continuous mapping. The authors proved a strong convergence result for the proposed method under mild conditions on the control parameters.Another area of interest in this study is the fixed point theory. Let U:H→HU:H\to Hbe a nonlinear map. The fixed point problem (FPP) is to find a point p∈Hp\in H(called the fixed point of UU) such that (12)Up=p.Up=p.In this work, we denote the set of fixed points of UUby F(U)F\left(U). Our interest in this study is to find a common element of the fixed point set, F(U)F\left(U), and the solution set of the variational inequality, VI(C,A)VI\left(C,A). That is, the problem of finding a point x∗∈H{x}^{\ast }\in Hsuch that (13)x∗∈VI(C,A)∩F(U).{x}^{\ast }\in VI\left(C,A)\cap F\left(U).Many algorithms have been proposed over the years and in recent times for solving the common solution problem (13) (see [30,31,32, 33,34,35, 36,37,38, 39,40] and references therein). Common solution problem of this type has drawn the attention of researchers because of its potential application to mathematical models whose constraints can be expressed as FPP and VIP. This arises in areas like signal processing, image recovery, and network resource allocation. An instance of this is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services is mathematically related (see [37,41,42] and references therein).Recently, Cai et al. [22] proposed the following inertial Tseng’s extragradient algorithm for approximating the common solution of pseudomonotone VIP and FPP for nonexpansive mappings in real Hilbert spaces: (14)x0,x1∈Hwn=xn+θn(xn−xn−1)yn=PC(wn−ψAwn)zn=yn−ψ(Ayn−Awn)xn+1=αnf(xn)+(1−αn)[βnTzn+(1−βn)zn],\left\{\begin{array}{l}{x}_{0},{x}_{1}\in H\\ {w}_{n}={x}_{n}+{\theta }_{n}\left({x}_{n}-{x}_{n-1})\\ {y}_{n}={P}_{C}\left({w}_{n}-\psi A{w}_{n})\\ {z}_{n}={y}_{n}-\psi \left(A{y}_{n}-A{w}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n})+\left(1-{\alpha }_{n})\left[{\beta }_{n}T{z}_{n}+\left(1-{\beta }_{n}){z}_{n}],\\ \end{array}\right.where ffis a contraction, TTis a nonexpansive mapping, AAis pseudomonotone, LL-Lipschitz and sequentially weakly continuous, and ψ∈0,1L\psi \in \left(0,\frac{1}{L}\right). They proved a strong convergence result for the proposed algorithm under some suitable conditions.One of the major drawbacks of Algorithm (14) is the fact that the step size ψ\psi of the algorithm depends on the Lipschitz constant of the cost operator. In many cases, this Lipschitz constant is unknown or even difficult to estimate. This makes it difficult to implement algorithms of this nature.Very recently, Thong and Hieu [23] proposed an iterative scheme for finding a common element of the solution set of monotone variational inequality and set of fixed points of demicontractive mappings as follows: (15)yn=PC(xn−ψnAxn)zn=yn−ψn(Ayn−Axn)xn+1=αnf(xn)+(1−αn)[βnUzn+(1−βn)zn],\hspace{5em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-{\psi }_{n}A{x}_{n})\\ {z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{x}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n})+\left(1-{\alpha }_{n})\left[{\beta }_{n}U{z}_{n}+\left(1-{\beta }_{n}){z}_{n}],\end{array}\right.\hspace{4.95em}ψn+1=minμ‖xn−yn‖‖Axn−Ayn‖,ψnifAxn−Ayn≠0ψn,otherwise,{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\mu \Vert {x}_{n}-{y}_{n}\Vert }{\Vert A{x}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}& {\rm{if}}\hspace{0.33em}A{x}_{n}-A{y}_{n}\ne 0\\ {\psi }_{n},& {\rm{otherwise}},\end{array}\right.\hspace{4.95em}where AAis monotone and LL-Lipschitz continuous, UUis a demicontractive mapping such that I−UI-Uis demiclosed at zero, and ffis a contraction. The authors proved a strong convergence result under suitable conditions for the proposed method.Motivated by the above results and the ongoing research activities in this direction, in this paper our aim is to introduce an effective iterative technique, which employs the efficient combination of the inertial technique, TEGM together with the viscosity method for finding a common solution of FPP of demicontractive mappings and pseudomonotone VIP with Lipschitz continuous and sequentially weakly continuous operator in Hilbert spaces. In line with this goal, we construct an algorithm with the following features: (i)Our algorithm approximates the solution of a more general class of VIP and FPP.(ii)The proposed method only requires one projection per iteration onto the feasible set, which guarantees the minimal cost of computation.(iii)Moreover, our method is computationally efficient. It employs an efficient self-adaptive step size technique which makes the algorithm independent of the Lipschitz constant of the cost operator.(iv)We employ the combination of the inertial technique together with the viscosity method, which are two of the efficient techniques for accelerating the rate of convergence of iterative schemes.(v)We prove a strong convergence theorem for the proposed algorithm without following the conventional “two-cases” approach often employed by researchers (e.g. see [22,23,29,43,44,45]). This makes our results in this paper to be more concise and precise.Furthermore, by several numerical experiments, we demonstrate the efficiency of our proposed method over many other existing methods in related literature.The remainder of this paper is organised as follows. In Section 2, useful definitions and lemmas employed in the study are presented. In Section 3, we present the proposed algorithm and highlight some of its notable features. Section 4 presents the convergence analysis of the proposed method. In Section 5, we carry out some numerical experiments to illustrate the computational advantage of our method over some of the existing methods in the literature. Finally, in Section 6 we give a concluding remark.2PreliminariesLet HHbe a real Hilbert space and CCbe a nonempty closed convex subset of HH. We denote the weak and strong convergence of sequence {xn}n=1∞{\left\{{x}_{n}\right\}}_{n=1}^{\infty }to xxby xn⇀x{x}_{n}\rightharpoonup x, as n→∞n\to \infty and xn→x{x}_{n}\to x, as n→∞n\to \infty .The metric projection [46,47], PC:H→C{P}_{C}:H\to Cis defined, for each x∈Hx\in H, as the unique element PCx∈C{P}_{C}x\in Csuch that ‖x−PCx‖=inf{‖x−z‖:z∈C}.\Vert x-{P}_{C}x\Vert =\inf \left\{\Vert x-z\Vert :z\in C\right\}.It is a known fact that PC{P}_{C}is nonexpansive, i.e. ‖PCx−PCy‖≤‖x−y‖∀x,y∈C\Vert {P}_{C}x-{P}_{C}y\Vert \le \Vert x-y\Vert \hspace{1em}\forall x,y\in C. Also, the mapping PC{P}_{C}is firmly nonexpansive, i.e.‖PCx−PCy‖2≤⟨PCx−PCy,x−y⟩,\Vert {P}_{C}x-{P}_{C}y{\Vert }^{2}\le \langle {P}_{C}x-{P}_{C}y,x-y\rangle ,for all x,y∈Hx,y\in H. Some results on the metric projection map are given below.Lemma 2.1[48] Let C be a nonempty closed convex subset of a real Hilbert space H. For any x∈Hx\in Hand z∈Cz\in C, Then, z=PCx⇔⟨x−z,z−y⟩≥0,forally∈C.z={P}_{C}x\iff \langle x-z,z-y\rangle \ge 0,\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}y\in C.Lemma 2.2[48,49] Let C be a nonempty, closed, and convex subset of a real Hilbert space H, x∈Hx\in H. Then: (1)∣∣PCx−PCy∣∣2≤⟨x−y,PCx−PCy⟩,∀y∈C| | {P}_{C}x-{P}_{C}y| {| }^{2}\le \langle x-y,{P}_{C}x-{P}_{C}y\rangle ,\hspace{1em}\forall y\in C.(2)∣∣x−PCx∣∣2+∣∣y−PCx∣∣2≤∣∣x−y∣∣2,∀y∈C| | x-{P}_{C}x| {| }^{2}+| | y-{P}_{C}x| {| }^{2}\le | | x-y| {| }^{2},\hspace{1em}\forall y\in C.(3)∣∣(I−PC)x−(I−PC)y∣∣2≤⟨x−y,(I−PC)x−(I−PC)y⟩,∀y∈C| | \left(I-{P}_{C})x-\left(I-{P}_{C})y| {| }^{2}\le \langle x-y,\left(I-{P}_{C})x-\left(I-{P}_{C})y\rangle ,\hspace{1em}\forall y\in C.Definition 2.3A mapping T:H→HT:H\to His said to be (1)Nonexpansive on HH, if there exists a constant L>0L\gt 0such that ‖Tx−Ty‖≤‖x−y‖,∀x,y∈H.\Vert Tx-Ty\Vert \le \Vert x-y\Vert ,\hspace{1em}\forall x,y\in H.(2)Quasi-nonexpansive on HH, if F(T)≠∅F\left(T)\ne \varnothing and ‖Tx−p‖≤‖x−p‖,∀p∈F(T),x∈H.\Vert Tx-p\Vert \le \Vert x-p\Vert ,\hspace{1em}\forall p\in F\left(T),x\in H.(3)λ\lambda -strictly pseudocontractive on HHwith 0≤λ<10\le \lambda \lt 1, if ‖Tx−Ty‖2≤‖x−y‖2+λ‖(I−T)x−(I−T)y‖2,∀x,y∈H.\Vert Tx-Ty{\Vert }^{2}\le \Vert x-y{\Vert }^{2}+\lambda \Vert \left(I-T)x-\left(I-T)y{\Vert }^{2},\hspace{1em}\forall x,y\in H.(4)β\beta -demicontractive with 0≤β<10\le \beta \lt 1if ‖Tx−p‖2≤‖x−p‖2+β‖(I−T)x‖2,∀p∈F(T),x∈H,\hspace{1em}\Vert Tx-p{\Vert }^{2}\le \Vert x-p{\Vert }^{2}+\beta \Vert \left(I-T)x{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H,or equivalently ⟨Tx−x,x−p⟩≤β−12‖x−Tx‖2,∀p∈F(T),x∈H,\langle Tx-x,x-p\rangle \le \frac{\beta -1}{2}\Vert x-Tx{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H,\hspace{4.6em}or equivalently ⟨Tx−p,x−p⟩≤‖x−p‖2+β−12‖x−Tx‖2,∀p∈F(T),x∈H.\langle Tx-p,x-p\rangle \le \Vert x-p{\Vert }^{2}+\frac{\beta -1}{2}\Vert x-Tx{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H.Remark 2.4It is known that every strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive. The class of demicontractive mappings includes all the other classes of mappings defined above (see [23]).Next, we give some examples of the class of demicontractive mappings, as shown in [23,50].Example 2.5(a)Let HHbe the real line and C=[−1,1]C=\left[-1,1]. Define TTon CCby: Tx=23xsin1x,x≠00ifx=0.Tx=\left\{\begin{array}{ll}\frac{2}{3}x\sin \frac{1}{x},& x\ne 0\\ 0& {\rm{if}}\hspace{0.33em}x=0.\end{array}\right.\hspace{6.05em}Then TTis demicontractive.(b)Consider a mapping T:[−2,1]→[−2,1]T:\left[-2,1]\to \left[-2,1]defined such that, Tx=−x2−x.Tx=-{x}^{2}-x.\hspace{6.9em}Then TTis a demicontractive map that is neither quasi-nonexpansive nor strictly pseudocontractive.We have the following lemmas which will be employed in our convergence analysis.Lemma 2.6[25] For each x,y∈Hx,y\in H, and δ∈R\delta \in {\mathbb{R}}, we have the following results: (1)∣∣x+y∣∣2≤∣∣x∣∣2+2⟨y,x+y⟩| | x+y| {| }^{2}\le | | x| {| }^{2}+2\langle y,x+y\rangle ;(2)∣∣x+y∣∣2=∣∣x∣∣2+2⟨x,y⟩+∣∣y∣∣2| | x+y| {| }^{2}=| | x| {| }^{2}+2\langle x,y\rangle +| | y| {| }^{2};(3)∣∣δx+(1−δ)y∣∣2=δ∣∣x∣∣2+(1−δ)∣∣y∣∣2−δ(1−δ)∣∣x−y∣∣2| | \delta x+\left(1-\delta )y| {| }^{2}=\delta | | x| {| }^{2}+\left(1-\delta )| | y| {| }^{2}-\delta \left(1-\delta )| | x-y| {| }^{2}.Lemma 2.7[51] Let {an}\left\{{a}_{n}\right\}be a sequence of nonnegative real numbers, {αn}\left\{{\alpha }_{n}\right\}be a sequence in (0,1)\left(0,1)with ∑n=1∞αn=∞{\sum }_{n=1}^{\infty }{\alpha }_{n}=\infty , and {bn}\left\{{b}_{n}\right\}be a sequence of real numbers. Assume thatan+1≤(1−αn)an+αnbn,foralln≥1,{a}_{n+1}\le \left(1-{\alpha }_{n}){a}_{n}+{\alpha }_{n}{b}_{n},\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}n\ge 1,if lim supk→∞bnk≤0{\mathrm{lim\; sup}}_{k\to \infty }{b}_{{n}_{k}}\le 0for every subsequence {ank}\left\{{a}_{{n}_{k}}\right\}of {an}\left\{{a}_{n}\right\}satisfying lim infk→∞(ank+1−ank)≥0{\mathrm{lim\; inf}}_{k\to \infty }\left({a}_{{n}_{k+1}}-{a}_{{n}_{k}})\ge 0, then limn→∞{\mathrm{lim}}_{n\to \infty }an=0{a}_{n}=0.Lemma 2.8[52] Assume that T:H→HT:H\to His a nonlinear operator with F(T)≠0F\left(T)\ne 0. Then, I−TI-Tis said to be demiclosed at zero if for any {xn}\left\{{x}_{n}\right\}in H, the following implication holds: xn⇀x{x}_{n}\rightharpoonup xand (I−T)xn→0⇒x∈F(T)\left(I-T){x}_{n}\to 0\Rightarrow x\in F\left(T).Lemma 2.9[53] Assume that D is a strongly positive bounded linear operator on a Hilbert space H with coefficient γ¯>0\bar{\gamma }\gt 0and 0<ρ≤∣∣D∣∣−10\lt \rho \le | | D| {| }^{-1}. Then ∣∣I−ρD∣∣≤1−ργ¯| | I-\rho D| | \le 1-\rho \bar{\gamma }.Lemma 2.10[54] Let U:H→HU:H\to Hbe β\beta -demicontractive with F(U)≠∅F\left(U)\ne \varnothing and set Uλ=(1−λ)+λU{U}_{\lambda }=\left(1-\lambda )+\lambda U, λ∈(0,1−β)\lambda \in \left(0,1-\beta ). Then, (i)F(U)=Fix(Uλ)F\left(U)=Fix\left({U}_{\lambda }).(ii)‖Uλx−z‖2≤‖x−z‖2−1λ(1−β−λ)‖(I−Uλ)x‖2,∀x∈H,z∈F(U)\Vert {U}_{\lambda }x-z{\Vert }^{2}\le \Vert x-z{\Vert }^{2}-\frac{1}{\lambda }\left(1-\beta -\lambda )\Vert \left(I-{U}_{\lambda })x{\Vert }^{2},\hspace{1em}\forall x\in H,z\in F\left(U).(iii)F(U)F\left(U)is a closed convex subset of H.Lemma 2.11[55] Consider the problem with C being a nonempty, closed, convex subset of a real Hilbert space H and A:C→HA:C\to Hbeing pseudomonotone and continuous. Then p is a solution of VIP (1) if and only if⟨Ax,x−p⟩≥0,∀x∈C.\langle Ax,x-p\rangle \ge 0,\hspace{1em}\forall x\in C.3Proposed algorithmIn this section, we propose an inertial viscosity-type Tseng’s extragradient algorithm with self adaptive step size and highlight some of its important features. We establish the convergence of the algorithm under the following conditions:Condition A(A1)The feasible set CCis closed, convex, and nonempty.(A2)The solution set denoted by Ω=VI(C,A)∩F(U)\Omega =VI\left(C,A)\cap F\left(U)is nonempty.(A3)The mapping AAis pseudomonotone, LL-Lipschitz continuous on HH, and sequentially weakly continuous on CC.(A4)The mapping U:H→HU:H\to His a τ\tau -demicontractive map such that I−UI-Uis demiclosed at zero.(A5)D:H→HD:H\to His a strongly positive bounded linear operator with coefficient γ¯\bar{\gamma }.(A6)f:H→Hf:H\to His a contraction with coefficient ρ∈(0,1)\rho \in \left(0,1)such that 0<γ<γ¯ρ0\lt \gamma \lt \frac{\bar{\gamma }}{\rho }.Condition B(B1){αn}⊂(0,1)\left\{{\alpha }_{n}\right\}\subset \left(0,1)such that limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0and ∑n=1∞αn=∞{\sum }_{n=1}^{\infty }{\alpha }_{n}=\infty .(B2)The positive sequence {εn}\left\{{\varepsilon }_{n}\right\}satisfies limn→∞εnαn=0,{βn}⊂(a,1−τ){\mathrm{lim}}_{n\to \infty }\frac{{\varepsilon }_{n}}{{\alpha }_{n}}=0,\left\{{\beta }_{n}\right\}\subset \left(a,1-\tau )for some a>0a\gt 0.Now, the algorithm is presented as follows:Algorithm 3.1Inertial TEGM with self-adaptive stepsizeStep 0.Given δ>0,ψ1>0,ϕ∈(0,1)\delta \gt 0,{\psi }_{1}\gt 0,\phi \in \left(0,1). Select initial data x0,x1∈H{x}_{0},{x}_{1}\in H, and set n=1n=1.Step 1.Given the (n−1n-1)th and nth iterates, choose δn{\delta }_{n}such that 0≤δn≤δˆn,∀n∈N0\le {\delta }_{n}\le {\hat{\delta }}_{n},\hspace{1em}\forall n\in {\mathbb{N}}with δˆn{\hat{\delta }}_{n}defined by (16)δˆn=minεn∣∣xn−xn−1∣∣,δ,ifxn≠xn−1,δ,otherwise.{\hat{\delta }}_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{| | {x}_{n}-{x}_{n-1}| | },\delta \right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{x}_{n}\ne {x}_{n-1},\\ \delta ,& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Step 2.Compute rn=xn+δn(xn−xn−1).{r}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}).Step 3.Compute yn=PC(rn−ψnArn).{y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}).\hspace{1.275em}If yn=rn{y}_{n}={r}_{n}, then set zn=rn{z}_{n}={r}_{n}and go to Step 5. Else go to Step 4.Step 4.Compute zn=yn−ψn(Ayn−Arn).{z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n}).Step 5.Compute xn+1=αnγf(rn)+(I−αnD)[(1−βn)zn+βnUzn].{x}_{n+1}={\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D)\left[\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}].\hspace{1.75em}Step 6.Compute (17)ψn+1=minϕ∣∣rn−yn∣∣∣∣Arn−Ayn∣∣,ψn,ifArn−Ayn≠0,ψn,otherwise.{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\phi | | {r}_{n}-{y}_{n}| | }{| | A{r}_{n}-A{y}_{n}| | },{\psi }_{n}\right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0,\\ {\psi }_{n},& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Set n≔n+1n:= n+1and return to Step 1.Below are some of the interesting features of our proposed algorithm.Remark 3.2(i)Observe that Algorithm 3.1 involves only one projection onto the feasible set CCper iteration, which makes the algorithm computationally efficient.(ii)The step size ψn{\psi }_{n}in (17) is self-adaptive and supports easy and simple computations, which makes it possible to implement our algorithm without prior knowledge of the Lipschitz constant of the cost operator.(iii)We also point out that in Step 1 of the algorithm, the inertial technique employed can easily be implemented in numerical computation, since the value of ∣∣xn−xn−1∣∣| | {x}_{n}-{x}_{n-1}| | is known prior to choosing δn{\delta }_{n}.Remark 3.3It can easily be seen from (16) and condition (B1) that limn→∞δn∣∣xn−xn−1∣∣=0andlimn→∞δnαn∣∣xn−xn−1∣∣=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | =0\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | =0.4Convergence analysisFirst, we establish some lemmas which will be employed in the convergence analysis of our proposed algorithm.Lemma 4.1The sequence {ψn}\left\{{\psi }_{n}\right\}generated by (17) is a nonincreasing sequence and limn→∞ψn=ψ≥{\mathrm{lim}}_{n\to \infty }{\psi }_{n}=\psi \ge minψ1,ϕL{\rm{\min }}\left\{{\psi }_{1},\frac{\phi }{L}\right\}.ProofIt follows from (17) that ψn+1≤ψn,∀n∈N{\psi }_{n+1}\le {\psi }_{n},\hspace{0.33em}\forall n\in {\mathbb{N}}. Hence, {ψn}\left\{{\psi }_{n}\right\}is nonincreasing. Also, since AAis Lipschitz continuous, we have ‖Arn−Ayn‖≤L‖rn−yn‖,\Vert A{r}_{n}-A{y}_{n}\Vert \le L\Vert {r}_{n}-{y}_{n}\Vert ,which implies that ‖rn−yn‖‖Arn−Ayn‖≥1L.\frac{\Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }\ge \frac{1}{L}.\hspace{3em}Consequently, we obtain ϕ‖rn−yn‖‖Arn−Ayn‖≥ϕL,whenArn−Ayn≠0.\frac{\phi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }\ge \frac{\phi }{L},\hspace{1em}\hspace{0.1em}\text{when}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0.Combining this together with (17), we obtain ψn≥minψ1,ϕL.{\psi }_{n}\ge {\rm{\min }}\left\{\phantom{\rule[-1.25em]{}{0ex}},{\psi }_{1},\frac{\phi }{L}\right\}.\hspace{2em}Since {ψn}\left\{{\psi }_{n}\right\}is nonincreasing and bounded below, we can conclude that limn→∞ψn=ψ≥minψ1,ϕL.□\hspace{15em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{\psi }_{n}=\psi \ge {\rm{\min }}\left\{\phantom{\rule[-1.25em]{}{0ex}},{\psi }_{1},\frac{\phi }{L}\right\}.\hspace{17.95em}\square Lemma 4.2Let {rn}\left\{{r}_{n}\right\}and {yn}\{{y}_{n}\}be two sequences generated by Algorithm 3.1, and suppose that conditions (A1)–(A3) hold. If there exists a subsequence {rnk}\left\{{r}_{{n}_{k}}\right\}of {rn}\left\{{r}_{n}\right\}convergent weakly to z∈Hz\in Hand limn→∞‖rnk−ynk‖=0{\mathrm{lim}}_{n\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, then z∈VI(C,A)z\in VI\left(C,A)ProofUsing the property of the projection map and yn=PC(rn−ψnArn){y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}), we obtain ⟨rnk−ψnkArnk−ynk,x−ynk⟩≤0∀x∈C,\langle {r}_{{n}_{k}}-{\psi }_{{n}_{k}}A{r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \le 0\hspace{1em}\forall x\in C,which implies that 1ψnk⟨rnk−ynk,x−ynk⟩≤⟨Arnk,x−ynk⟩∀x∈C.\frac{1}{{\psi }_{{n}_{k}}}\langle {r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \le \langle A{r}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \hspace{1em}\forall x\in C.From this we obtain (18)1ψnk⟨rnk−ynk,x−ynk⟩+⟨Arnk,ynk−rnk⟩≤⟨Arnk,x−rnk⟩∀x∈C.\frac{1}{{\psi }_{{n}_{k}}}\langle {r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle +\langle A{r}_{{n}_{k}},{y}_{{n}_{k}}-{r}_{{n}_{k}}\rangle \le \langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle \hspace{1em}\forall x\in C.Since {rnk}\left\{{r}_{{n}_{k}}\right\}converges weakly to z∈Hz\in H, we have that {rnk}\left\{{r}_{{n}_{k}}\right\}is bounded. Then, from the Lipschitz continuity of AAand ‖rnk−ynk‖→0\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0, we obtain that {Arnk}\left\{A{r}_{{n}_{k}}\right\}and {ynk}\{{y}_{{n}_{k}}\}are also bounded. Since ψnk≥ψ1,ϕL{\psi }_{{n}_{k}}\ge \left\{{\psi }_{1},\frac{\phi }{L}\right\}, from (18) it follows that (19)lim infk→∞⟨Arnk,x−rnk⟩≥0∀x∈C.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle \ge 0\hspace{1em}\forall x\in C.Moreover, we have that (20)⟨Aynk,x−ynk⟩=⟨Aynk−Arnk,x−rnk⟩+⟨Arnk,x−rnk⟩+⟨Aynk,rnk−ynk⟩.\langle A{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle =\langle A{y}_{{n}_{k}}-A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle +\langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle +\langle A{y}_{{n}_{k}},{r}_{{n}_{k}}-{y}_{{n}_{k}}\rangle .Since limk→∞‖rnk−ynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, then by the Lipschitz continuity of AAwe have limk→∞‖Arnk−Aynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert A{r}_{{n}_{k}}-A{y}_{{n}_{k}}\Vert =0. This together with (19) and (20) gives lim infk→∞⟨Aynk,x−ynk⟩≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle A{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \ge 0.Now, choose a decreasing sequence {θk}\left\{{\theta }_{k}\right\}of positive numbers such that θk→0{\theta }_{k}\to 0as k→∞k\to \infty . For any kk, we represent the smallest positive integer with Nk{N}_{k}such that: (21)⟨Aynj,x−ynj⟩+θk≥0∀j≥Nk.\langle A{y}_{{n}_{j}},x-{y}_{{n}_{j}}\rangle +{\theta }_{k}\ge 0\hspace{1em}\forall j\ge {N}_{k}.It is clear that the sequence {Nk}\left\{{N}_{k}\right\}is increasing since θk{\theta }_{k}is decreasing. Furthermore, for any kk, from {yNk}⊂C\{{y}_{{N}_{k}}\}\subset C, we can assume AyNk≠0A{y}_{{N}_{k}}\ne 0(otherwise, yNk{y}_{{N}_{k}}is a solution) and set: υNk=AyNk‖AyNk‖2.{\upsilon }_{{N}_{k}}=\frac{A{y}_{{N}_{k}}}{\Vert A{y}_{{N}_{k}}{\Vert }^{2}}.\hspace{4.65em}Consequently, we have ⟨AyNk,υNk⟩=1\langle A{y}_{{N}_{k}},{\upsilon }_{{N}_{k}}\rangle =1, for each kk. From (21), one can easily deduce that ⟨AyNk,x+θkυNk−yNk⟩≥0,∀k.\langle A{y}_{{N}_{k}},x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle \ge 0,\hspace{1em}\forall k.By the pseudomonotonicity of AA, we have ⟨A(x+θkυNk),x+θkυNk−yNk⟩≥0,\langle A\left(x+{\theta }_{k}{\upsilon }_{{N}_{k}}),x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle \ge 0,which implies that (22)⟨Ax,x−yNk⟩≥⟨Ax−A(x+θkυNk),x+θkυNk−yNk⟩−θk⟨Ax,υNk⟩.\langle Ax,x-{y}_{{N}_{k}}\rangle \ge \langle Ax-A\left(x+{\theta }_{k}{\upsilon }_{{N}_{k}}),x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle -{\theta }_{k}\langle Ax,{\upsilon }_{{N}_{k}}\rangle .Next, we show that limk→∞θkυNk=0{\mathrm{lim}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0. Indeed, since rnk⇀z{r}_{{n}_{k}}\rightharpoonup zand limk→∞‖rnk−ynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, we obtain yNk⇀z,k→∞{y}_{{N}_{k}}\rightharpoonup z,k\to \infty . Since {yn}⊂C\{{y}_{n}\}\subset C, we obtain z∈Cz\in C. By the sequentially weakly continuity of AAon CC, we have {Aynk}⇀Az\left\{A{y}_{{n}_{k}}\right\}\rightharpoonup Az. We can assume that Az≠0Az\ne 0(otherwise, zzis a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have 0<‖Az‖≤limk→∞‖Aynk‖.0\lt \Vert Az\Vert \le \mathop{\mathrm{lim}}\limits_{k\to \infty }\Vert A{y}_{{n}_{k}}\Vert .\hspace{10.5em}By the fact that {yNk}⊂{ynk}\{{y}_{{N}_{k}}\}\subset \{{y}_{{n}_{k}}\}and θk→0{\theta }_{k}\to 0as k→∞k\to \infty , we obtain 0≤lim supk→∞‖θkυNk‖=lim supk→∞θk‖AyNk‖≤lim supk→∞θklim infk→∞‖Aynk‖=0,0\le \mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\Vert {\theta }_{k}{\upsilon }_{{N}_{k}}\Vert =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\left(\frac{{\theta }_{k}}{\Vert A{y}_{{N}_{k}}\Vert }\right)\le \frac{{\mathrm{lim\; sup}}_{k\to \infty }{\theta }_{k}}{{\mathrm{lim\; inf}}_{k\to \infty }\Vert A{y}_{{n}_{k}}\Vert }=0,and this implies that lim supk→∞θkυNk=0{\mathrm{lim\; sup}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0. Now, by the facts that AAis Lipschitz continuous, sequences {yNk},{υNk}\{{y}_{{N}_{k}}\},\left\{{\upsilon }_{{N}_{k}}\right\}are bounded and limk→∞θkυNk=0{\mathrm{lim}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0, we conclude from (22) that lim infk→∞⟨Ax,x−yNk⟩≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle \ge 0.\hspace{9.7em}Consequently, we have ⟨Ax,x−z⟩=limk→∞⟨Ax,x−yNk⟩=lim infk→∞⟨Ax,x−yNk⟩≥0,∀x∈C.\langle Ax,x-z\rangle =\mathop{\mathrm{lim}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle =\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle \ge 0,\hspace{1em}\forall x\in C.Thus, by Lemma 2.11, z∈VI(C,A)z\in VI\left(C,A)as required.□Lemma 4.3Let sequences {zn}\left\{{z}_{n}\right\}and {yn}\{{y}_{n}\}be two sequences generated by Algorithm 3.1 such that conditions (A1)–(A3) hold. Then, for all p∈Ωp\in \Omega we have(23)‖zn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2,\Vert {z}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2},and(24)‖zn−yn‖≤ϕψnψn+1‖rn−yn‖.\Vert {z}_{n}-{y}_{n}\Vert \le \phi \frac{{\psi }_{n}}{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert .\hspace{8.3em}ProofBy applying the definition of {ψn}\left\{{\psi }_{n}\right\}, we have (25)‖Arn−Ayn‖≤ϕψn+1‖rn−yn‖,∀n∈N.\Vert A{r}_{n}-A{y}_{n}\Vert \le \frac{\phi }{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert ,\hspace{1em}\forall n\in {\mathbb{N}}.\hspace{3.35em}Clearly, if Arn=AynA{r}_{n}=A{y}_{n}, then inequality (25) holds. Otherwise, from (17) we have ψn+1=minψ‖rn−yn‖‖Arn−Ayn‖,ψn≤ϕ‖rn−yn‖‖Arn−Ayn‖.{\psi }_{n+1}={\rm{\min }}\left\{\frac{\psi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}\le \frac{\phi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }.\hspace{0.5em}It then follows that ‖Arn−Ayn‖≤ϕψn+1‖rn−yn‖.\Vert A{r}_{n}-A{y}_{n}\Vert \le \frac{\phi }{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert .\hspace{8em}Thus, the inequality (25) is valid both when Arn=AynA{r}_{n}=A{y}_{n}and Arn≠AynA{r}_{n}\ne A{y}_{n}. Now, from the definition of zn{z}_{n}and applying Lemma 2.6 we have (26)‖zn−p‖2=‖yn−ψn(Ayn−Arn)−p‖2=‖yn−p‖2+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2+‖yn−rn‖2+2⟨yn−rn,rn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2+‖yn−rn‖2−2⟨yn−rn,yn−rn⟩+2⟨yn−rn,yn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖−‖yn−rn‖+2⟨yn−rn,yn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩.\begin{array}{rcl}\Vert {z}_{n}-p{\Vert }^{2}& =& \Vert {y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n})-p{\Vert }^{2}\\ & =& \Vert {y}_{n}-p{\Vert }^{2}+{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}+\Vert {y}_{n}-{r}_{n}{\Vert }^{2}+2\langle {y}_{n}-{r}_{n},{r}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}+\Vert {y}_{n}-{r}_{n}{\Vert }^{2}-2\langle {y}_{n}-{r}_{n},{y}_{n}-{r}_{n}\rangle +2\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}\\ & & -2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p\Vert -\Vert {y}_{n}-{r}_{n}\Vert +2\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle yn-p,A{y}_{n}-A{r}_{n}\rangle .\end{array}Since yn=PC(rn−ψnArn){y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}), then by the projection property, we obtain ⟨yn−rn+ψnArn,yn−p⟩≤0,\langle {y}_{n}-{r}_{n}+{\psi }_{n}A{r}_{n},{y}_{n}-p\rangle \le 0,or equivalently, (27)⟨yn−rn,yn−p⟩≤−ψn⟨Arn,yn−p⟩.\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle \le -{\psi }_{n}\langle A{r}_{n},{y}_{n}-p\rangle .So, from (25), (26), and (27), we have (28)‖zn−p‖2≤‖rn−p‖2−‖yn−rn‖2−2ψn⟨Arn,yn−p⟩+ϕ2ψn2ψn+12‖rn−yn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−2ψn⟨yn−p,Ayn⟩.\begin{array}{rcl}\Vert {z}_{n}-p{\Vert }^{2}& \le & \Vert {r}_{n}-p{\Vert }^{2}-\Vert {y}_{n}-{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle A{r}_{n},{y}_{n}-p\rangle +{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}\rangle .\end{array}Now, from p∈VI(C,A)p\in VI\left(C,A), we have that ⟨Ap,yn−p⟩≥0,yn∈C.\langle Ap,{y}_{n}-p\rangle \ge 0,\hspace{1em}{y}_{n}\in C.Then, by the pseudomonotonicity of AA, we obtain (29)⟨Ayn,yn−p⟩≥0.\langle A{y}_{n},{y}_{n}-p\rangle \ge 0.\hspace{3.8em}Combining (28) and (29), we have that ‖zn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2.\Vert {z}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}.Moreover, from the definition of zn{z}_{n}and (25), we obtain ‖zn−yn‖≤ϕψnψn+1‖rn−yn‖,\Vert {z}_{n}-{y}_{n}\Vert \le \phi \frac{{\psi }_{n}}{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert ,\hspace{8.22em}which completes the proof.□Theorem 4.4Assume conditions (A)\left(A)and (B)\left(B)hold. Then, the sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 3.1 converges strongly to an element p∈Ωp\in \Omega , where p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p)is a solution of the variational inequality⟨(D−γf)p,p−q⟩≤0,∀q∈Ω.\langle \left(D-\gamma f)p,p-q\rangle \le 0,\hspace{1em}\forall q\in \Omega .\hspace{5.25em}ProofWe divide the proof of Theorem 4.4 as follows:Claim 1. The sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 3.1 is bounded.First, we show that PΩ(I−D+γf){P}_{\Omega }\left(I-D+\gamma f)is a contraction of HH. For all x,y∈Hx,y\in H, we have ‖PΩ(I−D+γf)(x)−PΩ(I−D+γf)(y)‖≤‖(I−D+γf)(x)−(I−D+γf)(y)‖≤‖(I−D)x−(I−D)y‖+γ‖fx−fy‖≤(1−γ¯)‖x−y‖+γρ‖x−y‖=(1−(γ¯−γρ))‖x−y‖.\begin{array}{rcl}\Vert {P}_{\Omega }\left(I-D+\gamma f)\left(x)-{P}_{\Omega }\left(I-D+\gamma f)(y)\Vert & \le & \Vert \left(I-D+\gamma f)\left(x)-\left(I-D+\gamma f)(y)\Vert \\ & \le & \Vert \left(I-D)x-\left(I-D)y\Vert +\gamma \Vert fx-fy\Vert \\ & \le & \left(1-\bar{\gamma })\Vert x-y\Vert +\gamma \rho \Vert x-y\Vert \\ & =& \left(1-\left(\bar{\gamma }-\gamma \rho ))\Vert x-y\Vert .\end{array}It shows that PΩ(I−D+γf){P}_{\Omega }\left(I-D+\gamma f)is a contraction. Thus, by the Banach contraction principle there exists an element p∈Ωp\in \Omega such that p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p). Next, setting gn=(1−βn)zn+βnUzn{g}_{n}=\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}and applying (23) we have (30)‖gn−p‖2=‖(1−βn)zn+βnUzn−p‖2=‖(1−βn)(zn−p)+βn(Uzn−p)‖2=(1−βn)2‖zn−p‖2+βn2‖Uzn−p‖2+2(1−βn)βn⟨Uzn−p,zn−p⟩\begin{array}{rcl}\Vert {g}_{n}-p{\Vert }^{2}& =& \Vert \left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}-p{\Vert }^{2}\\ & =& \Vert \left(1-{\beta }_{n})\left({z}_{n}-p)+{\beta }_{n}\left(U{z}_{n}-p){\Vert }^{2}\\ & =& {\left(1-{\beta }_{n})}^{2}\Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}^{2}\Vert U{z}_{n}-p{\Vert }^{2}+2\left(1-{\beta }_{n}){\beta }_{n}\langle U{z}_{n}-p,{z}_{n}-p\rangle \end{array}≤(1−βn)2‖zn−p‖2+βn2[‖zn−p‖2+τ‖zn−Uzn‖2]+2(1−βn)βn‖zn−p‖2−1−τ2‖zn−Uzn‖2=‖zn−p‖2+βn(βnτ−(1−βn)(1−τ)‖zn−Uzn‖2=‖zn−p‖2−βn(1−τ−βn)‖zn−Uzn‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2.\hspace{3em}\begin{array}{rcl}& \le & {\left(1-{\beta }_{n})}^{2}\Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}^{2}\left[\Vert {z}_{n}-p{\Vert }^{2}+\tau \Vert {z}_{n}-U{z}_{n}{\Vert }^{2}]+2\left(1-{\beta }_{n}){\beta }_{n}\left[\Vert {z}_{n}-p{\Vert }^{2}-\frac{1-\tau }{2}\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\right]\\ & =& \Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}({\beta }_{n}\tau -\left(1-{\beta }_{n})\left(1-\tau )\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\\ & =& \Vert {z}_{n}-p{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\\ & \le & \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}.\end{array}By the condition on βn{\beta }_{n}, from this we obtain (31)‖gn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2.\Vert {g}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}.From Lemma 4.1, we have that limn→∞1−ϕ2ψn2ψn+12=1−ϕ2>0.\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)=1-{\phi }^{2}\gt 0.\hspace{6.3em}This implies that there exists n0∈N{n}_{0}\in {\mathbb{N}}such that 1−ϕ2ψn2ψn+12>01-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\gt 0for all n≥n0n\ge {n}_{0}. Hence, from (31) we obtain (32)‖gn−p‖2≤‖rn−p‖2∀n≥n0.\Vert {g}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}\hspace{1em}\forall n\ge {n}_{0}.\hspace{6.45em}Also, by definition of rn{r}_{n}and triangle inequality, (33)‖rn−p‖=‖xn+δn(xn−xn−1−p)‖≤‖xn−p‖+δn‖xn−xn−1‖=‖xn−p‖+αnδnαn‖xn−xn−1‖.\Vert {r}_{n}-p\Vert =\Vert {x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}-p)\Vert \le \Vert {x}_{n}-p\Vert +{\delta }_{n}\Vert {x}_{n}-{x}_{n-1}\Vert =\Vert {x}_{n}-p\Vert +{\alpha }_{n}\hspace{1em}\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert .From Remark 3.3, we have δnαn‖xn−xn−1‖→0\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert \to 0as n→∞n\to \infty . Thus, there exists a constant G1>0{G}_{1}\gt 0that satisfies: (34)δnαn‖xn−xn−1‖≤G1,∀n≥1.\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert \le {G}_{1},\hspace{1em}\forall n\ge 1.\hspace{8.35em}So, from (32), (33), and (34) we obtain (35)‖gn−p‖≤‖rn−p‖≤‖xn−p‖+αnG1,∀n≥n0.\Vert {g}_{n}-p\Vert \le \Vert {r}_{n}-p\Vert \le \Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1},\hspace{1em}\forall n\ge {n}_{0}.Now, by applying Lemma 2.6 and (35), ∀n≥n0\forall n\ge {n}_{0}we have ‖xn+1−p‖=‖αnγf(rn)+(I−αnD)gn−p‖=‖αn(γf(rn)−Dp)+(I−αnD)(gn−p)‖≤αn‖γf(rn)−Dp‖+(1−αnγ¯)‖gn−p‖≤αn‖γf(rn)−γf(p)‖+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)≤αnγρ‖rn−p‖+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)≤αnγρ(‖xn−p‖+αnG1)+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)=(1−αn(γ¯−γρ))‖xn−p‖+αn‖γf(p)−Dp‖+(1−αn(γ¯−γρ))αnG1≤(1−αn(γ¯−γρ))‖xn−p‖+αn(γ¯−γρ)‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ≤max‖xn−p‖,‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ⋮≤max‖xn0−p‖,‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ.\begin{array}{rcl}\Vert {x}_{n+1}-p\Vert & =& \Vert {\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D){g}_{n}-p\Vert \\ & =& \Vert {\alpha }_{n}\left(\gamma f\left({r}_{n})-Dp)+\left(I-{\alpha }_{n}D)\left({g}_{n}-p)\Vert \\ & \le & {\alpha }_{n}\Vert \gamma f\left({r}_{n})-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\Vert {g}_{n}-p\Vert \\ & \le & {\alpha }_{n}\Vert \gamma f\left({r}_{n})-\gamma f\left(p)\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & \le & {\alpha }_{n}\gamma \rho \Vert {r}_{n}-p\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & \le & {\alpha }_{n}\gamma \rho (\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})+{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & =& \left(1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho ))\Vert {x}_{n}-p\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )){\alpha }_{n}{G}_{1}\\ & \le & (1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho ))\Vert {x}_{n}-p\Vert +{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )\left[\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right]\\ & \le & \max \left\{\Vert {x}_{n}-p\Vert ,\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right\}\\ & \vdots & \\ & \le & \max \left\{\Vert {x}_{{n}_{0}}-p\Vert ,\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right\}.\end{array}Hence, the sequence {xn}\left\{{x}_{n}\right\}is bounded, and so {rn}\left\{{r}_{n}\right\}, {yn}\{{y}_{n}\}, {zn}\left\{{z}_{n}\right\}are also bounded.Claim 2. The following inequality holds for all p∈Ωp\in \Omega and n∈Nn\in {\mathbb{N}}∣∣xn+1−p∣∣2≤1−2αn(γ¯−γρ)(1−αnγρ)∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2.\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \left(1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\right)| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}\\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}.\end{array}\hspace{7.65em}Using the Cauchy-Schwartz inequality and Lemma 2.6, we obtain (36)∣∣rn−p∣∣2=∣∣xn+δn(xn−xn−1)−p∣∣2=∣∣xn−p∣∣2+δn2∣∣xn−xn−1∣∣2+2δn⟨xn−p,xn−xn−1⟩≤∣∣xn−p∣∣2+δn2∣∣xn−xn−1∣∣2+2δn∣∣xn−xn−1∣∣∣∣xn−p∣∣=∣∣xn−p∣∣2+δn∣∣xn−xn−1∣∣(δn∣∣xn−xn−1∣∣+2∣∣xn−p∣∣)≤∣∣xn−p∣∣2+3G2δn∣∣xn−xn−1∣∣=∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣,\begin{array}{rcl}| | {r}_{n}-p| {| }^{2}& =& | | {x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1})-p| {| }^{2}\\ & =& | | {x}_{n}-p| {| }^{2}+{\delta }_{n}^{2}| | {x}_{n}-{x}_{n-1}| {| }^{2}+2{\delta }_{n}\langle {x}_{n}-p,{x}_{n}-{x}_{n-1}\rangle \\ & \le & | | {x}_{n}-p| {| }^{2}+{\delta }_{n}^{2}| | {x}_{n}-{x}_{n-1}| {| }^{2}+2{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | | | {x}_{n}-p| | \\ & =& | | {x}_{n}-p| {| }^{2}+{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | \left({\delta }_{n}| | {x}_{n}-{x}_{n-1}| | +2| | {x}_{n}-p| | )\\ & \le & | | {x}_{n}-p| {| }^{2}+3{G}_{2}{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | \\ & =& | | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | ,\end{array}where G2≔supn∈N{∣∣xn−p∣∣,θn∣∣xn−xn−1∣∣}>0{G}_{2}:= {\sup }_{n\in {\mathbb{N}}}\left\{| | {x}_{n}-p| | ,{\theta }_{n}| | {x}_{n}-{x}_{n-1}| | \right\}\gt 0.Now, by applying Lemma 2.6, (30), and (36) we have ∣∣xn+1−p∣∣2=∣∣αnγf(rn)+(I−αnD)gn−p∣∣2=∣∣αn(γf(rn)−Dp)+(I−αnD)(gn−p)∣∣2≤(1−αnγ¯)2∣∣gn−p∣∣2+2αn⟨γf(rn)−Dp,xn+1−p⟩≤(1−αnγ¯)2‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2+2αnγ⟨f(rn)−f(p),xn+1−p⟩+2αn⟨γf(p)−Dp,xn+1−p⟩≤(1−αnγ¯)2∣∣rn−p∣∣2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2+αnγρ(‖rn−p‖2+‖xn+1−p‖2)+2αn⟨γf(p)−Dp,xn+1−p⟩≤(1−αnγ¯)2∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2)+αnγρ∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣+‖xn+1−p‖2+2αn⟨γf(p)−Dp,xn+1−p⟩.\hspace{-45.45em}\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& =& | | {\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D){g}_{n}-p| {| }^{2}\\ & =& | | {\alpha }_{n}\left(\gamma f\left({r}_{n})-Dp)+\left(I-{\alpha }_{n}D)\left({g}_{n}-p)| {| }^{2}\\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}| | {g}_{n}-p| {| }^{2}+2{\alpha }_{n}\langle \gamma f\left({r}_{n})-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(\Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right)\\ & & +2{\alpha }_{n}\gamma \langle f\left({r}_{n})-f\left(p),{x}_{n+1}-p\rangle +2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(| | {r}_{n}-p| {| }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right)\\ & & +{\alpha }_{n}\gamma \rho \left(\Vert {r}_{n}-p{\Vert }^{2}+\Vert {x}_{n+1}-p{\Vert }^{2})+2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(| | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | -\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}\right.\\ & & -{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2})+{\alpha }_{n}\gamma \rho \left(| | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\Vert {x}_{n+1}-p{\Vert }^{2}\right)\\ & & +2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle .\end{array}Consequently, we obtain ∣∣xn+1−p∣∣2≤(1−2αnγ¯+(αnγ¯)2+αnγρ)(1−αnγρ)∣∣xn−p∣∣2+3G2((1−αnγ¯)2+αnγρ)(1−αnγρ)αnδnαn∣∣xn−xn−1∣∣+2αn(1−αnγρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2=(1−2αnγ¯+αnγρ)(1−αnγρ)∣∣xn−p∣∣2+(αnγ¯)2(1−αnγρ)∣∣xn−p∣∣2+3G2((1−αnγ¯)2+αnγρ)(1−αnγρ)αnδnαn∣∣xn−xn−1∣∣+2αn(1−αnγρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \frac{\left(1-2{\alpha }_{n}\bar{\gamma }+{\left({\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | \\ & & +\frac{2{\alpha }_{n}}{\left(1-{\alpha }_{n}\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}\\ & =& \frac{\left(1-2{\alpha }_{n}\bar{\gamma }+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}+\frac{{\left({\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}\\ & & +3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{2{\alpha }_{n}}{\left(1-{\alpha }_{n}\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}\end{array}≤(1−2αn(γ¯−γρ)(1−αnγρ))∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2,\hspace{1.75em}\begin{array}{rcl}& \le & (1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )})| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}\\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\},\end{array}where G3≔sup{∣∣xn−p∣∣2:n∈N}{G}_{3}:= \sup \left\{| | {x}_{n}-p| {| }^{2}:n\in {\mathbb{N}}\right\}. This gives the required inequality.Claim 3. The sequence {‖xn−p‖2}\left\{\Vert {x}_{n}-p{\Vert }^{2}\right\}converges to zero.Let p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p). From Claim 2, we obtain (37)∣∣xn+1−p∣∣2≤1−2αn(γ¯−γρ)(1−αnγρ)∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩.\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \left(1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\right)| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}.\end{array}To establish Claim 3, in view of Lemma 2.7, Remark 3.3, and the fact that limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0, it suffices to show that lim supk→∞⟨γf(p)−Dp,xnk+1−p⟩≤0{\mathrm{lim\; sup}}_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle \le 0for every subsequence {‖xnk−p‖}\left\{\Vert {x}_{{n}_{k}}-p\Vert \right\}of {‖xn−p‖}\left\{\Vert {x}_{n}-p\Vert \right\}satisfying lim infk→∞(‖xnk+1−p‖−‖xnk−p‖)≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\left(\Vert {x}_{{n}_{k}+1}-p\Vert -\Vert {x}_{{n}_{k}}-p\Vert )\ge 0.Suppose that {‖xnk−p‖}\left\{\Vert {x}_{{n}_{k}}-p\Vert \right\}is a subsequence of {‖xn−p‖}\left\{\Vert {x}_{n}-p\Vert \right\}such that (38)lim infk→∞(‖xnk+1−p‖−‖xnk−p‖)≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\left(\Vert {x}_{{n}_{k}+1}-p\Vert -\Vert {x}_{{n}_{k}}-p\Vert )\ge 0.Again, from Claim 2 we obtain (1−αnkγ¯)2(1−αnkγρ)1−ϕ2ψnk2ψnk+12‖rnk−ynk‖2≤1−2αnk(γ¯−γρ)(1−αnkγρ)∣∣xnk−p∣∣2−∣∣xnk+1−p∣∣2+2αnk(γ¯−γρ)(1−αnkγρ)αnkγ¯22(γ¯−γρ)G3+3G2((1−αnkγ¯)2+αnkγρ)2(γ¯−γρ)δnkαnk∣∣xnk−xnk−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xnk+1−p⟩.\begin{array}{l}\frac{{\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left(1-{\phi }^{2}\frac{{\psi }_{{n}_{k}}^{2}}{{\psi }_{{n}_{k}+1}^{2}}\right)\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}{\Vert }^{2}\\ \hspace{1.0em}\le \left(1-\frac{2{\alpha }_{{n}_{k}}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\right)| | {x}_{{n}_{k}}-p| {| }^{2}-| | {x}_{{n}_{k}+1}-p| {| }^{2}+\frac{2{\alpha }_{{n}_{k}}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left\{\frac{{\alpha }_{{n}_{k}}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ \hspace{1.0em}\hspace{1.0em}\left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}+{\alpha }_{{n}_{k}}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{{n}_{k}}}{{\alpha }_{{n}_{k}}}| | {x}_{{n}_{k}}-{x}_{{n}_{k}-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle \right\}.\end{array}Applying (38) and the fact that limk→∞αnk=0{\mathrm{lim}}_{k\to \infty }{\alpha }_{{n}_{k}}=0, we have (1−αnkγ¯)2(1−αnkγρ)1−ϕ2ψnk2ψnk+12‖rnk−ynk‖2→0,k→∞.\frac{{\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left(1-{\phi }^{2}\frac{{\psi }_{{n}_{k}}^{2}}{{\psi }_{{n}_{k}+1}^{2}}\right)\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}{\Vert }^{2}\to 0,\hspace{1em}k\to \infty .By the conditions on the control parameters, we obtain (39)‖rnk−ynk‖→0,k→∞.\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{0.9em}Following similar argument, from Claim 2 we have (40)‖Uznk−znk‖→0,k→∞.\Vert U{z}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .From (24) and (39), we obtain (41)‖znk−ynk‖→0,k→∞.\Vert {z}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{0.7em}Combining (39) and (41), we have (42)‖rnk−znk‖≤‖rnk−ynk‖+‖ynk−znk‖→0,k→∞.\Vert {r}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \le \Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert +\Vert {y}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .By Remark 3.3 and the definition of rn{r}_{n}, we obtain (43)‖xnk−rnk‖=δnk‖xnk−xnk−1‖→0,k→∞.\Vert {x}_{{n}_{k}}-{r}_{{n}_{k}}\Vert ={\delta }_{{n}_{k}}\Vert {x}_{{n}_{k}}-{x}_{{n}_{k}-1}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{5.25em}From (39), (42), and (43), we obtain (44)‖xnk−ynk‖→0,k→∞,‖xnk−znk‖→0,k→∞.\Vert {x}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty ,\hspace{1em}\Vert {x}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .Also, from (40) and (44), we obtain (45)‖xnk−Uznk‖→0,k→∞.\Vert {x}_{{n}_{k}}-U{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{12em}Using (44) and (45), we have (46)‖xnk−gnk‖≤(1−βnk)‖xnk−znk‖+βnk‖xnk−Uznk‖→0,k→∞.\Vert {x}_{{n}_{k}}-{g}_{{n}_{k}}\Vert \le \left(1-{\beta }_{{n}_{k}})\Vert {x}_{{n}_{k}}-{z}_{{n}_{k}}\Vert +{\beta }_{{n}_{k}}\Vert {x}_{{n}_{k}}-U{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .Combining this together with the fact that limk→∞αnk=0{\mathrm{lim}}_{k\to \infty }{\alpha }_{{n}_{k}}=0, we obtain (47)‖xnk+1−xnk‖≤αnk‖γf(rnk)−xnk‖+(1−αnkγ¯)‖gnk−xnk‖→0,k→∞.\Vert {x}_{{n}_{k}+1}-{x}_{{n}_{k}}\Vert \le {\alpha }_{{n}_{k}}\Vert \gamma f\left({r}_{{n}_{k}})-{x}_{{n}_{k}}\Vert +\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })\Vert {g}_{{n}_{k}}-{x}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .To complete the proof, we need to show that wω(xn)⊂Ω{w}_{\omega }\left({x}_{n})\subset \Omega . Since {xn}\left\{{x}_{n}\right\}is bounded, then wω(xn){w}_{\omega }\left({x}_{n})is nonempty. Let x∗∈wω(xn){x}^{\ast }\in {w}_{\omega }\left({x}_{n})be an arbitrary element. Then there exists a subsequence {xnk}\left\{{x}_{{n}_{k}}\right\}of {xn}\left\{{x}_{n}\right\}such that xnk⇀x∗{x}_{{n}_{k}}\rightharpoonup {x}^{\ast }as k→∞k\to \infty . By Lemma 4.2 and (39), it follows that x∗∈VI(C,A){x}^{\ast }\in VI\left(C,A). Consequently, we have wω(xn)⊂VI(C,A){w}_{\omega }\left({x}_{n})\subset VI\left(C,A). From (44), we have that znk⇀x∗{z}_{{n}_{k}}\rightharpoonup {x}^{\ast }as k→∞k\to \infty . Since I−UI-Uis demiclosed at zero, then it follows from (40) that x∗∈F(U){x}^{\ast }\in F\left(U). That is, wω(xn)⊂F(U){w}_{\omega }\left({x}_{n})\subset F\left(U). Therefore, we have wω(xn)⊂Ω{w}_{\omega }\left({x}_{n})\subset \Omega .Moreover, from (44) it follows that wω{yn}=wω{xn}=wω{zn}{w}_{\omega }\{{y}_{n}\}={w}_{\omega }\left\{{x}_{n}\right\}={w}_{\omega }\left\{{z}_{n}\right\}. By the boundedness of {xnk}\left\{{x}_{{n}_{k}}\right\}, there exists a subsequence {xnkj}\left\{{x}_{{n}_{{k}_{j}}}\right\}of {xnk}\left\{{x}_{{n}_{k}}\right\}such that xnkj⇀x†{x}_{{n}_{{k}_{j}}}\rightharpoonup {x}^{\dagger }and (48)limj→∞⟨γf(p)−Dp,xnkj−p⟩=lim supk→∞⟨γf(p)−Dp,xnk−p⟩=lim supk→∞⟨γf(p)−Dp,znk−p⟩.\mathop{\mathrm{lim}}\limits_{j\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{{k}_{j}}}-p\rangle =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{z}_{{n}_{k}}-p\rangle .Since p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p), it follows from (48) that (49)lim supk→∞⟨γf(p)−Dp,xnk−p⟩=limj→∞⟨γf(p)−Dp,xnkj−p⟩=⟨γf(p)−Dp,x†−p⟩≤0.\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle =\mathop{\mathrm{lim}}\limits_{j\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{{k}_{j}}}-p\rangle =\langle \gamma f\left(p)-Dp,{x}^{\dagger }-p\rangle \le 0.Hence, from (47) and (49), we obtain (50)lim supk→∞⟨γf(p)−Dp,xnk+1−p⟩=lim supk→∞⟨γf(p)−Dp,xnk+1−xnk⟩+lim supk→∞⟨γf(p)−Dp,xnk−p⟩=⟨γf(p)−Dp,x†−p⟩≤0.\begin{array}{rcl}\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle & =& \mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-{x}_{{n}_{k}}\rangle +\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle \\ & =& \langle \gamma f\left(p)-Dp,{x}^{\dagger }-p\rangle \le 0.\end{array}Applying Lemma 2.7 to (37), and using (50) together with the fact that limn→∞θnαn∣∣xn−xn−1∣∣=0{\mathrm{lim}}_{n\to \infty }\frac{{\theta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | =0and limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0, we deduce that limn→∞∣∣xn−p∣∣=0{\mathrm{lim}}_{n\to \infty }| | {x}_{n}-p| | =0as required.□Taking γ=1\gamma =1and D=ID=Iin Theorem 4.4, where IIis the identity mapping, then we have the following corollary.Corollary 4.5Let H be a Hilbert space and suppose U:H→HU:H\to His a τ\tau -demicontractive map. Let {xn}\left\{{x}_{n}\right\}be a sequence generated as follows:Algorithm 4.6Step 0.Given δ>0,ϕ∈(0,1)\delta \gt 0,\phi \in \left(0,1), select initial data x0,x1∈H{x}_{0},{x}_{1}\in H, λ0>0{\lambda }_{0}\gt 0, and set n=1n=1.Step 1.Given the (n−1n-1)th and nth iterates, choose δn{\delta }_{n}such that 0≤δ≤δn,∀n∈N0\le \delta \le {\delta }_{n},\hspace{1em}\forall n\in {\mathbb{N}}with δn{\delta }_{n}defined by: (51)δn=minεn∣∣xn−xn−1∣∣,δ,ifxn≠xn−1,δ,otherwise.{\delta }_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{| | {x}_{n}-{x}_{n-1}| | },\delta \right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{x}_{n}\ne {x}_{n-1},\\ \delta ,& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.\hspace{4.75em}Step 2.Compute (52)rn=xn+δn(xn−xn−1).{r}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}).\hspace{0.5em}Step 3.Compute the projection: (53)yn=PC(rn−ψnArn),{y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}),\hspace{1.85em}If yn=rn{y}_{n}={r}_{n}, then set yn=rn{y}_{n}={r}_{n}and go to Step 5. Else go to Step 4.Step 4.Compute (54)zn=yn−ψn(Ayn−Arn).{z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n}).Step 5.Compute (55)ψn+1=minϕ∣∣rn−yn∣∣∣∣Arn−Ayn∣∣,ψn,ifArn−Ayn≠0,ψn,otherwise.{\psi }_{n+1}=\left\{\begin{array}{ll}\min \left\{\frac{\phi | | {r}_{n}-{y}_{n}| | }{| | A{r}_{n}-A{y}_{n}| | },{\psi }_{n}\right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0,\\ {\psi }_{n},& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Step 6.Compute (56)xn+1=αnf(rn)+(1−αn)[(1−βn)zn+βnUzn].{x}_{n+1}={\alpha }_{n}f\left({r}_{n})+\left(1-{\alpha }_{n})\left[\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}].Set n≔n+1n:= n+1and return to Step 1.Assume that Ω=VI(C,A)∩F(U)≠0\Omega =VI\left(C,A)\cap F\left(U)\ne 0and other assumptions in conditions A and B are satisfied. Then the sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 4.6 converges strongly to a point p∈Ωp\in \Omega where p=PΩ∘f(p)p={P}_{\Omega }\circ f\left(p)is a solution of the variational inequalities.⟨(I−f)p,p−z⟩≤0forallz∈Ω.\langle \left(I-f)p,p-z\rangle \le 0\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}z\in \Omega .Remark 4.7The result in Corollary 4.5 complements the result of Tan and Qin [29], Gang et al. [22] and Thong and Hieu [23] in the following ways: (i)Our result in Corollary 4.5 extends the result of Tan and Qin [29] from pseudomonotone VIP to common solution problem of pseudomonotone variational inequality and FPPs of demicontractive maps.(ii)Corollary 4.5 result extends the result of Cai et al. [22] from FPP of nonexpansive maps to FPP of demicontractive maps.(iii)The result of Cai et al. [22] requires the knowledge of the Lipschitz constant of the cost operator while our result in Corollary 4.5 does not require any knowledge of the Lipschitz constant of the cost operator.(iv)The result of Corollary 4.5 extends the result of Thong and Hieu [23] from monotone VIP to pseudomonotone VIP.(v)Unlike the result of Thong and Hieu [23], our result in Corollary 4.5 employs inertial technique to speed up the rate of convergence of the algorithm.(vi)As shown in our convergence analysis, we did not adopt the conventional “two cases” approach employed in several papers to prove strong convergence. Our procedure is more concise and easy to comprehend.5Numerical examplesIn this section, we proceed to perform two numerical experiments to show the computational efficiency of our Algorithm 3.1 in comparison with some other algorithms in the literature. The graph of errors is plotted against the number of iterations in each case. All numerical computations were carried out using Matlab 2019(b). We use ‖xn+1−xn‖≤10−2\Vert {x}_{n+1}-{x}_{n}\Vert \le 1{0}^{-2}as the stopping criterion. The parameters are chosen as follows: Let f(x)=15xf\left(x)=\frac{1}{5}x, then ρ=15\rho =\frac{1}{5}is the Lipschitz constant for ff. Let D(x)=x3D\left(x)=\frac{x}{3}with constant γ¯=13\bar{\gamma }=\frac{1}{3}, then we take γ=1\gamma =1, which satisfies 0<γ<γ¯ρ0\lt \gamma \lt \frac{\bar{\gamma }}{\rho }. Let Ux=−32xUx=-\frac{3}{2}x. Choose δ=0.8,ψ1=0.6,ϕ=0.7,αn=1n+3,εn=1(n+3)3,βn=3n+15n+3\delta =0.8,{\psi }_{1}=0.6,\phi =0.7,{\alpha }_{n}=\frac{1}{n+3},{\varepsilon }_{n}=\frac{1}{{\left(n+3)}^{3}},{\beta }_{n}=\frac{3n+1}{5n+3}in our Algorithm 3.1.Take Tx=x2,ψ=0.8L,θn=1(n+3)2Tx=\frac{x}{2},\psi =\frac{0.8}{L},{\theta }_{n}=\frac{1}{{\left(n+3)}^{2}}in Algorithm (14).Let Gx=x−x1,γn=1n+1,ω=0.09,ρn=n2n+1Gx=x-{x}_{1},{\gamma }_{n}=\frac{1}{n+1},\omega =0.09,{\rho }_{n}=\frac{n}{2n+1}in Appendix 6.1.Take Tnx=−2nmod5x,λ=m=μ=12,σn=1n+3,τn=13,γn=16,μn=12{T}_{n}x=-\frac{2}{n\hspace{0.3em}\mathrm{mod}\hspace{0.3em}5}x,\lambda =m=\mu =\frac{1}{2},{\sigma }_{n}=\frac{1}{n+3},{\tau }_{n}=\frac{1}{3},{\gamma }_{n}=\frac{1}{6},{\mu }_{n}=\frac{1}{2}, in Appendices 6.2 and 6.3.Example 5.1Consider the linear operator A:Rm→Rm(m=5,10,15,20)A:{{\mathbb{R}}}^{m}\to {{\mathbb{R}}}^{m}\hspace{0.33em}\left(m=5,10,15,20)as follows: A(x)=Fx+gA\left(x)=Fx+g, where g∈Rmg\in {{\mathbb{R}}}^{m}and F=BBT+M+EF=B{B}^{T}+M+E, matrix B∈Rm×mB\in {{\mathbb{R}}}^{m\times m}, matrix M∈Rm×mM\in {{\mathbb{R}}}^{m\times m}, is skew symmetric, and matrix E∈Rm×mE\in {{\mathbb{R}}}^{m\times m}is a diagonal matrix whose diagonal terms are nonnegative (which implies that FFis positive symmetric definite). We choose the feasible set as C={x∈Rm:−2≤xi≤5,i=1,…,m}C=\left\{x\in {{\mathbb{R}}}^{m}:-2\le {x}_{i}\le 5,\hspace{0.33em}i=1,\ldots ,m\right\}. It can easily be verified that the mapping AAis strongly pseudomonotone and Lipschitz continuous with L=‖F‖L=\Vert F\Vert . In this example, both BBand MMentries are generated randomly in [−2,2]\left[-2,2], EEis generated randomly in [0,2]\left[0,2], and g=0g=0. The initial values x0=x1{x}_{0}={x}_{1}are generated randomly by rand(m,1){\rm{rand}}\left(m,1).The stopping criterion used for our computation is ‖xn+1−xn‖<10−2\Vert {x}_{n+1}-{x}_{n}\Vert \lt 1{0}^{-2}. We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 1 and Table 1.Figure 1Top left: m=5m=5; top right: m=10m=10; bottom left: m=15m=15; bottom right: m=20m=20.Table 1Numerical results for Example 5.1Algorithm 14Appendix 6.1Appendix 6.2Appendix 6.3Algorithm 3.1m=5m=5No. of Iter.101111226CPU time (s)1.71480.98800.88461.93790.4792m=10m=10No. of Iter.111111226CPU time (s)1.43751.10260.99231.97120.5256m=15m=15No. of Iter.111111246CPU time (s)1.41070.93371.05541.98880.6284m=20m=20No. of Iter.111112256CPU time (s)1.29530.81840.97711.61420.4390Example 5.2We consider the next example in the infinite dimensional Hilbert space H=L2([0,1])H={L}^{2}\left(\left[0,1])with inner product (57)⟨x,y⟩≔∫01x(t)y(t)dtforallx,y∈H,\langle x,y\rangle := {\int }_{0}^{1}x\left(t)y\left(t){\rm{d}}t\hspace{1em}{\rm{for}}\hspace{0.33em}{\rm{all}}\hspace{0.33em}x,y\in H,and induced norm (58)∣∣x∣∣≔∫01∣x(t)∣2dt12forallx∈H.\hspace{0.25em}| | x| | := {\left({\int }_{0}^{1}| x\left(t){| }^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}\hspace{1em}{\rm{for}}\hspace{0.33em}{\rm{all}}\hspace{0.33em}x\in H.Now, define A:H→HA:H\to Hby A(x)(t)=max{0,x(t)}A\left(x)\left(t)=\hspace{0.1em}\text{max}\hspace{0.1em}\left\{0,x\left(t)\right\}, for all t∈[0,1],x∈Ht\in \left[0,1],x\in H. It is easy to see that AAis pseudomonotone and 1-Lipschitz continuous on HH. It can easily be verified that all the conditions of Theorem 4.4 are satisfied.We choose four different initial values as follows:Case I: x0=2t3+13{x}_{0}=\frac{2{t}^{3}+1}{3}and x1=3t5+t2+1{x}_{1}=3{t}^{5}+{t}^{2}+1;Case II: x0=exp(−t){x}_{0}=\exp \left(-t)and x1=cos2t{x}_{1}=\cos 2t;Case III: x0=t3+t+5{x}_{0}={t}^{3}+t+5and x1=exp(−2t){x}_{1}=\exp \left(-2t);Case IV: x0=2t5+t2+3{x}_{0}=2{t}^{5}+{t}^{2}+3and x1=2t3−t2+3{x}_{1}=2{t}^{3}-{t}^{2}+3.The stopping criterion used for our computation is ‖xn+1−xn‖<10−2\Vert {x}_{n+1}-{x}_{n}\Vert \lt 1{0}^{-2}. We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 2 and Table 2.Figure 2Top left: Case I; top right: Case II; bottom left: Case III; bottom right: Case IV.Table 2Numerical results for Example 5.2Algorithm 14Appendix 6.1Appendix 6.2Appendix 6.3Algorithm 3.1No. of Iter.681254No. of Iter.681254No. of Iter.681254No. of Iter.91117856ConclusionWe studied the pseudomonotone VIP with a fixed point constraint. We introduced a new inertial TEGM with an adaptive step size for approximating a solution of the pseudomonotone VIP, which is also a fixed point of demicontractive mappings. We proved strong convergence results for the proposed algorithm without the knowledge of the Lipschitz constant of the cost operator. Finally, we presented several numerical experiments to demonstrate the efficiency of our proposed method in comparison with some of the existing methods in the literature. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

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de Gruyter
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© 2022 Victor Amarachi Uzor et al., published by De Gruyter
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10.1515/math-2022-0030
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Abstract

1IntroductionLet HHbe a real Hilbert space with inner product ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle and induced norm ‖⋅‖\Vert \cdot \Vert . In this paper, we consider the variational inequality problem (VIP) of finding a point p∈Cp\in Csuch that (1)⟨Ap,x−p⟩≥0,∀x∈C,\langle Ap,x-p\rangle \ge 0,\hspace{1em}\forall x\in C,where CCis a nonempty closed convex subset of HH, and A:H→HA:H\to His a nonlinear operator. We denote by VI(C,A)VI\left(C,A)the solution set of the VIP (1).Variational inequality theory, which was first introduced independently by Fichera [1] and Stampacchia [2], is a vital tool in mathematical analysis, and has a vast application across several fields of study, such as optimisation theory, engineering, physics, operator theory, economics, and many others (see [3,4, 5,6] and references therein). Over the years, several iterative methods have been formulated and adopted in solving VIP (1) (see [7,8,9, 10,11] and references therein). There are two common approaches to solving the VIP, namely, the regularised methods and the projection methods. These approaches usually require that the nonlinear operator AAin VIP (1) has certain monotonicity. In this study, we adopt the projection method and consider the case in which the associated nonlinear operator is pseudomonotone (see definition below) – a larger class than monotone mappings.Now, we review some nonlinear operators in nonlinear analysis.Definition 1.1A mapping A:H→HA:H\to His said to be (1)γ\gamma -strongly monotone on HHif there exists a constant γ>0\gamma \gt 0such that (2)⟨Ax−Ay,x−y⟩≥γ‖x−y‖2,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge \gamma \Vert x-y{\Vert }^{2},\hspace{1em}\forall x,y\in H.\hspace{1.4em}(2)γ\gamma -inverse strongly monotone on HHif there exists a constant γ>0\gamma \gt 0such that ⟨Ax−Ay,x−y⟩≥γ‖Ax−Ay‖2,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge \gamma \Vert Ax-Ay{\Vert }^{2},\hspace{1em}\forall x,y\in H.(3)Monotone on HH, if (3)⟨Ax−Ay,x−y⟩≥0,∀x,y∈H.\langle Ax-Ay,x-y\rangle \ge 0,\hspace{1em}\forall x,y\in H.\hspace{4.75em}(4)γ\gamma -strongly pseudomonotone on HH, if there exists a constant γ>0\gamma \gt 0such that (4)⟨Ay,x−y⟩≥0⇒⟨Ax,x−y⟩≥γ‖x−y‖2,∀x,y∈H.\langle Ay,x-y\rangle \ge 0\Rightarrow \langle Ax,x-y\rangle \ge \gamma \Vert x-y{\Vert }^{2},\hspace{1em}\forall x,y\in H.(5)Pseudomonotone on HH, if (5)⟨Ay,x−y⟩≥0⇒⟨Ax,x−y⟩≥0,∀x,y∈H.\langle Ay,x-y\rangle \ge 0\Rightarrow \langle Ax,x-y\rangle \ge 0,\hspace{1em}\forall x,y\in H.(6)Lipschitz-continuous on HH, if there exists a constant L>0L\gt 0such that (6)‖Ax−Ay‖≤L‖x−y‖,∀x,y∈H.\Vert Ax-Ay\Vert \le L\Vert x-y\Vert ,\hspace{1em}\forall x,y\in H.\hspace{1.55em}If L∈[0,1)L\in {[}0,1), then AAis said to be a contraction mapping.(7)Sequentially weakly continuous on HH, if for each sequence {xn}\left\{{x}_{n}\right\}, xn⇀ximpliesTxn⇀Tx,x∈H.{x}_{n}\rightharpoonup x\hspace{1em}{\rm{implies}}\hspace{1em}T{x}_{n}\rightharpoonup Tx,\hspace{1em}x\in H.From the above definitions, we observe that (1)⇒(3)⇒(5)\left(1)\Rightarrow \left(3)\Rightarrow \left(5)and (1)⇒(4)⇒(5)\left(1)\Rightarrow \left(4)\Rightarrow \left(5). However, the converses are not generally true. Moreover, if AAis γ\gamma - strongly monotone and LL- Lipschitz continuous, then AAis γL2\frac{\gamma }{{L}^{2}}- inverse strongly monotone (see [12,13]).The simplest known projection method for solving VIP is the gradient method (GM), which involves a single projection onto the feasible set CCper iteration. However, the algorithm only converges weakly under some strict conditions that the operator is either strongly monotone or inverse strongly monotone, but fails to converge if AAis monotone. The classical gradient projection algorithm proposed by Sibony [14] is given as follows: (7)xn+1=PC(xn−λAxn),{x}_{n+1}={P}_{C}\left({x}_{n}-\lambda A{x}_{n}),where AAis strongly monotone and LL-Lipschitz continuous, with step size λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right).Korpelevich [15] and Antipin [16] proposed the extragradient method (EGM) for solving VIP (1), thereby relaxing the conditions placed in (7). The initial algorithm proposed by Korpelevich was employed in solving saddle point problems, but was later extended to VIPs in both Euclidean space and infinite dimensional Hilbert spaces. The EGM method is given as follows: (8)x0∈Cyn=PC(xn−λAxn)xn+1=PC(xn−λAyn),\left\{\begin{array}{l}{x}_{0}\in C\\ {y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {x}_{n+1}={P}_{C}\left({x}_{n}-\lambda A{y}_{n}),\end{array}\right.where λ∈0,1L\lambda \in \left(0,\frac{1}{L}\right), AAis monotone and LL-Lipschitz continuous, and PC{P}_{C}denotes the metric projection from HHonto CC. If the set VI(C,A)VI\left(C,A)is nonempty, then the algorithm only converges weakly to an element in VI(C,A)VI\left(C,A).Over the years, EGM has been of interest to several researchers. Also, many results and variants have been developed from this method, using the assumptions of Lipschitz continuity, monotonicity, and pseudomonotonicity, see [17,18, 19,20] and references therein.Due to the extensive amount of time required in executing the EGM method, as a result of calculating two projections onto the closed convex set CCin each iteration, Censor et al. [8] proposed the subgradient extragradient method (SEGM) in which they replaced the second projection onto CCby a projection onto a half-space, thus, making computation easier and convergence rate faster. The SEGM is presented as follows: (9)yn=PC(xn−λAxn)Tn={w∈H:⟨xn−λAxn−yn,w−yn⟩≤0}xn+1=PTn(xn−λAyn),∀n≥0,\hspace{3.75em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {T}_{n}=\left\{w\in H:\langle {x}_{n}-\lambda A{x}_{n}-{y}_{n},w-{y}_{n}\rangle \le 0\right\}\\ {x}_{n+1}={P}_{{T}_{n}}\left({x}_{n}-\lambda A{y}_{n}),\hspace{1em}\forall n\ge 0,\end{array}\right.where λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right). The authors only obtained a weak convergence result for the proposed method. However, they later introduced a hybrid SEGM in [7] and obtained a strong convergence result. Likewise, Tseng [21], in the bid to improve on the EGM, proposed Tseng’s extragradient method (TEGM), which only requires one projection per iteration, as follows: (10)yn=PC(xn−λAxn)xn+1=yn+λ(Axn−Ayn),∀n≥0,\hspace{0.6em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-\lambda A{x}_{n})\\ {x}_{n+1}={y}_{n}+\lambda \left(A{x}_{n}-A{y}_{n}),\hspace{1em}\forall n\ge 0,\end{array}\right.where AAis monotone, LL-Lipschitz continuous, and λ∈0,2L\lambda \in \left(0,\frac{2}{L}\right). The TEGM (10) converges to a weak solution of the VIP with the assumption that VI(C,A)VI\left(C,A)is nonempty. The TEGM is also known as the forward-backward method. Recently, some authors have carried out some interesting works on the TEGM (see [22,23] and references therein).In this work, we consider the inertial algorithm, which is a two-step iteration process and a technique for accelerating the speed of convergence of iterative schemes. The inertial extrapolation technique was derived by Polyak [24] from a dynamic system called the heavy ball with friction. Due to its efficiency, the inertial technique has become a centre of attraction and interest to many researchers in this field. Over the years, researchers have studied the inertial algorithm and applied it to solve different optimisation problems, see [25,26, 27,28] and references therein.Very recently, Tan and Qin [29] proposed the following Tseng’s extragradient algorithm for solving pseudomonotone VIP: (11)sn=xn+δn(xn−xn−1)yn=PC(sn−ψnAsn)zn=yn−ψn(Ayn−Asn)xn+1=αnf(zn)+(1−αn)zn,\left\{\begin{array}{l}{s}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1})\\ {y}_{n}={P}_{C}\left({s}_{n}-{\psi }_{n}A{s}_{n})\\ {z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{s}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({z}_{n})+\left(1-{\alpha }_{n}){z}_{n},\end{array}\right.\hspace{2.4em}δn=minεn‖xn−xn−1‖,δifxn≠xn−1δ,otherwise.{\delta }_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{\Vert {x}_{n}-{x}_{n-1}\Vert },\delta \right\}& {\rm{if}}\hspace{0.33em}{x}_{n}\ne {x}_{n-1}\\ \delta ,& {\rm{otherwise}}.\end{array}\right.\hspace{2.1em}ψn+1=minϕ‖sn−yn‖‖Asn−Ayn‖,ψnifAsn−Ayn≠0ψn,otherwise,{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\phi \Vert {s}_{n}-{y}_{n}\Vert }{\Vert A{s}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}& {\rm{if}}\hspace{0.33em}A{s}_{n}-A{y}_{n}\ne 0\\ {\psi }_{n},& {\rm{otherwise}},\end{array}\right.where ffis a contraction and AAis a pseudomonotone, Lipschitz continuous, and sequentially weakly continuous mapping. The authors proved a strong convergence result for the proposed method under mild conditions on the control parameters.Another area of interest in this study is the fixed point theory. Let U:H→HU:H\to Hbe a nonlinear map. The fixed point problem (FPP) is to find a point p∈Hp\in H(called the fixed point of UU) such that (12)Up=p.Up=p.In this work, we denote the set of fixed points of UUby F(U)F\left(U). Our interest in this study is to find a common element of the fixed point set, F(U)F\left(U), and the solution set of the variational inequality, VI(C,A)VI\left(C,A). That is, the problem of finding a point x∗∈H{x}^{\ast }\in Hsuch that (13)x∗∈VI(C,A)∩F(U).{x}^{\ast }\in VI\left(C,A)\cap F\left(U).Many algorithms have been proposed over the years and in recent times for solving the common solution problem (13) (see [30,31,32, 33,34,35, 36,37,38, 39,40] and references therein). Common solution problem of this type has drawn the attention of researchers because of its potential application to mathematical models whose constraints can be expressed as FPP and VIP. This arises in areas like signal processing, image recovery, and network resource allocation. An instance of this is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services is mathematically related (see [37,41,42] and references therein).Recently, Cai et al. [22] proposed the following inertial Tseng’s extragradient algorithm for approximating the common solution of pseudomonotone VIP and FPP for nonexpansive mappings in real Hilbert spaces: (14)x0,x1∈Hwn=xn+θn(xn−xn−1)yn=PC(wn−ψAwn)zn=yn−ψ(Ayn−Awn)xn+1=αnf(xn)+(1−αn)[βnTzn+(1−βn)zn],\left\{\begin{array}{l}{x}_{0},{x}_{1}\in H\\ {w}_{n}={x}_{n}+{\theta }_{n}\left({x}_{n}-{x}_{n-1})\\ {y}_{n}={P}_{C}\left({w}_{n}-\psi A{w}_{n})\\ {z}_{n}={y}_{n}-\psi \left(A{y}_{n}-A{w}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n})+\left(1-{\alpha }_{n})\left[{\beta }_{n}T{z}_{n}+\left(1-{\beta }_{n}){z}_{n}],\\ \end{array}\right.where ffis a contraction, TTis a nonexpansive mapping, AAis pseudomonotone, LL-Lipschitz and sequentially weakly continuous, and ψ∈0,1L\psi \in \left(0,\frac{1}{L}\right). They proved a strong convergence result for the proposed algorithm under some suitable conditions.One of the major drawbacks of Algorithm (14) is the fact that the step size ψ\psi of the algorithm depends on the Lipschitz constant of the cost operator. In many cases, this Lipschitz constant is unknown or even difficult to estimate. This makes it difficult to implement algorithms of this nature.Very recently, Thong and Hieu [23] proposed an iterative scheme for finding a common element of the solution set of monotone variational inequality and set of fixed points of demicontractive mappings as follows: (15)yn=PC(xn−ψnAxn)zn=yn−ψn(Ayn−Axn)xn+1=αnf(xn)+(1−αn)[βnUzn+(1−βn)zn],\hspace{5em}\left\{\begin{array}{l}{y}_{n}={P}_{C}\left({x}_{n}-{\psi }_{n}A{x}_{n})\\ {z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{x}_{n})\\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n})+\left(1-{\alpha }_{n})\left[{\beta }_{n}U{z}_{n}+\left(1-{\beta }_{n}){z}_{n}],\end{array}\right.\hspace{4.95em}ψn+1=minμ‖xn−yn‖‖Axn−Ayn‖,ψnifAxn−Ayn≠0ψn,otherwise,{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\mu \Vert {x}_{n}-{y}_{n}\Vert }{\Vert A{x}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}& {\rm{if}}\hspace{0.33em}A{x}_{n}-A{y}_{n}\ne 0\\ {\psi }_{n},& {\rm{otherwise}},\end{array}\right.\hspace{4.95em}where AAis monotone and LL-Lipschitz continuous, UUis a demicontractive mapping such that I−UI-Uis demiclosed at zero, and ffis a contraction. The authors proved a strong convergence result under suitable conditions for the proposed method.Motivated by the above results and the ongoing research activities in this direction, in this paper our aim is to introduce an effective iterative technique, which employs the efficient combination of the inertial technique, TEGM together with the viscosity method for finding a common solution of FPP of demicontractive mappings and pseudomonotone VIP with Lipschitz continuous and sequentially weakly continuous operator in Hilbert spaces. In line with this goal, we construct an algorithm with the following features: (i)Our algorithm approximates the solution of a more general class of VIP and FPP.(ii)The proposed method only requires one projection per iteration onto the feasible set, which guarantees the minimal cost of computation.(iii)Moreover, our method is computationally efficient. It employs an efficient self-adaptive step size technique which makes the algorithm independent of the Lipschitz constant of the cost operator.(iv)We employ the combination of the inertial technique together with the viscosity method, which are two of the efficient techniques for accelerating the rate of convergence of iterative schemes.(v)We prove a strong convergence theorem for the proposed algorithm without following the conventional “two-cases” approach often employed by researchers (e.g. see [22,23,29,43,44,45]). This makes our results in this paper to be more concise and precise.Furthermore, by several numerical experiments, we demonstrate the efficiency of our proposed method over many other existing methods in related literature.The remainder of this paper is organised as follows. In Section 2, useful definitions and lemmas employed in the study are presented. In Section 3, we present the proposed algorithm and highlight some of its notable features. Section 4 presents the convergence analysis of the proposed method. In Section 5, we carry out some numerical experiments to illustrate the computational advantage of our method over some of the existing methods in the literature. Finally, in Section 6 we give a concluding remark.2PreliminariesLet HHbe a real Hilbert space and CCbe a nonempty closed convex subset of HH. We denote the weak and strong convergence of sequence {xn}n=1∞{\left\{{x}_{n}\right\}}_{n=1}^{\infty }to xxby xn⇀x{x}_{n}\rightharpoonup x, as n→∞n\to \infty and xn→x{x}_{n}\to x, as n→∞n\to \infty .The metric projection [46,47], PC:H→C{P}_{C}:H\to Cis defined, for each x∈Hx\in H, as the unique element PCx∈C{P}_{C}x\in Csuch that ‖x−PCx‖=inf{‖x−z‖:z∈C}.\Vert x-{P}_{C}x\Vert =\inf \left\{\Vert x-z\Vert :z\in C\right\}.It is a known fact that PC{P}_{C}is nonexpansive, i.e. ‖PCx−PCy‖≤‖x−y‖∀x,y∈C\Vert {P}_{C}x-{P}_{C}y\Vert \le \Vert x-y\Vert \hspace{1em}\forall x,y\in C. Also, the mapping PC{P}_{C}is firmly nonexpansive, i.e.‖PCx−PCy‖2≤⟨PCx−PCy,x−y⟩,\Vert {P}_{C}x-{P}_{C}y{\Vert }^{2}\le \langle {P}_{C}x-{P}_{C}y,x-y\rangle ,for all x,y∈Hx,y\in H. Some results on the metric projection map are given below.Lemma 2.1[48] Let C be a nonempty closed convex subset of a real Hilbert space H. For any x∈Hx\in Hand z∈Cz\in C, Then, z=PCx⇔⟨x−z,z−y⟩≥0,forally∈C.z={P}_{C}x\iff \langle x-z,z-y\rangle \ge 0,\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}y\in C.Lemma 2.2[48,49] Let C be a nonempty, closed, and convex subset of a real Hilbert space H, x∈Hx\in H. Then: (1)∣∣PCx−PCy∣∣2≤⟨x−y,PCx−PCy⟩,∀y∈C| | {P}_{C}x-{P}_{C}y| {| }^{2}\le \langle x-y,{P}_{C}x-{P}_{C}y\rangle ,\hspace{1em}\forall y\in C.(2)∣∣x−PCx∣∣2+∣∣y−PCx∣∣2≤∣∣x−y∣∣2,∀y∈C| | x-{P}_{C}x| {| }^{2}+| | y-{P}_{C}x| {| }^{2}\le | | x-y| {| }^{2},\hspace{1em}\forall y\in C.(3)∣∣(I−PC)x−(I−PC)y∣∣2≤⟨x−y,(I−PC)x−(I−PC)y⟩,∀y∈C| | \left(I-{P}_{C})x-\left(I-{P}_{C})y| {| }^{2}\le \langle x-y,\left(I-{P}_{C})x-\left(I-{P}_{C})y\rangle ,\hspace{1em}\forall y\in C.Definition 2.3A mapping T:H→HT:H\to His said to be (1)Nonexpansive on HH, if there exists a constant L>0L\gt 0such that ‖Tx−Ty‖≤‖x−y‖,∀x,y∈H.\Vert Tx-Ty\Vert \le \Vert x-y\Vert ,\hspace{1em}\forall x,y\in H.(2)Quasi-nonexpansive on HH, if F(T)≠∅F\left(T)\ne \varnothing and ‖Tx−p‖≤‖x−p‖,∀p∈F(T),x∈H.\Vert Tx-p\Vert \le \Vert x-p\Vert ,\hspace{1em}\forall p\in F\left(T),x\in H.(3)λ\lambda -strictly pseudocontractive on HHwith 0≤λ<10\le \lambda \lt 1, if ‖Tx−Ty‖2≤‖x−y‖2+λ‖(I−T)x−(I−T)y‖2,∀x,y∈H.\Vert Tx-Ty{\Vert }^{2}\le \Vert x-y{\Vert }^{2}+\lambda \Vert \left(I-T)x-\left(I-T)y{\Vert }^{2},\hspace{1em}\forall x,y\in H.(4)β\beta -demicontractive with 0≤β<10\le \beta \lt 1if ‖Tx−p‖2≤‖x−p‖2+β‖(I−T)x‖2,∀p∈F(T),x∈H,\hspace{1em}\Vert Tx-p{\Vert }^{2}\le \Vert x-p{\Vert }^{2}+\beta \Vert \left(I-T)x{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H,or equivalently ⟨Tx−x,x−p⟩≤β−12‖x−Tx‖2,∀p∈F(T),x∈H,\langle Tx-x,x-p\rangle \le \frac{\beta -1}{2}\Vert x-Tx{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H,\hspace{4.6em}or equivalently ⟨Tx−p,x−p⟩≤‖x−p‖2+β−12‖x−Tx‖2,∀p∈F(T),x∈H.\langle Tx-p,x-p\rangle \le \Vert x-p{\Vert }^{2}+\frac{\beta -1}{2}\Vert x-Tx{\Vert }^{2},\hspace{1em}\forall p\in F\left(T),x\in H.Remark 2.4It is known that every strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive. The class of demicontractive mappings includes all the other classes of mappings defined above (see [23]).Next, we give some examples of the class of demicontractive mappings, as shown in [23,50].Example 2.5(a)Let HHbe the real line and C=[−1,1]C=\left[-1,1]. Define TTon CCby: Tx=23xsin1x,x≠00ifx=0.Tx=\left\{\begin{array}{ll}\frac{2}{3}x\sin \frac{1}{x},& x\ne 0\\ 0& {\rm{if}}\hspace{0.33em}x=0.\end{array}\right.\hspace{6.05em}Then TTis demicontractive.(b)Consider a mapping T:[−2,1]→[−2,1]T:\left[-2,1]\to \left[-2,1]defined such that, Tx=−x2−x.Tx=-{x}^{2}-x.\hspace{6.9em}Then TTis a demicontractive map that is neither quasi-nonexpansive nor strictly pseudocontractive.We have the following lemmas which will be employed in our convergence analysis.Lemma 2.6[25] For each x,y∈Hx,y\in H, and δ∈R\delta \in {\mathbb{R}}, we have the following results: (1)∣∣x+y∣∣2≤∣∣x∣∣2+2⟨y,x+y⟩| | x+y| {| }^{2}\le | | x| {| }^{2}+2\langle y,x+y\rangle ;(2)∣∣x+y∣∣2=∣∣x∣∣2+2⟨x,y⟩+∣∣y∣∣2| | x+y| {| }^{2}=| | x| {| }^{2}+2\langle x,y\rangle +| | y| {| }^{2};(3)∣∣δx+(1−δ)y∣∣2=δ∣∣x∣∣2+(1−δ)∣∣y∣∣2−δ(1−δ)∣∣x−y∣∣2| | \delta x+\left(1-\delta )y| {| }^{2}=\delta | | x| {| }^{2}+\left(1-\delta )| | y| {| }^{2}-\delta \left(1-\delta )| | x-y| {| }^{2}.Lemma 2.7[51] Let {an}\left\{{a}_{n}\right\}be a sequence of nonnegative real numbers, {αn}\left\{{\alpha }_{n}\right\}be a sequence in (0,1)\left(0,1)with ∑n=1∞αn=∞{\sum }_{n=1}^{\infty }{\alpha }_{n}=\infty , and {bn}\left\{{b}_{n}\right\}be a sequence of real numbers. Assume thatan+1≤(1−αn)an+αnbn,foralln≥1,{a}_{n+1}\le \left(1-{\alpha }_{n}){a}_{n}+{\alpha }_{n}{b}_{n},\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}n\ge 1,if lim supk→∞bnk≤0{\mathrm{lim\; sup}}_{k\to \infty }{b}_{{n}_{k}}\le 0for every subsequence {ank}\left\{{a}_{{n}_{k}}\right\}of {an}\left\{{a}_{n}\right\}satisfying lim infk→∞(ank+1−ank)≥0{\mathrm{lim\; inf}}_{k\to \infty }\left({a}_{{n}_{k+1}}-{a}_{{n}_{k}})\ge 0, then limn→∞{\mathrm{lim}}_{n\to \infty }an=0{a}_{n}=0.Lemma 2.8[52] Assume that T:H→HT:H\to His a nonlinear operator with F(T)≠0F\left(T)\ne 0. Then, I−TI-Tis said to be demiclosed at zero if for any {xn}\left\{{x}_{n}\right\}in H, the following implication holds: xn⇀x{x}_{n}\rightharpoonup xand (I−T)xn→0⇒x∈F(T)\left(I-T){x}_{n}\to 0\Rightarrow x\in F\left(T).Lemma 2.9[53] Assume that D is a strongly positive bounded linear operator on a Hilbert space H with coefficient γ¯>0\bar{\gamma }\gt 0and 0<ρ≤∣∣D∣∣−10\lt \rho \le | | D| {| }^{-1}. Then ∣∣I−ρD∣∣≤1−ργ¯| | I-\rho D| | \le 1-\rho \bar{\gamma }.Lemma 2.10[54] Let U:H→HU:H\to Hbe β\beta -demicontractive with F(U)≠∅F\left(U)\ne \varnothing and set Uλ=(1−λ)+λU{U}_{\lambda }=\left(1-\lambda )+\lambda U, λ∈(0,1−β)\lambda \in \left(0,1-\beta ). Then, (i)F(U)=Fix(Uλ)F\left(U)=Fix\left({U}_{\lambda }).(ii)‖Uλx−z‖2≤‖x−z‖2−1λ(1−β−λ)‖(I−Uλ)x‖2,∀x∈H,z∈F(U)\Vert {U}_{\lambda }x-z{\Vert }^{2}\le \Vert x-z{\Vert }^{2}-\frac{1}{\lambda }\left(1-\beta -\lambda )\Vert \left(I-{U}_{\lambda })x{\Vert }^{2},\hspace{1em}\forall x\in H,z\in F\left(U).(iii)F(U)F\left(U)is a closed convex subset of H.Lemma 2.11[55] Consider the problem with C being a nonempty, closed, convex subset of a real Hilbert space H and A:C→HA:C\to Hbeing pseudomonotone and continuous. Then p is a solution of VIP (1) if and only if⟨Ax,x−p⟩≥0,∀x∈C.\langle Ax,x-p\rangle \ge 0,\hspace{1em}\forall x\in C.3Proposed algorithmIn this section, we propose an inertial viscosity-type Tseng’s extragradient algorithm with self adaptive step size and highlight some of its important features. We establish the convergence of the algorithm under the following conditions:Condition A(A1)The feasible set CCis closed, convex, and nonempty.(A2)The solution set denoted by Ω=VI(C,A)∩F(U)\Omega =VI\left(C,A)\cap F\left(U)is nonempty.(A3)The mapping AAis pseudomonotone, LL-Lipschitz continuous on HH, and sequentially weakly continuous on CC.(A4)The mapping U:H→HU:H\to His a τ\tau -demicontractive map such that I−UI-Uis demiclosed at zero.(A5)D:H→HD:H\to His a strongly positive bounded linear operator with coefficient γ¯\bar{\gamma }.(A6)f:H→Hf:H\to His a contraction with coefficient ρ∈(0,1)\rho \in \left(0,1)such that 0<γ<γ¯ρ0\lt \gamma \lt \frac{\bar{\gamma }}{\rho }.Condition B(B1){αn}⊂(0,1)\left\{{\alpha }_{n}\right\}\subset \left(0,1)such that limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0and ∑n=1∞αn=∞{\sum }_{n=1}^{\infty }{\alpha }_{n}=\infty .(B2)The positive sequence {εn}\left\{{\varepsilon }_{n}\right\}satisfies limn→∞εnαn=0,{βn}⊂(a,1−τ){\mathrm{lim}}_{n\to \infty }\frac{{\varepsilon }_{n}}{{\alpha }_{n}}=0,\left\{{\beta }_{n}\right\}\subset \left(a,1-\tau )for some a>0a\gt 0.Now, the algorithm is presented as follows:Algorithm 3.1Inertial TEGM with self-adaptive stepsizeStep 0.Given δ>0,ψ1>0,ϕ∈(0,1)\delta \gt 0,{\psi }_{1}\gt 0,\phi \in \left(0,1). Select initial data x0,x1∈H{x}_{0},{x}_{1}\in H, and set n=1n=1.Step 1.Given the (n−1n-1)th and nth iterates, choose δn{\delta }_{n}such that 0≤δn≤δˆn,∀n∈N0\le {\delta }_{n}\le {\hat{\delta }}_{n},\hspace{1em}\forall n\in {\mathbb{N}}with δˆn{\hat{\delta }}_{n}defined by (16)δˆn=minεn∣∣xn−xn−1∣∣,δ,ifxn≠xn−1,δ,otherwise.{\hat{\delta }}_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{| | {x}_{n}-{x}_{n-1}| | },\delta \right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{x}_{n}\ne {x}_{n-1},\\ \delta ,& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Step 2.Compute rn=xn+δn(xn−xn−1).{r}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}).Step 3.Compute yn=PC(rn−ψnArn).{y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}).\hspace{1.275em}If yn=rn{y}_{n}={r}_{n}, then set zn=rn{z}_{n}={r}_{n}and go to Step 5. Else go to Step 4.Step 4.Compute zn=yn−ψn(Ayn−Arn).{z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n}).Step 5.Compute xn+1=αnγf(rn)+(I−αnD)[(1−βn)zn+βnUzn].{x}_{n+1}={\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D)\left[\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}].\hspace{1.75em}Step 6.Compute (17)ψn+1=minϕ∣∣rn−yn∣∣∣∣Arn−Ayn∣∣,ψn,ifArn−Ayn≠0,ψn,otherwise.{\psi }_{n+1}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{\phi | | {r}_{n}-{y}_{n}| | }{| | A{r}_{n}-A{y}_{n}| | },{\psi }_{n}\right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0,\\ {\psi }_{n},& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Set n≔n+1n:= n+1and return to Step 1.Below are some of the interesting features of our proposed algorithm.Remark 3.2(i)Observe that Algorithm 3.1 involves only one projection onto the feasible set CCper iteration, which makes the algorithm computationally efficient.(ii)The step size ψn{\psi }_{n}in (17) is self-adaptive and supports easy and simple computations, which makes it possible to implement our algorithm without prior knowledge of the Lipschitz constant of the cost operator.(iii)We also point out that in Step 1 of the algorithm, the inertial technique employed can easily be implemented in numerical computation, since the value of ∣∣xn−xn−1∣∣| | {x}_{n}-{x}_{n-1}| | is known prior to choosing δn{\delta }_{n}.Remark 3.3It can easily be seen from (16) and condition (B1) that limn→∞δn∣∣xn−xn−1∣∣=0andlimn→∞δnαn∣∣xn−xn−1∣∣=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | =0\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | =0.4Convergence analysisFirst, we establish some lemmas which will be employed in the convergence analysis of our proposed algorithm.Lemma 4.1The sequence {ψn}\left\{{\psi }_{n}\right\}generated by (17) is a nonincreasing sequence and limn→∞ψn=ψ≥{\mathrm{lim}}_{n\to \infty }{\psi }_{n}=\psi \ge minψ1,ϕL{\rm{\min }}\left\{{\psi }_{1},\frac{\phi }{L}\right\}.ProofIt follows from (17) that ψn+1≤ψn,∀n∈N{\psi }_{n+1}\le {\psi }_{n},\hspace{0.33em}\forall n\in {\mathbb{N}}. Hence, {ψn}\left\{{\psi }_{n}\right\}is nonincreasing. Also, since AAis Lipschitz continuous, we have ‖Arn−Ayn‖≤L‖rn−yn‖,\Vert A{r}_{n}-A{y}_{n}\Vert \le L\Vert {r}_{n}-{y}_{n}\Vert ,which implies that ‖rn−yn‖‖Arn−Ayn‖≥1L.\frac{\Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }\ge \frac{1}{L}.\hspace{3em}Consequently, we obtain ϕ‖rn−yn‖‖Arn−Ayn‖≥ϕL,whenArn−Ayn≠0.\frac{\phi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }\ge \frac{\phi }{L},\hspace{1em}\hspace{0.1em}\text{when}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0.Combining this together with (17), we obtain ψn≥minψ1,ϕL.{\psi }_{n}\ge {\rm{\min }}\left\{\phantom{\rule[-1.25em]{}{0ex}},{\psi }_{1},\frac{\phi }{L}\right\}.\hspace{2em}Since {ψn}\left\{{\psi }_{n}\right\}is nonincreasing and bounded below, we can conclude that limn→∞ψn=ψ≥minψ1,ϕL.□\hspace{15em}\mathop{\mathrm{lim}}\limits_{n\to \infty }{\psi }_{n}=\psi \ge {\rm{\min }}\left\{\phantom{\rule[-1.25em]{}{0ex}},{\psi }_{1},\frac{\phi }{L}\right\}.\hspace{17.95em}\square Lemma 4.2Let {rn}\left\{{r}_{n}\right\}and {yn}\{{y}_{n}\}be two sequences generated by Algorithm 3.1, and suppose that conditions (A1)–(A3) hold. If there exists a subsequence {rnk}\left\{{r}_{{n}_{k}}\right\}of {rn}\left\{{r}_{n}\right\}convergent weakly to z∈Hz\in Hand limn→∞‖rnk−ynk‖=0{\mathrm{lim}}_{n\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, then z∈VI(C,A)z\in VI\left(C,A)ProofUsing the property of the projection map and yn=PC(rn−ψnArn){y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}), we obtain ⟨rnk−ψnkArnk−ynk,x−ynk⟩≤0∀x∈C,\langle {r}_{{n}_{k}}-{\psi }_{{n}_{k}}A{r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \le 0\hspace{1em}\forall x\in C,which implies that 1ψnk⟨rnk−ynk,x−ynk⟩≤⟨Arnk,x−ynk⟩∀x∈C.\frac{1}{{\psi }_{{n}_{k}}}\langle {r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \le \langle A{r}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \hspace{1em}\forall x\in C.From this we obtain (18)1ψnk⟨rnk−ynk,x−ynk⟩+⟨Arnk,ynk−rnk⟩≤⟨Arnk,x−rnk⟩∀x∈C.\frac{1}{{\psi }_{{n}_{k}}}\langle {r}_{{n}_{k}}-{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle +\langle A{r}_{{n}_{k}},{y}_{{n}_{k}}-{r}_{{n}_{k}}\rangle \le \langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle \hspace{1em}\forall x\in C.Since {rnk}\left\{{r}_{{n}_{k}}\right\}converges weakly to z∈Hz\in H, we have that {rnk}\left\{{r}_{{n}_{k}}\right\}is bounded. Then, from the Lipschitz continuity of AAand ‖rnk−ynk‖→0\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0, we obtain that {Arnk}\left\{A{r}_{{n}_{k}}\right\}and {ynk}\{{y}_{{n}_{k}}\}are also bounded. Since ψnk≥ψ1,ϕL{\psi }_{{n}_{k}}\ge \left\{{\psi }_{1},\frac{\phi }{L}\right\}, from (18) it follows that (19)lim infk→∞⟨Arnk,x−rnk⟩≥0∀x∈C.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle \ge 0\hspace{1em}\forall x\in C.Moreover, we have that (20)⟨Aynk,x−ynk⟩=⟨Aynk−Arnk,x−rnk⟩+⟨Arnk,x−rnk⟩+⟨Aynk,rnk−ynk⟩.\langle A{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle =\langle A{y}_{{n}_{k}}-A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle +\langle A{r}_{{n}_{k}},x-{r}_{{n}_{k}}\rangle +\langle A{y}_{{n}_{k}},{r}_{{n}_{k}}-{y}_{{n}_{k}}\rangle .Since limk→∞‖rnk−ynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, then by the Lipschitz continuity of AAwe have limk→∞‖Arnk−Aynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert A{r}_{{n}_{k}}-A{y}_{{n}_{k}}\Vert =0. This together with (19) and (20) gives lim infk→∞⟨Aynk,x−ynk⟩≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle A{y}_{{n}_{k}},x-{y}_{{n}_{k}}\rangle \ge 0.Now, choose a decreasing sequence {θk}\left\{{\theta }_{k}\right\}of positive numbers such that θk→0{\theta }_{k}\to 0as k→∞k\to \infty . For any kk, we represent the smallest positive integer with Nk{N}_{k}such that: (21)⟨Aynj,x−ynj⟩+θk≥0∀j≥Nk.\langle A{y}_{{n}_{j}},x-{y}_{{n}_{j}}\rangle +{\theta }_{k}\ge 0\hspace{1em}\forall j\ge {N}_{k}.It is clear that the sequence {Nk}\left\{{N}_{k}\right\}is increasing since θk{\theta }_{k}is decreasing. Furthermore, for any kk, from {yNk}⊂C\{{y}_{{N}_{k}}\}\subset C, we can assume AyNk≠0A{y}_{{N}_{k}}\ne 0(otherwise, yNk{y}_{{N}_{k}}is a solution) and set: υNk=AyNk‖AyNk‖2.{\upsilon }_{{N}_{k}}=\frac{A{y}_{{N}_{k}}}{\Vert A{y}_{{N}_{k}}{\Vert }^{2}}.\hspace{4.65em}Consequently, we have ⟨AyNk,υNk⟩=1\langle A{y}_{{N}_{k}},{\upsilon }_{{N}_{k}}\rangle =1, for each kk. From (21), one can easily deduce that ⟨AyNk,x+θkυNk−yNk⟩≥0,∀k.\langle A{y}_{{N}_{k}},x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle \ge 0,\hspace{1em}\forall k.By the pseudomonotonicity of AA, we have ⟨A(x+θkυNk),x+θkυNk−yNk⟩≥0,\langle A\left(x+{\theta }_{k}{\upsilon }_{{N}_{k}}),x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle \ge 0,which implies that (22)⟨Ax,x−yNk⟩≥⟨Ax−A(x+θkυNk),x+θkυNk−yNk⟩−θk⟨Ax,υNk⟩.\langle Ax,x-{y}_{{N}_{k}}\rangle \ge \langle Ax-A\left(x+{\theta }_{k}{\upsilon }_{{N}_{k}}),x+{\theta }_{k}{\upsilon }_{{N}_{k}}-{y}_{{N}_{k}}\rangle -{\theta }_{k}\langle Ax,{\upsilon }_{{N}_{k}}\rangle .Next, we show that limk→∞θkυNk=0{\mathrm{lim}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0. Indeed, since rnk⇀z{r}_{{n}_{k}}\rightharpoonup zand limk→∞‖rnk−ynk‖=0{\mathrm{lim}}_{k\to \infty }\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert =0, we obtain yNk⇀z,k→∞{y}_{{N}_{k}}\rightharpoonup z,k\to \infty . Since {yn}⊂C\{{y}_{n}\}\subset C, we obtain z∈Cz\in C. By the sequentially weakly continuity of AAon CC, we have {Aynk}⇀Az\left\{A{y}_{{n}_{k}}\right\}\rightharpoonup Az. We can assume that Az≠0Az\ne 0(otherwise, zzis a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have 0<‖Az‖≤limk→∞‖Aynk‖.0\lt \Vert Az\Vert \le \mathop{\mathrm{lim}}\limits_{k\to \infty }\Vert A{y}_{{n}_{k}}\Vert .\hspace{10.5em}By the fact that {yNk}⊂{ynk}\{{y}_{{N}_{k}}\}\subset \{{y}_{{n}_{k}}\}and θk→0{\theta }_{k}\to 0as k→∞k\to \infty , we obtain 0≤lim supk→∞‖θkυNk‖=lim supk→∞θk‖AyNk‖≤lim supk→∞θklim infk→∞‖Aynk‖=0,0\le \mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\Vert {\theta }_{k}{\upsilon }_{{N}_{k}}\Vert =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\left(\frac{{\theta }_{k}}{\Vert A{y}_{{N}_{k}}\Vert }\right)\le \frac{{\mathrm{lim\; sup}}_{k\to \infty }{\theta }_{k}}{{\mathrm{lim\; inf}}_{k\to \infty }\Vert A{y}_{{n}_{k}}\Vert }=0,and this implies that lim supk→∞θkυNk=0{\mathrm{lim\; sup}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0. Now, by the facts that AAis Lipschitz continuous, sequences {yNk},{υNk}\{{y}_{{N}_{k}}\},\left\{{\upsilon }_{{N}_{k}}\right\}are bounded and limk→∞θkυNk=0{\mathrm{lim}}_{k\to \infty }{\theta }_{k}{\upsilon }_{{N}_{k}}=0, we conclude from (22) that lim infk→∞⟨Ax,x−yNk⟩≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle \ge 0.\hspace{9.7em}Consequently, we have ⟨Ax,x−z⟩=limk→∞⟨Ax,x−yNk⟩=lim infk→∞⟨Ax,x−yNk⟩≥0,∀x∈C.\langle Ax,x-z\rangle =\mathop{\mathrm{lim}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle =\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\langle Ax,x-{y}_{{N}_{k}}\rangle \ge 0,\hspace{1em}\forall x\in C.Thus, by Lemma 2.11, z∈VI(C,A)z\in VI\left(C,A)as required.□Lemma 4.3Let sequences {zn}\left\{{z}_{n}\right\}and {yn}\{{y}_{n}\}be two sequences generated by Algorithm 3.1 such that conditions (A1)–(A3) hold. Then, for all p∈Ωp\in \Omega we have(23)‖zn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2,\Vert {z}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2},and(24)‖zn−yn‖≤ϕψnψn+1‖rn−yn‖.\Vert {z}_{n}-{y}_{n}\Vert \le \phi \frac{{\psi }_{n}}{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert .\hspace{8.3em}ProofBy applying the definition of {ψn}\left\{{\psi }_{n}\right\}, we have (25)‖Arn−Ayn‖≤ϕψn+1‖rn−yn‖,∀n∈N.\Vert A{r}_{n}-A{y}_{n}\Vert \le \frac{\phi }{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert ,\hspace{1em}\forall n\in {\mathbb{N}}.\hspace{3.35em}Clearly, if Arn=AynA{r}_{n}=A{y}_{n}, then inequality (25) holds. Otherwise, from (17) we have ψn+1=minψ‖rn−yn‖‖Arn−Ayn‖,ψn≤ϕ‖rn−yn‖‖Arn−Ayn‖.{\psi }_{n+1}={\rm{\min }}\left\{\frac{\psi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert },{\psi }_{n}\right\}\le \frac{\phi \Vert {r}_{n}-{y}_{n}\Vert }{\Vert A{r}_{n}-A{y}_{n}\Vert }.\hspace{0.5em}It then follows that ‖Arn−Ayn‖≤ϕψn+1‖rn−yn‖.\Vert A{r}_{n}-A{y}_{n}\Vert \le \frac{\phi }{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert .\hspace{8em}Thus, the inequality (25) is valid both when Arn=AynA{r}_{n}=A{y}_{n}and Arn≠AynA{r}_{n}\ne A{y}_{n}. Now, from the definition of zn{z}_{n}and applying Lemma 2.6 we have (26)‖zn−p‖2=‖yn−ψn(Ayn−Arn)−p‖2=‖yn−p‖2+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2+‖yn−rn‖2+2⟨yn−rn,rn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2+‖yn−rn‖2−2⟨yn−rn,yn−rn⟩+2⟨yn−rn,yn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖−‖yn−rn‖+2⟨yn−rn,yn−p⟩+ψn2‖Ayn−Arn‖2−2ψn⟨yn−p,Ayn−Arn⟩.\begin{array}{rcl}\Vert {z}_{n}-p{\Vert }^{2}& =& \Vert {y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n})-p{\Vert }^{2}\\ & =& \Vert {y}_{n}-p{\Vert }^{2}+{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}+\Vert {y}_{n}-{r}_{n}{\Vert }^{2}+2\langle {y}_{n}-{r}_{n},{r}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}+\Vert {y}_{n}-{r}_{n}{\Vert }^{2}-2\langle {y}_{n}-{r}_{n},{y}_{n}-{r}_{n}\rangle +2\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}\\ & & -2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p\Vert -\Vert {y}_{n}-{r}_{n}\Vert +2\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle +{\psi }_{n}^{2}\Vert A{y}_{n}-A{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle yn-p,A{y}_{n}-A{r}_{n}\rangle .\end{array}Since yn=PC(rn−ψnArn){y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}), then by the projection property, we obtain ⟨yn−rn+ψnArn,yn−p⟩≤0,\langle {y}_{n}-{r}_{n}+{\psi }_{n}A{r}_{n},{y}_{n}-p\rangle \le 0,or equivalently, (27)⟨yn−rn,yn−p⟩≤−ψn⟨Arn,yn−p⟩.\langle {y}_{n}-{r}_{n},{y}_{n}-p\rangle \le -{\psi }_{n}\langle A{r}_{n},{y}_{n}-p\rangle .So, from (25), (26), and (27), we have (28)‖zn−p‖2≤‖rn−p‖2−‖yn−rn‖2−2ψn⟨Arn,yn−p⟩+ϕ2ψn2ψn+12‖rn−yn‖2−2ψn⟨yn−p,Ayn−Arn⟩=‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−2ψn⟨yn−p,Ayn⟩.\begin{array}{rcl}\Vert {z}_{n}-p{\Vert }^{2}& \le & \Vert {r}_{n}-p{\Vert }^{2}-\Vert {y}_{n}-{r}_{n}{\Vert }^{2}-2{\psi }_{n}\langle A{r}_{n},{y}_{n}-p\rangle +{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}-A{r}_{n}\rangle \\ & =& \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-2{\psi }_{n}\langle {y}_{n}-p,A{y}_{n}\rangle .\end{array}Now, from p∈VI(C,A)p\in VI\left(C,A), we have that ⟨Ap,yn−p⟩≥0,yn∈C.\langle Ap,{y}_{n}-p\rangle \ge 0,\hspace{1em}{y}_{n}\in C.Then, by the pseudomonotonicity of AA, we obtain (29)⟨Ayn,yn−p⟩≥0.\langle A{y}_{n},{y}_{n}-p\rangle \ge 0.\hspace{3.8em}Combining (28) and (29), we have that ‖zn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2.\Vert {z}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}.Moreover, from the definition of zn{z}_{n}and (25), we obtain ‖zn−yn‖≤ϕψnψn+1‖rn−yn‖,\Vert {z}_{n}-{y}_{n}\Vert \le \phi \frac{{\psi }_{n}}{{\psi }_{n+1}}\Vert {r}_{n}-{y}_{n}\Vert ,\hspace{8.22em}which completes the proof.□Theorem 4.4Assume conditions (A)\left(A)and (B)\left(B)hold. Then, the sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 3.1 converges strongly to an element p∈Ωp\in \Omega , where p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p)is a solution of the variational inequality⟨(D−γf)p,p−q⟩≤0,∀q∈Ω.\langle \left(D-\gamma f)p,p-q\rangle \le 0,\hspace{1em}\forall q\in \Omega .\hspace{5.25em}ProofWe divide the proof of Theorem 4.4 as follows:Claim 1. The sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 3.1 is bounded.First, we show that PΩ(I−D+γf){P}_{\Omega }\left(I-D+\gamma f)is a contraction of HH. For all x,y∈Hx,y\in H, we have ‖PΩ(I−D+γf)(x)−PΩ(I−D+γf)(y)‖≤‖(I−D+γf)(x)−(I−D+γf)(y)‖≤‖(I−D)x−(I−D)y‖+γ‖fx−fy‖≤(1−γ¯)‖x−y‖+γρ‖x−y‖=(1−(γ¯−γρ))‖x−y‖.\begin{array}{rcl}\Vert {P}_{\Omega }\left(I-D+\gamma f)\left(x)-{P}_{\Omega }\left(I-D+\gamma f)(y)\Vert & \le & \Vert \left(I-D+\gamma f)\left(x)-\left(I-D+\gamma f)(y)\Vert \\ & \le & \Vert \left(I-D)x-\left(I-D)y\Vert +\gamma \Vert fx-fy\Vert \\ & \le & \left(1-\bar{\gamma })\Vert x-y\Vert +\gamma \rho \Vert x-y\Vert \\ & =& \left(1-\left(\bar{\gamma }-\gamma \rho ))\Vert x-y\Vert .\end{array}It shows that PΩ(I−D+γf){P}_{\Omega }\left(I-D+\gamma f)is a contraction. Thus, by the Banach contraction principle there exists an element p∈Ωp\in \Omega such that p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p). Next, setting gn=(1−βn)zn+βnUzn{g}_{n}=\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}and applying (23) we have (30)‖gn−p‖2=‖(1−βn)zn+βnUzn−p‖2=‖(1−βn)(zn−p)+βn(Uzn−p)‖2=(1−βn)2‖zn−p‖2+βn2‖Uzn−p‖2+2(1−βn)βn⟨Uzn−p,zn−p⟩\begin{array}{rcl}\Vert {g}_{n}-p{\Vert }^{2}& =& \Vert \left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}-p{\Vert }^{2}\\ & =& \Vert \left(1-{\beta }_{n})\left({z}_{n}-p)+{\beta }_{n}\left(U{z}_{n}-p){\Vert }^{2}\\ & =& {\left(1-{\beta }_{n})}^{2}\Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}^{2}\Vert U{z}_{n}-p{\Vert }^{2}+2\left(1-{\beta }_{n}){\beta }_{n}\langle U{z}_{n}-p,{z}_{n}-p\rangle \end{array}≤(1−βn)2‖zn−p‖2+βn2[‖zn−p‖2+τ‖zn−Uzn‖2]+2(1−βn)βn‖zn−p‖2−1−τ2‖zn−Uzn‖2=‖zn−p‖2+βn(βnτ−(1−βn)(1−τ)‖zn−Uzn‖2=‖zn−p‖2−βn(1−τ−βn)‖zn−Uzn‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2.\hspace{3em}\begin{array}{rcl}& \le & {\left(1-{\beta }_{n})}^{2}\Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}^{2}\left[\Vert {z}_{n}-p{\Vert }^{2}+\tau \Vert {z}_{n}-U{z}_{n}{\Vert }^{2}]+2\left(1-{\beta }_{n}){\beta }_{n}\left[\Vert {z}_{n}-p{\Vert }^{2}-\frac{1-\tau }{2}\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\right]\\ & =& \Vert {z}_{n}-p{\Vert }^{2}+{\beta }_{n}({\beta }_{n}\tau -\left(1-{\beta }_{n})\left(1-\tau )\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\\ & =& \Vert {z}_{n}-p{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert {z}_{n}-U{z}_{n}{\Vert }^{2}\\ & \le & \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}.\end{array}By the condition on βn{\beta }_{n}, from this we obtain (31)‖gn−p‖2≤‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2.\Vert {g}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}.From Lemma 4.1, we have that limn→∞1−ϕ2ψn2ψn+12=1−ϕ2>0.\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)=1-{\phi }^{2}\gt 0.\hspace{6.3em}This implies that there exists n0∈N{n}_{0}\in {\mathbb{N}}such that 1−ϕ2ψn2ψn+12>01-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\gt 0for all n≥n0n\ge {n}_{0}. Hence, from (31) we obtain (32)‖gn−p‖2≤‖rn−p‖2∀n≥n0.\Vert {g}_{n}-p{\Vert }^{2}\le \Vert {r}_{n}-p{\Vert }^{2}\hspace{1em}\forall n\ge {n}_{0}.\hspace{6.45em}Also, by definition of rn{r}_{n}and triangle inequality, (33)‖rn−p‖=‖xn+δn(xn−xn−1−p)‖≤‖xn−p‖+δn‖xn−xn−1‖=‖xn−p‖+αnδnαn‖xn−xn−1‖.\Vert {r}_{n}-p\Vert =\Vert {x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}-p)\Vert \le \Vert {x}_{n}-p\Vert +{\delta }_{n}\Vert {x}_{n}-{x}_{n-1}\Vert =\Vert {x}_{n}-p\Vert +{\alpha }_{n}\hspace{1em}\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert .From Remark 3.3, we have δnαn‖xn−xn−1‖→0\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert \to 0as n→∞n\to \infty . Thus, there exists a constant G1>0{G}_{1}\gt 0that satisfies: (34)δnαn‖xn−xn−1‖≤G1,∀n≥1.\frac{{\delta }_{n}}{{\alpha }_{n}}\Vert {x}_{n}-{x}_{n-1}\Vert \le {G}_{1},\hspace{1em}\forall n\ge 1.\hspace{8.35em}So, from (32), (33), and (34) we obtain (35)‖gn−p‖≤‖rn−p‖≤‖xn−p‖+αnG1,∀n≥n0.\Vert {g}_{n}-p\Vert \le \Vert {r}_{n}-p\Vert \le \Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1},\hspace{1em}\forall n\ge {n}_{0}.Now, by applying Lemma 2.6 and (35), ∀n≥n0\forall n\ge {n}_{0}we have ‖xn+1−p‖=‖αnγf(rn)+(I−αnD)gn−p‖=‖αn(γf(rn)−Dp)+(I−αnD)(gn−p)‖≤αn‖γf(rn)−Dp‖+(1−αnγ¯)‖gn−p‖≤αn‖γf(rn)−γf(p)‖+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)≤αnγρ‖rn−p‖+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)≤αnγρ(‖xn−p‖+αnG1)+αn‖γf(p)−Dp‖+(1−αnγ¯)(‖xn−p‖+αnG1)=(1−αn(γ¯−γρ))‖xn−p‖+αn‖γf(p)−Dp‖+(1−αn(γ¯−γρ))αnG1≤(1−αn(γ¯−γρ))‖xn−p‖+αn(γ¯−γρ)‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ≤max‖xn−p‖,‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ⋮≤max‖xn0−p‖,‖γf(p)−Dp‖γ¯−γρ+G1γ¯−γρ.\begin{array}{rcl}\Vert {x}_{n+1}-p\Vert & =& \Vert {\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D){g}_{n}-p\Vert \\ & =& \Vert {\alpha }_{n}\left(\gamma f\left({r}_{n})-Dp)+\left(I-{\alpha }_{n}D)\left({g}_{n}-p)\Vert \\ & \le & {\alpha }_{n}\Vert \gamma f\left({r}_{n})-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\Vert {g}_{n}-p\Vert \\ & \le & {\alpha }_{n}\Vert \gamma f\left({r}_{n})-\gamma f\left(p)\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & \le & {\alpha }_{n}\gamma \rho \Vert {r}_{n}-p\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & \le & {\alpha }_{n}\gamma \rho (\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})+{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\bar{\gamma })\left(\Vert {x}_{n}-p\Vert +{\alpha }_{n}{G}_{1})\\ & =& \left(1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho ))\Vert {x}_{n}-p\Vert +{\alpha }_{n}\Vert \gamma f\left(p)-Dp\Vert +\left(1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )){\alpha }_{n}{G}_{1}\\ & \le & (1-{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho ))\Vert {x}_{n}-p\Vert +{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )\left[\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right]\\ & \le & \max \left\{\Vert {x}_{n}-p\Vert ,\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right\}\\ & \vdots & \\ & \le & \max \left\{\Vert {x}_{{n}_{0}}-p\Vert ,\frac{\Vert \gamma f\left(p)-Dp\Vert }{\bar{\gamma }-\gamma \rho }+\frac{{G}_{1}}{\bar{\gamma }-\gamma \rho }\right\}.\end{array}Hence, the sequence {xn}\left\{{x}_{n}\right\}is bounded, and so {rn}\left\{{r}_{n}\right\}, {yn}\{{y}_{n}\}, {zn}\left\{{z}_{n}\right\}are also bounded.Claim 2. The following inequality holds for all p∈Ωp\in \Omega and n∈Nn\in {\mathbb{N}}∣∣xn+1−p∣∣2≤1−2αn(γ¯−γρ)(1−αnγρ)∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2.\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \left(1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\right)| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}\\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}.\end{array}\hspace{7.65em}Using the Cauchy-Schwartz inequality and Lemma 2.6, we obtain (36)∣∣rn−p∣∣2=∣∣xn+δn(xn−xn−1)−p∣∣2=∣∣xn−p∣∣2+δn2∣∣xn−xn−1∣∣2+2δn⟨xn−p,xn−xn−1⟩≤∣∣xn−p∣∣2+δn2∣∣xn−xn−1∣∣2+2δn∣∣xn−xn−1∣∣∣∣xn−p∣∣=∣∣xn−p∣∣2+δn∣∣xn−xn−1∣∣(δn∣∣xn−xn−1∣∣+2∣∣xn−p∣∣)≤∣∣xn−p∣∣2+3G2δn∣∣xn−xn−1∣∣=∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣,\begin{array}{rcl}| | {r}_{n}-p| {| }^{2}& =& | | {x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1})-p| {| }^{2}\\ & =& | | {x}_{n}-p| {| }^{2}+{\delta }_{n}^{2}| | {x}_{n}-{x}_{n-1}| {| }^{2}+2{\delta }_{n}\langle {x}_{n}-p,{x}_{n}-{x}_{n-1}\rangle \\ & \le & | | {x}_{n}-p| {| }^{2}+{\delta }_{n}^{2}| | {x}_{n}-{x}_{n-1}| {| }^{2}+2{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | | | {x}_{n}-p| | \\ & =& | | {x}_{n}-p| {| }^{2}+{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | \left({\delta }_{n}| | {x}_{n}-{x}_{n-1}| | +2| | {x}_{n}-p| | )\\ & \le & | | {x}_{n}-p| {| }^{2}+3{G}_{2}{\delta }_{n}| | {x}_{n}-{x}_{n-1}| | \\ & =& | | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | ,\end{array}where G2≔supn∈N{∣∣xn−p∣∣,θn∣∣xn−xn−1∣∣}>0{G}_{2}:= {\sup }_{n\in {\mathbb{N}}}\left\{| | {x}_{n}-p| | ,{\theta }_{n}| | {x}_{n}-{x}_{n-1}| | \right\}\gt 0.Now, by applying Lemma 2.6, (30), and (36) we have ∣∣xn+1−p∣∣2=∣∣αnγf(rn)+(I−αnD)gn−p∣∣2=∣∣αn(γf(rn)−Dp)+(I−αnD)(gn−p)∣∣2≤(1−αnγ¯)2∣∣gn−p∣∣2+2αn⟨γf(rn)−Dp,xn+1−p⟩≤(1−αnγ¯)2‖rn−p‖2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2+2αnγ⟨f(rn)−f(p),xn+1−p⟩+2αn⟨γf(p)−Dp,xn+1−p⟩≤(1−αnγ¯)2∣∣rn−p∣∣2−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2+αnγρ(‖rn−p‖2+‖xn+1−p‖2)+2αn⟨γf(p)−Dp,xn+1−p⟩≤(1−αnγ¯)2∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣−1−ϕ2ψn2ψn+12‖rn−yn‖2−βn(1−τ−βn)‖Uzn−zn‖2)+αnγρ∣∣xn−p∣∣2+3G2αnδnαn∣∣xn−xn−1∣∣+‖xn+1−p‖2+2αn⟨γf(p)−Dp,xn+1−p⟩.\hspace{-45.45em}\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& =& | | {\alpha }_{n}\gamma f\left({r}_{n})+\left(I-{\alpha }_{n}D){g}_{n}-p| {| }^{2}\\ & =& | | {\alpha }_{n}\left(\gamma f\left({r}_{n})-Dp)+\left(I-{\alpha }_{n}D)\left({g}_{n}-p)| {| }^{2}\\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}| | {g}_{n}-p| {| }^{2}+2{\alpha }_{n}\langle \gamma f\left({r}_{n})-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(\Vert {r}_{n}-p{\Vert }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right)\\ & & +2{\alpha }_{n}\gamma \langle f\left({r}_{n})-f\left(p),{x}_{n+1}-p\rangle +2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(| | {r}_{n}-p| {| }^{2}-\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}-{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right)\\ & & +{\alpha }_{n}\gamma \rho \left(\Vert {r}_{n}-p{\Vert }^{2}+\Vert {x}_{n+1}-p{\Vert }^{2})+2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & \le & {\left(1-{\alpha }_{n}\bar{\gamma })}^{2}\left(| | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | -\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}\right.\\ & & -{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2})+{\alpha }_{n}\gamma \rho \left(| | {x}_{n}-p| {| }^{2}+3{G}_{2}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\Vert {x}_{n+1}-p{\Vert }^{2}\right)\\ & & +2{\alpha }_{n}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle .\end{array}Consequently, we obtain ∣∣xn+1−p∣∣2≤(1−2αnγ¯+(αnγ¯)2+αnγρ)(1−αnγρ)∣∣xn−p∣∣2+3G2((1−αnγ¯)2+αnγρ)(1−αnγρ)αnδnαn∣∣xn−xn−1∣∣+2αn(1−αnγρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2=(1−2αnγ¯+αnγρ)(1−αnγρ)∣∣xn−p∣∣2+(αnγ¯)2(1−αnγρ)∣∣xn−p∣∣2+3G2((1−αnγ¯)2+αnγρ)(1−αnγρ)αnδnαn∣∣xn−xn−1∣∣+2αn(1−αnγρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \frac{\left(1-2{\alpha }_{n}\bar{\gamma }+{\left({\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | \\ & & +\frac{2{\alpha }_{n}}{\left(1-{\alpha }_{n}\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}\\ & =& \frac{\left(1-2{\alpha }_{n}\bar{\gamma }+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}+\frac{{\left({\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}| | {x}_{n}-p| {| }^{2}\\ & & +3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}{\alpha }_{n}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{2{\alpha }_{n}}{\left(1-{\alpha }_{n}\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\}\end{array}≤(1−2αn(γ¯−γρ)(1−αnγρ))∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩−(1−αnγ¯)2(1−αnγρ)1−ϕ2ψn2ψn+12‖rn−yn‖2+βn(1−τ−βn)‖Uzn−zn‖2,\hspace{1.75em}\begin{array}{rcl}& \le & (1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )})| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}\\ & & -\frac{{\left(1-{\alpha }_{n}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\left(1-{\phi }^{2}\frac{{\psi }_{n}^{2}}{{\psi }_{n+1}^{2}}\right)\Vert {r}_{n}-{y}_{n}{\Vert }^{2}+{\beta }_{n}\left(1-\tau -{\beta }_{n})\Vert U{z}_{n}-{z}_{n}{\Vert }^{2}\right\},\end{array}where G3≔sup{∣∣xn−p∣∣2:n∈N}{G}_{3}:= \sup \left\{| | {x}_{n}-p| {| }^{2}:n\in {\mathbb{N}}\right\}. This gives the required inequality.Claim 3. The sequence {‖xn−p‖2}\left\{\Vert {x}_{n}-p{\Vert }^{2}\right\}converges to zero.Let p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p). From Claim 2, we obtain (37)∣∣xn+1−p∣∣2≤1−2αn(γ¯−γρ)(1−αnγρ)∣∣xn−p∣∣2+2αn(γ¯−γρ)(1−αnγρ)αnγ¯22(γ¯−γρ)G3+3G2((1−αnγ¯)2+αnγρ)2(γ¯−γρ)δnαn∣∣xn−xn−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xn+1−p⟩.\begin{array}{rcl}| | {x}_{n+1}-p| {| }^{2}& \le & \left(1-\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\right)| | {x}_{n}-p| {| }^{2}+\frac{2{\alpha }_{n}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{n}\gamma \rho )}\left\{\frac{{\alpha }_{n}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ & & \left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{n}\bar{\gamma })}^{2}+{\alpha }_{n}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{n+1}-p\rangle \right\}.\end{array}To establish Claim 3, in view of Lemma 2.7, Remark 3.3, and the fact that limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0, it suffices to show that lim supk→∞⟨γf(p)−Dp,xnk+1−p⟩≤0{\mathrm{lim\; sup}}_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle \le 0for every subsequence {‖xnk−p‖}\left\{\Vert {x}_{{n}_{k}}-p\Vert \right\}of {‖xn−p‖}\left\{\Vert {x}_{n}-p\Vert \right\}satisfying lim infk→∞(‖xnk+1−p‖−‖xnk−p‖)≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\left(\Vert {x}_{{n}_{k}+1}-p\Vert -\Vert {x}_{{n}_{k}}-p\Vert )\ge 0.Suppose that {‖xnk−p‖}\left\{\Vert {x}_{{n}_{k}}-p\Vert \right\}is a subsequence of {‖xn−p‖}\left\{\Vert {x}_{n}-p\Vert \right\}such that (38)lim infk→∞(‖xnk+1−p‖−‖xnk−p‖)≥0.\mathop{\mathrm{lim\; inf}}\limits_{k\to \infty }\left(\Vert {x}_{{n}_{k}+1}-p\Vert -\Vert {x}_{{n}_{k}}-p\Vert )\ge 0.Again, from Claim 2 we obtain (1−αnkγ¯)2(1−αnkγρ)1−ϕ2ψnk2ψnk+12‖rnk−ynk‖2≤1−2αnk(γ¯−γρ)(1−αnkγρ)∣∣xnk−p∣∣2−∣∣xnk+1−p∣∣2+2αnk(γ¯−γρ)(1−αnkγρ)αnkγ¯22(γ¯−γρ)G3+3G2((1−αnkγ¯)2+αnkγρ)2(γ¯−γρ)δnkαnk∣∣xnk−xnk−1∣∣+1(γ¯−γρ)⟨γf(p)−Dp,xnk+1−p⟩.\begin{array}{l}\frac{{\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left(1-{\phi }^{2}\frac{{\psi }_{{n}_{k}}^{2}}{{\psi }_{{n}_{k}+1}^{2}}\right)\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}{\Vert }^{2}\\ \hspace{1.0em}\le \left(1-\frac{2{\alpha }_{{n}_{k}}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\right)| | {x}_{{n}_{k}}-p| {| }^{2}-| | {x}_{{n}_{k}+1}-p| {| }^{2}+\frac{2{\alpha }_{{n}_{k}}\left(\bar{\gamma }-\gamma \rho )}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left\{\frac{{\alpha }_{{n}_{k}}{\bar{\gamma }}^{2}}{2\left(\bar{\gamma }-\gamma \rho )}{G}_{3}\right.\\ \hspace{1.0em}\hspace{1.0em}\left.+3{G}_{2}\frac{\left({\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}+{\alpha }_{{n}_{k}}\gamma \rho )}{2\left(\bar{\gamma }-\gamma \rho )}\frac{{\delta }_{{n}_{k}}}{{\alpha }_{{n}_{k}}}| | {x}_{{n}_{k}}-{x}_{{n}_{k}-1}| | +\frac{1}{\left(\bar{\gamma }-\gamma \rho )}\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle \right\}.\end{array}Applying (38) and the fact that limk→∞αnk=0{\mathrm{lim}}_{k\to \infty }{\alpha }_{{n}_{k}}=0, we have (1−αnkγ¯)2(1−αnkγρ)1−ϕ2ψnk2ψnk+12‖rnk−ynk‖2→0,k→∞.\frac{{\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })}^{2}}{\left(1-{\alpha }_{{n}_{k}}\gamma \rho )}\left(1-{\phi }^{2}\frac{{\psi }_{{n}_{k}}^{2}}{{\psi }_{{n}_{k}+1}^{2}}\right)\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}{\Vert }^{2}\to 0,\hspace{1em}k\to \infty .By the conditions on the control parameters, we obtain (39)‖rnk−ynk‖→0,k→∞.\Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{0.9em}Following similar argument, from Claim 2 we have (40)‖Uznk−znk‖→0,k→∞.\Vert U{z}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .From (24) and (39), we obtain (41)‖znk−ynk‖→0,k→∞.\Vert {z}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{0.7em}Combining (39) and (41), we have (42)‖rnk−znk‖≤‖rnk−ynk‖+‖ynk−znk‖→0,k→∞.\Vert {r}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \le \Vert {r}_{{n}_{k}}-{y}_{{n}_{k}}\Vert +\Vert {y}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .By Remark 3.3 and the definition of rn{r}_{n}, we obtain (43)‖xnk−rnk‖=δnk‖xnk−xnk−1‖→0,k→∞.\Vert {x}_{{n}_{k}}-{r}_{{n}_{k}}\Vert ={\delta }_{{n}_{k}}\Vert {x}_{{n}_{k}}-{x}_{{n}_{k}-1}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{5.25em}From (39), (42), and (43), we obtain (44)‖xnk−ynk‖→0,k→∞,‖xnk−znk‖→0,k→∞.\Vert {x}_{{n}_{k}}-{y}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty ,\hspace{1em}\Vert {x}_{{n}_{k}}-{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .Also, from (40) and (44), we obtain (45)‖xnk−Uznk‖→0,k→∞.\Vert {x}_{{n}_{k}}-U{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .\hspace{12em}Using (44) and (45), we have (46)‖xnk−gnk‖≤(1−βnk)‖xnk−znk‖+βnk‖xnk−Uznk‖→0,k→∞.\Vert {x}_{{n}_{k}}-{g}_{{n}_{k}}\Vert \le \left(1-{\beta }_{{n}_{k}})\Vert {x}_{{n}_{k}}-{z}_{{n}_{k}}\Vert +{\beta }_{{n}_{k}}\Vert {x}_{{n}_{k}}-U{z}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .Combining this together with the fact that limk→∞αnk=0{\mathrm{lim}}_{k\to \infty }{\alpha }_{{n}_{k}}=0, we obtain (47)‖xnk+1−xnk‖≤αnk‖γf(rnk)−xnk‖+(1−αnkγ¯)‖gnk−xnk‖→0,k→∞.\Vert {x}_{{n}_{k}+1}-{x}_{{n}_{k}}\Vert \le {\alpha }_{{n}_{k}}\Vert \gamma f\left({r}_{{n}_{k}})-{x}_{{n}_{k}}\Vert +\left(1-{\alpha }_{{n}_{k}}\bar{\gamma })\Vert {g}_{{n}_{k}}-{x}_{{n}_{k}}\Vert \to 0,\hspace{1em}k\to \infty .To complete the proof, we need to show that wω(xn)⊂Ω{w}_{\omega }\left({x}_{n})\subset \Omega . Since {xn}\left\{{x}_{n}\right\}is bounded, then wω(xn){w}_{\omega }\left({x}_{n})is nonempty. Let x∗∈wω(xn){x}^{\ast }\in {w}_{\omega }\left({x}_{n})be an arbitrary element. Then there exists a subsequence {xnk}\left\{{x}_{{n}_{k}}\right\}of {xn}\left\{{x}_{n}\right\}such that xnk⇀x∗{x}_{{n}_{k}}\rightharpoonup {x}^{\ast }as k→∞k\to \infty . By Lemma 4.2 and (39), it follows that x∗∈VI(C,A){x}^{\ast }\in VI\left(C,A). Consequently, we have wω(xn)⊂VI(C,A){w}_{\omega }\left({x}_{n})\subset VI\left(C,A). From (44), we have that znk⇀x∗{z}_{{n}_{k}}\rightharpoonup {x}^{\ast }as k→∞k\to \infty . Since I−UI-Uis demiclosed at zero, then it follows from (40) that x∗∈F(U){x}^{\ast }\in F\left(U). That is, wω(xn)⊂F(U){w}_{\omega }\left({x}_{n})\subset F\left(U). Therefore, we have wω(xn)⊂Ω{w}_{\omega }\left({x}_{n})\subset \Omega .Moreover, from (44) it follows that wω{yn}=wω{xn}=wω{zn}{w}_{\omega }\{{y}_{n}\}={w}_{\omega }\left\{{x}_{n}\right\}={w}_{\omega }\left\{{z}_{n}\right\}. By the boundedness of {xnk}\left\{{x}_{{n}_{k}}\right\}, there exists a subsequence {xnkj}\left\{{x}_{{n}_{{k}_{j}}}\right\}of {xnk}\left\{{x}_{{n}_{k}}\right\}such that xnkj⇀x†{x}_{{n}_{{k}_{j}}}\rightharpoonup {x}^{\dagger }and (48)limj→∞⟨γf(p)−Dp,xnkj−p⟩=lim supk→∞⟨γf(p)−Dp,xnk−p⟩=lim supk→∞⟨γf(p)−Dp,znk−p⟩.\mathop{\mathrm{lim}}\limits_{j\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{{k}_{j}}}-p\rangle =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle =\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{z}_{{n}_{k}}-p\rangle .Since p=PΩ(I−D+γf)(p)p={P}_{\Omega }\left(I-D+\gamma f)\left(p), it follows from (48) that (49)lim supk→∞⟨γf(p)−Dp,xnk−p⟩=limj→∞⟨γf(p)−Dp,xnkj−p⟩=⟨γf(p)−Dp,x†−p⟩≤0.\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle =\mathop{\mathrm{lim}}\limits_{j\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{{k}_{j}}}-p\rangle =\langle \gamma f\left(p)-Dp,{x}^{\dagger }-p\rangle \le 0.Hence, from (47) and (49), we obtain (50)lim supk→∞⟨γf(p)−Dp,xnk+1−p⟩=lim supk→∞⟨γf(p)−Dp,xnk+1−xnk⟩+lim supk→∞⟨γf(p)−Dp,xnk−p⟩=⟨γf(p)−Dp,x†−p⟩≤0.\begin{array}{rcl}\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-p\rangle & =& \mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}+1}-{x}_{{n}_{k}}\rangle +\mathop{\mathrm{lim\; sup}}\limits_{k\to \infty }\langle \gamma f\left(p)-Dp,{x}_{{n}_{k}}-p\rangle \\ & =& \langle \gamma f\left(p)-Dp,{x}^{\dagger }-p\rangle \le 0.\end{array}Applying Lemma 2.7 to (37), and using (50) together with the fact that limn→∞θnαn∣∣xn−xn−1∣∣=0{\mathrm{lim}}_{n\to \infty }\frac{{\theta }_{n}}{{\alpha }_{n}}| | {x}_{n}-{x}_{n-1}| | =0and limn→∞αn=0{\mathrm{lim}}_{n\to \infty }{\alpha }_{n}=0, we deduce that limn→∞∣∣xn−p∣∣=0{\mathrm{lim}}_{n\to \infty }| | {x}_{n}-p| | =0as required.□Taking γ=1\gamma =1and D=ID=Iin Theorem 4.4, where IIis the identity mapping, then we have the following corollary.Corollary 4.5Let H be a Hilbert space and suppose U:H→HU:H\to His a τ\tau -demicontractive map. Let {xn}\left\{{x}_{n}\right\}be a sequence generated as follows:Algorithm 4.6Step 0.Given δ>0,ϕ∈(0,1)\delta \gt 0,\phi \in \left(0,1), select initial data x0,x1∈H{x}_{0},{x}_{1}\in H, λ0>0{\lambda }_{0}\gt 0, and set n=1n=1.Step 1.Given the (n−1n-1)th and nth iterates, choose δn{\delta }_{n}such that 0≤δ≤δn,∀n∈N0\le \delta \le {\delta }_{n},\hspace{1em}\forall n\in {\mathbb{N}}with δn{\delta }_{n}defined by: (51)δn=minεn∣∣xn−xn−1∣∣,δ,ifxn≠xn−1,δ,otherwise.{\delta }_{n}=\left\{\begin{array}{ll}{\rm{\min }}\left\{\frac{{\varepsilon }_{n}}{| | {x}_{n}-{x}_{n-1}| | },\delta \right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{x}_{n}\ne {x}_{n-1},\\ \delta ,& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.\hspace{4.75em}Step 2.Compute (52)rn=xn+δn(xn−xn−1).{r}_{n}={x}_{n}+{\delta }_{n}\left({x}_{n}-{x}_{n-1}).\hspace{0.5em}Step 3.Compute the projection: (53)yn=PC(rn−ψnArn),{y}_{n}={P}_{C}\left({r}_{n}-{\psi }_{n}A{r}_{n}),\hspace{1.85em}If yn=rn{y}_{n}={r}_{n}, then set yn=rn{y}_{n}={r}_{n}and go to Step 5. Else go to Step 4.Step 4.Compute (54)zn=yn−ψn(Ayn−Arn).{z}_{n}={y}_{n}-{\psi }_{n}\left(A{y}_{n}-A{r}_{n}).Step 5.Compute (55)ψn+1=minϕ∣∣rn−yn∣∣∣∣Arn−Ayn∣∣,ψn,ifArn−Ayn≠0,ψn,otherwise.{\psi }_{n+1}=\left\{\begin{array}{ll}\min \left\{\frac{\phi | | {r}_{n}-{y}_{n}| | }{| | A{r}_{n}-A{y}_{n}| | },{\psi }_{n}\right\},& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}A{r}_{n}-A{y}_{n}\ne 0,\\ {\psi }_{n},& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.Step 6.Compute (56)xn+1=αnf(rn)+(1−αn)[(1−βn)zn+βnUzn].{x}_{n+1}={\alpha }_{n}f\left({r}_{n})+\left(1-{\alpha }_{n})\left[\left(1-{\beta }_{n}){z}_{n}+{\beta }_{n}U{z}_{n}].Set n≔n+1n:= n+1and return to Step 1.Assume that Ω=VI(C,A)∩F(U)≠0\Omega =VI\left(C,A)\cap F\left(U)\ne 0and other assumptions in conditions A and B are satisfied. Then the sequence {xn}\left\{{x}_{n}\right\}generated by Algorithm 4.6 converges strongly to a point p∈Ωp\in \Omega where p=PΩ∘f(p)p={P}_{\Omega }\circ f\left(p)is a solution of the variational inequalities.⟨(I−f)p,p−z⟩≤0forallz∈Ω.\langle \left(I-f)p,p-z\rangle \le 0\hspace{1em}{for}\hspace{0.33em}{all}\hspace{0.33em}z\in \Omega .Remark 4.7The result in Corollary 4.5 complements the result of Tan and Qin [29], Gang et al. [22] and Thong and Hieu [23] in the following ways: (i)Our result in Corollary 4.5 extends the result of Tan and Qin [29] from pseudomonotone VIP to common solution problem of pseudomonotone variational inequality and FPPs of demicontractive maps.(ii)Corollary 4.5 result extends the result of Cai et al. [22] from FPP of nonexpansive maps to FPP of demicontractive maps.(iii)The result of Cai et al. [22] requires the knowledge of the Lipschitz constant of the cost operator while our result in Corollary 4.5 does not require any knowledge of the Lipschitz constant of the cost operator.(iv)The result of Corollary 4.5 extends the result of Thong and Hieu [23] from monotone VIP to pseudomonotone VIP.(v)Unlike the result of Thong and Hieu [23], our result in Corollary 4.5 employs inertial technique to speed up the rate of convergence of the algorithm.(vi)As shown in our convergence analysis, we did not adopt the conventional “two cases” approach employed in several papers to prove strong convergence. Our procedure is more concise and easy to comprehend.5Numerical examplesIn this section, we proceed to perform two numerical experiments to show the computational efficiency of our Algorithm 3.1 in comparison with some other algorithms in the literature. The graph of errors is plotted against the number of iterations in each case. All numerical computations were carried out using Matlab 2019(b). We use ‖xn+1−xn‖≤10−2\Vert {x}_{n+1}-{x}_{n}\Vert \le 1{0}^{-2}as the stopping criterion. The parameters are chosen as follows: Let f(x)=15xf\left(x)=\frac{1}{5}x, then ρ=15\rho =\frac{1}{5}is the Lipschitz constant for ff. Let D(x)=x3D\left(x)=\frac{x}{3}with constant γ¯=13\bar{\gamma }=\frac{1}{3}, then we take γ=1\gamma =1, which satisfies 0<γ<γ¯ρ0\lt \gamma \lt \frac{\bar{\gamma }}{\rho }. Let Ux=−32xUx=-\frac{3}{2}x. Choose δ=0.8,ψ1=0.6,ϕ=0.7,αn=1n+3,εn=1(n+3)3,βn=3n+15n+3\delta =0.8,{\psi }_{1}=0.6,\phi =0.7,{\alpha }_{n}=\frac{1}{n+3},{\varepsilon }_{n}=\frac{1}{{\left(n+3)}^{3}},{\beta }_{n}=\frac{3n+1}{5n+3}in our Algorithm 3.1.Take Tx=x2,ψ=0.8L,θn=1(n+3)2Tx=\frac{x}{2},\psi =\frac{0.8}{L},{\theta }_{n}=\frac{1}{{\left(n+3)}^{2}}in Algorithm (14).Let Gx=x−x1,γn=1n+1,ω=0.09,ρn=n2n+1Gx=x-{x}_{1},{\gamma }_{n}=\frac{1}{n+1},\omega =0.09,{\rho }_{n}=\frac{n}{2n+1}in Appendix 6.1.Take Tnx=−2nmod5x,λ=m=μ=12,σn=1n+3,τn=13,γn=16,μn=12{T}_{n}x=-\frac{2}{n\hspace{0.3em}\mathrm{mod}\hspace{0.3em}5}x,\lambda =m=\mu =\frac{1}{2},{\sigma }_{n}=\frac{1}{n+3},{\tau }_{n}=\frac{1}{3},{\gamma }_{n}=\frac{1}{6},{\mu }_{n}=\frac{1}{2}, in Appendices 6.2 and 6.3.Example 5.1Consider the linear operator A:Rm→Rm(m=5,10,15,20)A:{{\mathbb{R}}}^{m}\to {{\mathbb{R}}}^{m}\hspace{0.33em}\left(m=5,10,15,20)as follows: A(x)=Fx+gA\left(x)=Fx+g, where g∈Rmg\in {{\mathbb{R}}}^{m}and F=BBT+M+EF=B{B}^{T}+M+E, matrix B∈Rm×mB\in {{\mathbb{R}}}^{m\times m}, matrix M∈Rm×mM\in {{\mathbb{R}}}^{m\times m}, is skew symmetric, and matrix E∈Rm×mE\in {{\mathbb{R}}}^{m\times m}is a diagonal matrix whose diagonal terms are nonnegative (which implies that FFis positive symmetric definite). We choose the feasible set as C={x∈Rm:−2≤xi≤5,i=1,…,m}C=\left\{x\in {{\mathbb{R}}}^{m}:-2\le {x}_{i}\le 5,\hspace{0.33em}i=1,\ldots ,m\right\}. It can easily be verified that the mapping AAis strongly pseudomonotone and Lipschitz continuous with L=‖F‖L=\Vert F\Vert . In this example, both BBand MMentries are generated randomly in [−2,2]\left[-2,2], EEis generated randomly in [0,2]\left[0,2], and g=0g=0. The initial values x0=x1{x}_{0}={x}_{1}are generated randomly by rand(m,1){\rm{rand}}\left(m,1).The stopping criterion used for our computation is ‖xn+1−xn‖<10−2\Vert {x}_{n+1}-{x}_{n}\Vert \lt 1{0}^{-2}. We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 1 and Table 1.Figure 1Top left: m=5m=5; top right: m=10m=10; bottom left: m=15m=15; bottom right: m=20m=20.Table 1Numerical results for Example 5.1Algorithm 14Appendix 6.1Appendix 6.2Appendix 6.3Algorithm 3.1m=5m=5No. of Iter.101111226CPU time (s)1.71480.98800.88461.93790.4792m=10m=10No. of Iter.111111226CPU time (s)1.43751.10260.99231.97120.5256m=15m=15No. of Iter.111111246CPU time (s)1.41070.93371.05541.98880.6284m=20m=20No. of Iter.111112256CPU time (s)1.29530.81840.97711.61420.4390Example 5.2We consider the next example in the infinite dimensional Hilbert space H=L2([0,1])H={L}^{2}\left(\left[0,1])with inner product (57)⟨x,y⟩≔∫01x(t)y(t)dtforallx,y∈H,\langle x,y\rangle := {\int }_{0}^{1}x\left(t)y\left(t){\rm{d}}t\hspace{1em}{\rm{for}}\hspace{0.33em}{\rm{all}}\hspace{0.33em}x,y\in H,and induced norm (58)∣∣x∣∣≔∫01∣x(t)∣2dt12forallx∈H.\hspace{0.25em}| | x| | := {\left({\int }_{0}^{1}| x\left(t){| }^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}\hspace{1em}{\rm{for}}\hspace{0.33em}{\rm{all}}\hspace{0.33em}x\in H.Now, define A:H→HA:H\to Hby A(x)(t)=max{0,x(t)}A\left(x)\left(t)=\hspace{0.1em}\text{max}\hspace{0.1em}\left\{0,x\left(t)\right\}, for all t∈[0,1],x∈Ht\in \left[0,1],x\in H. It is easy to see that AAis pseudomonotone and 1-Lipschitz continuous on HH. It can easily be verified that all the conditions of Theorem 4.4 are satisfied.We choose four different initial values as follows:Case I: x0=2t3+13{x}_{0}=\frac{2{t}^{3}+1}{3}and x1=3t5+t2+1{x}_{1}=3{t}^{5}+{t}^{2}+1;Case II: x0=exp(−t){x}_{0}=\exp \left(-t)and x1=cos2t{x}_{1}=\cos 2t;Case III: x0=t3+t+5{x}_{0}={t}^{3}+t+5and x1=exp(−2t){x}_{1}=\exp \left(-2t);Case IV: x0=2t5+t2+3{x}_{0}=2{t}^{5}+{t}^{2}+3and x1=2t3−t2+3{x}_{1}=2{t}^{3}-{t}^{2}+3.The stopping criterion used for our computation is ‖xn+1−xn‖<10−2\Vert {x}_{n+1}-{x}_{n}\Vert \lt 1{0}^{-2}. We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 2 and Table 2.Figure 2Top left: Case I; top right: Case II; bottom left: Case III; bottom right: Case IV.Table 2Numerical results for Example 5.2Algorithm 14Appendix 6.1Appendix 6.2Appendix 6.3Algorithm 3.1No. of Iter.681254No. of Iter.681254No. of Iter.681254No. of Iter.91117856ConclusionWe studied the pseudomonotone VIP with a fixed point constraint. We introduced a new inertial TEGM with an adaptive step size for approximating a solution of the pseudomonotone VIP, which is also a fixed point of demicontractive mappings. We proved strong convergence results for the proposed algorithm without the knowledge of the Lipschitz constant of the cost operator. Finally, we presented several numerical experiments to demonstrate the efficiency of our proposed method in comparison with some of the existing methods in the literature.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: Tseng’s extragradient method; pseudomonotone; demicontractive; variational inequalities; fixed point; strong convergence; adaptive step size; inertial technique; 65K15; 47J25; 65J15; 90C33

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