The split common fixed point problem for infinite families of demicontractive mappings

The split common fixed point problem for infinite families of demicontractive mappings Center of Excellence in In this paper, we propose a new algorithm for solving the split common fixed point Mathematics and Applied Mathematics, Department of problem for infinite families of demicontractive mappings. Strong convergence of the Mathematics, Faculty of Science, proposed method is established under suitable control conditions. We apply our Chiang Mai University, Chiang Mai, main results to study the split common null point problem, the split variational Thailand Center of Excellence in inequality problem, and the split equilibrium problem in the framework of a real Mathematics, CHE, Bangkok, Hilbert space. A numerical example supporting our main result is also given. Thailand Full list of author information is MSC: 47H09; 47J05; 47J25; 47N10 available at the end of the article Keywords: Fixed point problem; Demicontractive mappings; Null point problem; Variational inequality problem; Equilibrium problem 1 Introduction Let H be a real Hilbert space with inner product ·, · and norm ·.Let I denote the identity mapping. Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H ,respectively.Let A : H → H be a bounded linear operator with adjoint oper- 1 2 1 2 ator A . The split feasibility problem (SFP), which was first introduced by Censor and Elfving [1], is to find ∗ ∗ v ∈ C such that Av ∈ Q.(1) Let P and P be the orthogonal projections onto the sets C and Q,respectively. Assume C Q that (1) has a solution. It known that v ∈ H solves (1) if and only if it solves the fixed point equation ∗ ∗ ∗ v = P I + γ A (P – I)A v , C Q where γ > 0 is any positive constant. SFP has been used to model significant real-world inverse problems in sensor networks, radiation therapy treatment planning, antenna design, immaterial science, computerized tomography, etc. (see [2–4]). © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 2 of 21 The split common fixed point problem (SCFP) for mappings T and S,which wasfirst introduced by Censor and Segal [5], is to find ∗ ∗ v ∈ F(T)suchthat Av ∈ F(S), (2) where T : H → H and S : H → H are two mappings satisfying F(T)= {x ∈ H : 1 1 2 2 1 Tx = x}= ∅ and F(S)= {x ∈ H : Sx = x}= ∅, respectively. Since each closed and convex subset may be considered as a fixed point set of a projection onto the subset, the SCFP is a generalization of the SFP. Recently, the SFP and SCFP have been studied by many authors; see, for example, [6–11]. In 2010, Moudafi [11] introduced the following algorithm for solving (2)for twodemi- contractive mappings: x ∈ H choose arbitrarily, 1 1 (3) u = x + γαA (S – I)Ax , n n n x =(1 – β )u + β Tu , n ∈ N. n+1 n n n n He proved that {x } converges weakly to some solution of SCFP. The multiple set split feasibility problem (MSSFP), which was first introduced by Censor et al. [4], is to find m r ∗ ∗ v ∈ C such that Av ∈ Q,(4) i i i=1 i=1 m r where {C } and {Q } are families of nonempty closed convex subsets of real Hilbert i i i=1 i=1 spaces H and H ,respectively. We seethatif m = r =1, then problem (4)reduces to prob- 1 2 lem (1). Recently, Eslamian [12] considered the problem of finding a point m m m ∗ ∗ ∗ v ∈ F(U)suchthat A v ∈ F(S)and A v ∈ F(T ), (5) i 1 i 2 i i=1 i=1 i=1 where A , A : H → H are bounded linear operators, and U : H → H , T : H → H 1 2 1 2 i 1 1 i 2 2 and S : H → H , i = 1,2,..., m.Healsopresented anew algorithmtosolve (5)for finite i 2 2 families of quasi-nonexpansive mappings: x ∈ H choose arbitrarily, 1 1 1 ∗ u = x + ηβA (S – I)A x , n n i 1 n ⎨ 1 i=1 m y = u + η β A (T – I)A u , (6) n n i 2 n i=1 m 2 ⎪ m z = α y + α U y , n n,0 n n,i i n i=1 x = θ γ f (x )+(I – θ B)z , n ∈ N. n+1 n n n n He proved that {x } converges strongly to some solution of (5)under some controlcon- ditions. Question. Can we modify algorithm (6) to a simple one for solving the problem of finding ∞ ∞ ∞ ∗ ∗ ∗ v ∈ F(U)suchthat A v ∈ F(S)and A v ∈ F(T ), (7) i 1 i 2 i i=1 i=1 i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 3 of 21 where A , A : H → H are bounded linear operators, and {U : H → H : i ∈ N}, 1 2 1 2 i 1 1 {T : H → H : i ∈ N} and {S : H → H : i ∈ N} are infinite families of k -, k -, and k - i 2 2 i 2 2 3 2 1 demicontractive mappings, respectively. In this work, we introduce a new algorithm for solving problem (7) for infinite families of demicontractive mappings and prove its strong convergence to a solution of problem (7). 2Preliminaries Throughout this paper, we adopt the following notations. (i) “→”and “” denote the strong and weak convergence, respectively. (ii) ω (x ) denotes the set of the cluster points of {x } in the weak topology, that is, ω n n ∃{x } of {x } such that x  x. n n n i i (iii) is the solution set of problem (7), that is, ∞ ∞ ∞ ∗ ∗ ∗ = v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T ) . i 1 i 2 i i=1 i=1 i=1 A mapping P is said to be a metric projection of H onto C if for every x ∈ H,there exists auniquenearest pointin C,denoted by P x,suchthat x – P x≤x – z, ∀z ∈ C. It is known that P is a firmly nonexpansive mapping. Moreover, P is characterized by the C C following property: x – P x, y – P x≤ 0 for all x ∈ H, y ∈ C. A bounded linear operator C C B : H → H is said to be strongly positive if there is a constant ξ >0 such that Bx, x≥ ξ x for all x ∈ H. Definition 2.1 The mapping T : H → H is said to be (i) L-Lipschitzian if there exists L >0 such that Tu – Tv≤ Lu – v for all u, v ∈ H; (ii) α-contraction if T is α-Lipschitzian with α ∈ [0, 1),thatis, Tu – Tv≤ αu – v for all u, v ∈ H; (iii) nonexpansive if T is 1-Lipschitzian; (iv) quasi-nonexpansive if F(T) = ∅ and Tu – v≤u – v for all u ∈ H, v ∈ F(T); (v) firmly nonexpansive if 2 2 Tu – Tv ≤u – v – (u – v)–(Tu – Tv) for all u, v ∈ H; or equivalently, for all u, v ∈ H, Tu – Tv ≤Tu – Tv, u – v; Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 4 of 21 (vi) λ-inverse strongly monotone if there exists λ >0 such that u – v, Tu – Tv≥ λTu – Tv for all u, v ∈ H; (vii) k-demicontractive if F(T) = ∅ and there exists k ∈ [0, 1) such that 2 2 2 Tu – v ≤u – v + ku – Tu for all u ∈ H, v ∈ F(T). The following example is an infinite family of k-demicontractive mappings in R . 2 2 Example 2.2 For i ∈ N,let U : R → R be defined for all x , x ∈ R by i 1 2 –2i U (x , x )= x , x , i 1 2 1 2 i +1 and · is the Euclidean norm on R .Observe that F(U )=0 × R for all i ∈ N,thatis, if x =(x , x ) ∈ R × R and p =(0, p ) ∈ F(U ), then 1 2 2 i –2i U x – p = x , x –(0, p ) i 1 2 2 i +1 –2i 2 2 = |x | + |x – p | 1 2 2 i +1 2 2 ≤ 4|x | + |x – p | 1 2 2 2 2 2 2 = |x | + (1 + 1) |x | + |x – p | 1 1 2 2 3 2i 2 2 ≤x – p + 1+ |x | 4 i +1 2 2 = x – p + U x – x . So, U are -demicontractive mappings for all i ∈ N. Definition 2.3 The mapping T : H → H is said to be demiclosed at zero if for any sequence {u }⊂ H with u  u and Tu → 0, we have Tu =0. n n n Lemma 2.4 ([13]) Assume that B is a self-adjoint strongly positive bounded linear operator –1 on a Hilbert space H with coefficient ξ >0 and 0< μ ≤B . Then I – μB≤ 1– ξμ. Lemma 2.5 ([14]) Let H be a real Hilbert space. Then the following results hold: 2 2 2 (i) u + v = u +2u, v + v ∀u, v ∈ H; 2 2 (ii) u + v ≤u +2v, u + v∀u, v ∈ H. Lemma 2.6 ([15]) Let {a } be a sequence of nonnegative real numbers satisfying the fol- lowing relation: a ≤ (1 – γ )a + δ , n ∈ N, n+1 n n n where Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 5 of 21 (i) {γ }⊂ (0, 1), γ = ∞; n n n=1 δ ∞ (ii) lim sup ≤ 0 or |δ | < ∞. n→∞ γ n=1 Then lim a =0. n→∞ n Lemma 2.7 ([16]) Let {κ } be a sequence of real numbers that does not decrease at infinity, that is, there exists at a subsequence {κ } of {κ } that satisfies κ < κ for all i ∈ N. For n n n n +1 i i i every n ≥ n , define the integer sequence {τ(n)} as follows: τ(n)= max{l ∈ N : l ≤ n, κ < κ }, l l+1 where n ∈ N is such that {l ≤ n : κ < κ }= ∅. Then: o o l l+1 (i) τ(n ) ≤ τ(n +1) ≤ ··· , and τ(n) →∞; o o (ii) for all n ≥ n , max{κ , κ }≤ κ . o n τ(n) τ(n)+1 3 Results and discussion In this section, we propose a new algorithm, which is a modification of (6)and proveits strong convergence under some suitable conditions. We start with the following important lemma. Lemma 3.1 For two real Hilbert spaces H and H , let A : H → H be a bounded linear 1 2 1 2 operator with adjoint operator A . If T : H → H is a k-demicontractive mapping, then 2 2 2 2 2 ∗ ∗ ∗ 2 x + δA (T – I)Ax – x ≤ x – x – δ 1– k – δA (T – I)Ax ∗ ∗ for all x ∈ H such that Ax ∈ F(T). Proof Suppose that T : H → H is a k-demicontractive mapping and let x ∈ H be such 2 2 1 that Ax ∈ F(T). Then we have 2 2 ∗ ∗ ∗ ∗ ∗ x – x + δA (T – I)Ax ≤ x – x +2δ x – x , A (T – I)Ax 2 2 + δ A (T – I)Ax.(8) Since A is a bounded linear operator with adjoint operator A and T is a k-demicontractive mapping, by Lemma 2.5(ii) we deduce that ∗ ∗ ∗ x – x , A (T – I)Ax = Ax – Ax ,(T – I)Ax = TAx – Ax , TAx – Ax – (T – I)Ax 2 2 ∗ 2 ∗ = TAx – Ax + TAx – Ax – Ax – Ax – (T – I)Ax ∗ 2 ≤ Ax – Ax + kTAx – Ax 2 2 2 ∗ + TAx – Ax – Ax – Ax – (T – I)Ax k –1 = (T – I)Ax.(9) 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 6 of 21 From (8)and (9)weget 2 2 2 ∗ ∗ ∗ 2 x + δA (T – I)Ax – x ≤ x – x – δ 1– k – δA (T – I)Ax . This completes the proof. Lemma 3.2 For two real Hilbert spaces H and H , let A : H → H be a bounded linear 1 2 1 2 operator with adjoint operator A , and let {T : H → H : i ∈ N} be an infinite family of i 2 2 k-demicontractive mappings. Let {x } be sequence in H , and let n 1 u = x + α δ A (T – I)Ax , ∀n ∈ N, (10) n n n,i n i n i=1 where {α } is a real sequence in [0, 1] satisfying α =1. Then we have n,i n,i i=1 2 2 2 ∗ ∗ 2 u – x ≤ x – x – α δ 1– k – δ A (T – I)Ax n n n,i n n i n i=1 ∗ ∗ for all x ∈ H such that Ax ∈ F(T ). 1 i i=1 ∗ ∗ Proof Let x ∈ H be such that Ax ∈ F(T ). From (10) and Lemma 3.1 we obtain 1 i i=1 2 2 ∗ ∗ ∗ u – x ≤ α x – x + δ A (T – I)Ax n n,i n n i n i=1 2 2 ∗ 2 ≤ α x – x – δ 1– k – δ A (T – I)Ax n,i n n n i n i=1 2 2 ∗ 2 = x – x – α δ 1– k – δ A (T – I)Ax . n n,i n n i n i=1 This completes the proof. Lemma 3.3 Let {T : H → H : i ∈ N} be an infinite family of k-demicontractive mappings i 1 1 from a Hilbert space H to itself. Let {x } be sequence in H , and let 1 n 1 u = x + α δ (T – I)x , ∀n ∈ N, (11) n n n,i n i n i=1 where {α } is a real sequence in [0, 1] satisfying α =1. Then we have n,i n,i i=1 2 2 2 ∗ ∗ u – x ≤ x – x – α δ (1 – k – δ ) (T – I)x n n n,i n n i n i=1 for all x ∈ F(T ). i=1 Proof The statement directly follows from Lemma 3.2 by putting H = H and A = I. 1 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 7 of 21 Now, we introduce a new algorithm for solving problem (7) for an infinite family of demicontractive mappings and then prove its strong convergence. Theorem 3.4 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let f : H → H 1 1 1 2 be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear op- erator on H with coefficient ξ >2ρ and B =1. Let {S : H → H : i ∈ N}, {T : H → H : 1 i 2 2 i 2 2 i ∈ N}, and {U : H → H : i ∈ N} be infinite families of k -, k -, and k -demicontractive i 1 1 1 2 3 mappings such that S –I, T –I, and U –I are demiclosed at zero, respectively. Suppose that i i i ∞ ∞ ∞ ∗ ∗ ∗ = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= ∅. For arbitrary x ∈ H , i 1 i 2 i 1 1 i=1 i=1 i=1 let {u }, {v }, {y }, and {x } be generated by n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (12) y = v + γ τ (U – I)v , ⎪ n n n,i n i n i=1 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 1–k (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Proof For any u, v ∈ H , by Lemma 2.4 we have P (f + I – B)u – P (f + I – B)v ≤ (f + I – B)u –(f + I – B)v ≤ f (u)– f (v) + I – Bu – v ≤ ρu – v +(1– ξ)u – v ≤ (1 – ρ)u – v, that is, the mapping P (f + I – B) is a contraction. So, by the Banach contraction principle ∗ ∗ ∗ there is a unique element x ∈ H such that x = P (f + I – B)x . ∞ ∞ ∗ ∗ ∗ ∗ ∗ Let x = P (f + I – B)x ,thatis, x ∈ F(U )is such that A x ∈ F(S )and A x ∈ i 1 i 2 i=1 i=1 F(T ). From Lemmas 3.2 and 3.3 and from (12)weobtain i=1 2 2 2 ∗ ∗ 2 u – x ≤ x – x – α δ 1– k – δ A  (S – I)A x , (13) n n n,i n 1 n 1 i 1 n i=1 2 2 2 ∗ ∗ 2 v – x ≤ u – x – β θ 1– k – θ A  (T – I)A u , (14) n n n,i n 2 n 2 i 2 n i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 8 of 21 and 2 2 2 ∗ ∗ y – x ≤ v – x – λ τ (1 – k – τ ) (U – I)v . (15) n n n,i n 3 n i n i=1 Therefore 2 2 2 ∗ ∗ 2 y – x ≤ x – x – α δ 1– k – δ A  (S – I)A x n n n,i n 1 n 1 i 1 n i=1 – β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 – λ τ (1 – k – τ ) (U – I)v . (16) n,i n 3 n i n i=1 By conditions (C4), (C5),and (C6) we have ∗ ∗ y – x ≤ x – x . (17) n n –1 By condition (C3) we may assume that σ ∈ (0, B ) for all n ∈ N. By Lemma 2.4 we get I – σ B≤ 1– σ ξ.From(12)and (17)weget n n ∗ ∗ x – x = σ f (y )+(I – σ B)y – x n+1 n n n n ∗ ∗ = σ f (y )– Bx +(I – σ B) y – x n n n n ∗ ∗ ∗ ∗ ≤ σ f (y )– f x + f x – Bx + I – σ B y – x n n n n ∗ ∗ ∗ ∗ ≤ σ ρ y – x + σ f x – Bx +(1– σ ξ) y – x n n n n n ∗ ∗ f (x )– Bx ≤ 1– σ (ξ – ρ) x – x + σ (ξ – ρ) n n n ξ – ρ ∗ ∗ f (x )– Bx ≤ max x – x , ξ – ρ ∗ ∗ f (x )– Bx ≤ max x – x , . (18) ξ – ρ Therefore {x } is bounded, and we also have that {y } and {f (y )} are bounded. To this end, n n n we consider the following two cases. ∗ ∞ Case 1. Suppose that {x – x } is nonincreasing for some n ∈ N.Thenweget that n o n=n lim x – x  exists. By (16), (17), and Lemma 2.5(i) we get n→∞ n 2 2 ∗ ∗ ∗ x – x = σ f (y )– Bx +(I – σ B) y – x n+1 n n n n 2 2 ∗ ∗ ≤ σ f (y )– Bx +(1– σ ξ) y – x n n n n ∗ ∗ +2σ (1 – σ ξ) f (y )– Bx y – x n n n n ≤ σ M + x – x n n Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 9 of 21 – α δ 1– k – δ A  (S – I)A x n,i n 1 n 1 i 1 n i=1 – β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 – λ τ (1 – k – τ ) (U – I)v , n,i n 3 n i n i=1 where ∗ ∗ ∗ M = sup f (y )– Bx +2 f (y )– Bx x – x . n n n This implies, for j = 1,2,..., n, α δ 1– k – δ A  (S – I)A x n,j n 1 n 1 j 1 n ≤ α δ 1– k – δ A  (S – I)A x n,i n 1 n 1 i 1 n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x , (19) n n n+1 β θ 1– k – θ A  (T – I)A u n,j n 2 n 2 j 2 n ≤ β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x , (20) n n n+1 and 2 2 λ τ (1 – k – τ ) (U – I)v ≤ λ τ (1 – k – τ ) (U – I)v n,j n 3 n j n n,i n 3 n i n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x . (21) n n n+1 From (19), (20), (21), and conditions (C2)–(C6) we obtain lim (S – I)A x = lim α (S – I)A x = 0, (22) j 1 n n,i i 1 n n→∞ n→∞ i=1 lim (T – I)A u = lim β (T – I)A u = 0, (23) j 2 n n,i i 2 n n→∞ n→∞ i=1 and lim (U – I)v = lim λ (U – I)v = 0. (24) j n n,i i n n→∞ n→∞ i=1 Next, we show that ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x ≤ 0, where x = P (f + I – B)x . n→∞ Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 10 of 21 To see this, choose a subsequence {x } of {x } such that n n ∗ ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x = lim f x – Bx , x – x . n n p→∞ n→∞ Since the sequence {x } is bounded, there exists a subsequence {x } of {x } such that n n n p p p x  z ∈ H . Without loss of generality, we may assume that x  z ∈ H .Since A is a n 1 n 1 1 p p bounded linear operator, this yields that A x  A z. By the demiclosedness principle of 1 n 1 S – I at zero and (22)weget A z ∈ F(S ). By (12)and (22)wehave i 1 i i=1 2 ∗ u – x  = x + α δ A (S – I)A x – x n n n n,i n i 1 n n i=1 ≤ α δ A  (S – I)A x →0as n →∞. n,i n 1 i 1 n i=1 Similarly, we also have v – u → 0as n →∞.Using thefactthat x  z and n n n u – x → 0, we conclude that u  z.Since A is a bounded linear operator, we n n n 2 get that A u  A z. By the demiclosedness principle of T – I at zero and (23)weget 2 n 2 i A z ∈ F(T ). Again, since u  z and v –u → 0, we conclude that v  z.Bythe 2 i n n n n i=1 p p demiclosedness principle of U – I at zero and (24)wealsohave z ∈ F(U ). Therefore i i i=1 z ∈ . ∗ ∗ Since x = P (f + I – B)x and z ∈ ,weget ∗ ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x = lim f x – Bx , x – x n n p→∞ n→∞ ∗ ∗ ∗ = f x – Bx , z – x ≤ 0. (25) Using Lemma 2.5 and (17), we have 2 2 ∗ ∗ ∗ x – x = σ f (y )– Bx +(I – σ B) y – x n+1 n n n n ∗ ∗ ∗ ≤ (1 – σ ξ) y – x +2σ f (y )– Bx , x – x n n n n n+1 ∗ ∗ ∗ ≤ (1 – σ ξ) x – x +2ρσ y – x x – x n n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x n n+1 2 2 2 ∗ ∗ ∗ ≤ (1 – σ ξ) x – x + ρσ x – x + x – x n n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x n n+1 2 2 ∗ ∗ = 1– σ (ξ – ρ) x – x + ρσ x – x n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x . n n+1 This implies that σ (ξ – ρ) 2σ 2 2 n n ∗ ∗ ∗ ∗ ∗ x – x ≤ 1– x – x + f x – Bx , x – x . (26) n+1 n n+1 1– σ ρ 1– σ ρ n n By (25), (26), and Lemma 2.6 we conclude that x → x as n →∞. n Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 11 of 21 Case 2. Suppose that there exists an integer m such that ∗ ∗ x – x ≤ x – x . m m +1 o o Put κ = x – x  for all n ≥ m .Thenwehave κ ≤ κ .Let {μ(n)} be the sequence n n o m m +1 o o defined by μ(n)= max{l ∈ N : l ≤ n, κ ≤ κ } l l+1 for all n ≥ m . By Lemma 2.7 we obtain that {μ(n)} is a nondecreasing sequence such that lim μ(n)= ∞ and κ ≤ κ for all n ≥ m . μ(n) μ(n)+1 o n→∞ By thesameargumentasincase1we obtain lim (S – I)A x =0, lim (T – I)A u =0, i 1 μ(n) i 2 μ(n) n→∞ n→∞ and lim (U – I)v =0. i μ(n) n→∞ By the demiclosedness principle of S – I, T – I,and U – I at zero, we have ω (x ) ⊂ . i i i ω μ(n) This implies that ∗ ∗ ∗ lim sup f x – Bx , x – x ≤ 0. μ(n) n→∞ By a similar argument from (26)wealsohave σ (ξ – ρ) 2σ μ(n) μ(n) 2 2 ∗ ∗ ∗ κ ≤ 1– κ + f x – Bx , x – x . μ(n)+1 μ(n)+1 μ(n) 1– σ ρ 1– σ ρ μ(n) μ(n) So, we get lim κ =0 and also have lim κ = 0. By Lemma 2.7 we have n→∞ μ(n) n→∞ μ(n)+1 0 ≤ κ ≤ max{κ , κ }≤ κ . n n μ(n) μ(n)+1 Therefore x → x as n →∞. This completes the proof. By setting T = I for all i ∈ N in Theorem 3.4 we obtain the following result. Corollary 3.5 Let H and H be two real Hilbert spaces, let A : H → H be a bounded 1 2 1 1 2 linear operator with adjoint operator A . Let f : H → H be a ρ-contraction mapping, 1 1 and let B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {S : H → H : i ∈ N} and {U : H → H : i ∈ N} be infinite fami- i 2 2 i 1 1 lies of k -and k -demicontractive mappings such that S – Iand U – Iare demiclosed at 1 3 i i ∞ ∞ ∗ ∗ zero, respectively. Suppose that  = {v ∈ F(U ): A v ∈ F(S )}= ∅. For arbitrary i 1 i i=1 i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 12 of 21 x ∈ H , let {u }, {y }, and {x } be generated by 1 1 n n n ⎪ u = x + α δ A (S – I)A x , n n n,i n i 1 n ⎨ i=1 1 y = u + γ τ (U – I)u , (27) n n n,i n i n i=1 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {τ }, {σ }, {α }, and {γ } are sequences in [0, 1] satisfying the following condi- n n n n,i n,i tions: n n (C1) α = γ =1 for all n ∈ N; n,i n,i i=1 i=1 (C2) lim inf α >0 and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Remark 3.6 By the same setting as in Corollary 3.5,Eslamian[17] used another algorithm for solving the same problem as in Corollary 3.5;see [17], Theorem 3.3. Note that each step of our algorithm is much easier for computation than that of Eslamian [17]because our algorithm concerns only the finite sum. By setting f (y)= v for all y ∈ H and B = I in Theorem 3.4 we obtain the following result. Corollary 3.7 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let {S : H → H : i 2 2 1 2 i ∈ N}, {T : H → H : i ∈ N}, and {U : H → H : i ∈ N} be infinite families of k -, k -, and i 2 2 i 1 1 1 2 k -demicontractive mappings such that S – I, T – I, and U – I are demiclosed at zero, re- 3 i i i ∞ ∞ ∞ ∗ ∗ ∗ spectively. Suppose that = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= i 1 i 2 i i=1 i=1 i=1 ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (28) y = v + γ τ (U – I)v , ⎪ n n n,i n i n i=1 x = σ v +(1– σ )y , n ∈ N, n+1 n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 1–k (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 Then the sequence {x } converges strongly to x = P (v). It is known that every quasi-nonexpansive mapping is 0-demicontractive mapping, so the following result is directly obtained by Theorem 3.2. Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 13 of 21 Corollary 3.8 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let {S : H → i 2 1 2 H : i ∈ N}, {T : H → H : i ∈ N}, and {U : H → H : i ∈ N} be infinite families of quasi- 2 i 2 2 i 1 1 nonexpansive mappings such that S – I, T – I, and U – I are demiclosed at zero, respec- i i i ∞ ∞ ∞ ∗ ∗ ∗ tively. Suppose that = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= ∅. i 1 i 2 i i=1 i=1 i=1 For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (29) y = v + γ τ (U – I)v , n n n,i n i n ⎪ i=1 x = σ v +(1– σ )y , n ∈ N, n+1 n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 Then the sequence {x } converges strongly to x = P (v). 4 Applications 4.1 The split common null point problem Let M be the set-valued mapping of H into 2 .The effective domain of M is denoted by D(M), that is, D(M)= {x ∈ H : Mx = ∅}. The mapping M is said to be monotone if x – y, u – v≥ 0, ∀x, y ∈ D(M), u ∈ Mx, v ∈ My. A monotone mapping M is said to be maximal if the graph G(M) is not properly contained in the graph of any other monotone map, where G(M)= {(x, y) ∈ H × H : y ∈ Mx}.Itis known that M is maximal if and only if for (x, u) ∈ H × H, x – y, u – v≥ 0for every (y, v) ∈ G(M)implies u ∈ Mx. For the maximal monotone operator M,wecan associateits resolvent J defined by M –1 J ≡ (I + δM) : H → D(M), where δ >0. It is known that if M is a maximal monotone operator, then the resolvent J is firmly M –1 nonexpansive, and F(J )= M 0 ≡{x ∈ H :0 ∈ Mx} for every δ >0. H H 1 2 Let H and H be two real Hilbert spaces. Let M : H → 2 , O : H → 2 ,and P : 1 2 i 1 i 2 i H → 2 be multivalued mappings. The split common null point problem (SCNPP) [18] is to find a point u ∈ H such that 0 ∈ M u (30) i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 14 of 21 ∗ ∗ and the points v = A u ∈ H satisfy j 2 0 ∈ O v , (31) j=1 where A : H → H (1 ≤ j ≤ q) are bounded linear operators. j 1 2 Now, we apply Theorem 3.4 to solve the problem of finding a point u ∈ H such that 0 ∈ M u (32) i=1 ∗ ∗ ∗ ∗ and the points v = A u ∈ H and s = A u ∈ H satisfy 1 2 2 2 ∞ ∞ ∗ ∗ 0 ∈ O v and 0 ∈ P s , (33) i i i=1 i=1 where A , A : H → H are bounded linear operators. 1 2 1 2 Since every firmly nonexpansive mapping is a 0-demicontractive mapping, we obtain the following theorem for problem (32)–(33). Theorem 4.1 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let f : H → H 1 1 1 2 be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {M : H → 2 : i ∈ N}, {O : 1 i 1 i H H 2 2 H → 2 : i ∈ N}, and {P : H → 2 : i ∈ N} be maximal monotone mappings. Sup- 2 i 2 ∞ ∞ ∞ ∗ –1 ∗ –1 ∗ –1 pose that  = {v ∈ M 0: A v ∈ O 0 and A v ∈ P 0}= ∅. For arbitrary 1 2 i=1 i i=1 i i=1 i x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n n O ∗ i u = x + α δ A (J – I)A x , ⎪ n n n,i n r 1 n i=1 1 1 n P ⎨ ∗ i v = u + β θ A (J – I)A u , n n n,i n 2 n i=1 2 2 (34) y = v + γ τ (J – I)v , ⎪ n n n,i n r n i=1 3 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where r , r , r >0 and {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satis- 1 2 3 n n n n n,i n,i n,i fying the following conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . 4.2 The split variational inequality problem Let C and Q be nonempty closed convex subsets of two real Hilbert spaces H and H , 1 2 respectively. Let A : H → H be a bounded linear operator, g : H → H ,and h : H → H . 1 2 1 1 2 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 15 of 21 The split variational inequality problem (SVIP) is to find a point u ∈ C such that ∗ ∗ g u , x – u ≥ 0, ∀x ∈ C, (35) ∗ ∗ and the point v = Au ∈ Q satisfy ∗ ∗ h v , y – v ≥ 0, ∀y ∈ Q. (36) We denote the solution set of the SVIP by Ω = SVIP(C, Q, g, h, A). The set of all solu- tions of variational inequality problem (35)isdenoted by VIP(C, g), and it is known that VIP(C, g)= F(P (I – λg)) for all λ >0. Let A , A : H → H be two bounded linear operators, g : H → H ,and h , l : H → H . 1 2 1 2 i 1 1 i i 2 2 In this section, we apply Theorem 3.4 to solve the problem of finding a point u ∈ C i=1 such that ∗ ∗ g u , x – u ≥ 0, ∀x ∈ C , (37) i i i=1 ∞ ∞ ∗ ∗ ∗ ∗ and the point v = A u ∈ Q , s = A u ∈ K satisfy 1 i 2 i i=1 i=1 ∞ ∞ ∗ ∗ ∗ ∗ h v , y – v ≥ 0, ∀y ∈ Q,and l s , z – s ≥ 0, ∀z ∈ K , (38) i i i i i=1 i=1 where {C } is a family of nonempty closed convex subsets of a real Hilbert space H ,and i i∈N 1 {Q } and {K } are two families of nonempty closed convex subsets of a real Hilbert i i∈N i i∈N space H . We now prove a strong convergence theorem for problem (37)–(38). Theorem 4.2 Let {C } be the family of nonempty closed convex subsets of a real Hilbert i i∈N space H , let {Q } and {K } be two families of nonempty closed convex subsets of a real 1 i i∈N i i∈N Hilbert space H , and let A , A : H → H be two bounded linear operators with adjoint 2 1 2 1 2 ∗ ∗ operators A and A , respectively. Let f : H → H be a ρ-contraction mapping, and let 1 1 1 2 B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {g : H → H : i ∈ N}, {h : H → H ; i ∈ N}, and {l : H → H ; i ∈ N} be i 1 1 i 2 2 i 2 2 r -, r -, and r -inverse strongly monotone mappings, respectively. Let r = min{r , r , r } and 1 2 3 1 2 3 ∞ ∞ ∗ ∗ ∗ μ ∈ (0, 2r). Suppose that  = {v ∈ VIP(C , g ): A v ∈ VIP(Q , h ) and A v ∈ i i 1 i i 2 i=1 i=1 VIP(K , l )}= ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by i i 1 1 n n n n i=1 u = x + α δ A (P (I – μh )– I)A x , n n n,i n Q i 1 n i=1 1 i ⎨ n v = u + β θ A (P (I – μl )– I)A u , n n n,i n K i 2 n i=1 2 i (39) y = v + γ τ (P (I – μg )– I)v , n n n,i n C i n i=1 i x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satisfying the following n n n n n,i n,i n,i conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 16 of 21 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Proof It is known that S := P (I – μh ), T =: P (I – μl ), and U := P (I – μg ) are nonex- i Q i i K i i C i i i i pensive mappings for all μ ∈ (0, 2r), and hence they are 0-demicontractive mappings. We obtain the desired result from Theorem 3.4. 4.3 The split equilibrium problem Let H and H be two real Hilbert spaces, and let C and Q be nonempty closed convex 1 2 subsets of H and H ,respectively. Let A : H → H be a bounded linear operator, and let 1 2 1 2 g : C × C → R and h : Q × Q → R be two bifunctions. The split equilibrium problem (SEP) is to find a point u ∈ C such that g u , x ≥ 0, ∀x ∈ C, (40) and Au ∈ Q satisfy h Au , y ≥ 0, ∀y ∈ Q. (41) The set of all solutions of equilibrium problem (40)isdenoted by EP(g). Lemma 4.3 ([19]) Let C be a nonempty closed convex subset of H, and let g be a bifunction of C ×Cinto R satisfying the following conditions: (A1) g(x, x)=0 for all x ∈ C; (A2) g is monotone, that is, g(x, y)+ g(y, x) ≤ 0 for all x, y ∈ C; (A3) for all x, y, z ∈ C, lim sup g tz +(1– t)x, y ≤ g(x, y); t↓0 (A4) g(x, ·) is convex and lower semicontinuous for all x ∈ C. Let g : C × C → R be a bifunction satisfying conditions (A1)–(A4), and let r >0 and x ∈ H. Then there exists z ∈ Csuch that g(z, y)+ y – z, z – x≥ 0 for all y ∈ C. Lemma 4.4 ([20]) Let C be a nonempty closed convex subset of H, and let g be a bifunction of C × Cinto R satisfying conditions (A1)–(A4). For r >0 and x ∈ H, define the mapping T : H → Cof g by T x = z ∈ C : g(z, y)+ y – z, z – x≥ 0, ∀y ∈ C , ∀x ∈ H. Then the following hold: (i) T is single-valued; (ii) T is firmly nonexpansive; r Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 17 of 21 (iii) F(T )= EP(g); (iv) EP(g) is closed and convex. Let A , A : H → H be two bounded linear operators, and let g : C × C → R and 1 2 1 2 i i i h , l : Q × Q → R be bifunctions for all i ∈ N. In this section, we apply Theorem 3.4 to i i i i solve the problem of finding a point ∞ ∞ ∞ ∗ ∗ ∗ u ∈ EP(g)suchthat A v ∈ EP(h)and A v ∈ EP(l ). (42) i 1 i 2 i i=1 i=1 i=1 h l g i i i By Lemma 4.4(iii) we have that T , T ,and T are firmly nonexpansive mappings, and r r r 1 2 3 hence they are 0-demicontractive mappings. We obtain the following result from Theo- rem 3.4. Theorem 4.5 Let {C } be a family of nonempty closed convex subsets of a real Hilbert i i∈N space H , let {Q } and {K } be two families of nonempty closed convex subsets of a real 1 i i∈N i i∈N Hilbert space H , and let A , A : H → H be two bounded linear operators with adjoint 2 1 2 1 2 ∗ ∗ operators A and A , respectively. Let f : H → H be a ρ-contraction mapping, and let B 1 1 1 2 be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let g : C × C → R and h , l : Q × Q → R be bifunctions satisfying conditions i i i i i i i ∞ ∞ ∗ ∗ ∗ (A1)–(A4) for all i ∈ N. Suppose that  = {v ∈ EP(g ): A v ∈ EP(h ) and A v ∈ i 1 i 2 i=1 i=1 EP(l )}= ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by i 1 1 n n n n i=1 ∗ i u = x + α δ A (T – I)A x , n n n,i n r 1 n i=1 1 1 ⎨ n ∗ i v = u + β θ A (T – I)A u , n n n,i n r 2 n i=1 2 2 (43) y = v + γ τ (T – I)v , ⎪ n n n,i n r n i=1 3 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where r , r , r >0 and {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satis- 1 2 3 n n n n n,i n,i n,i fying the following conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . 5 Numerical example for the main result We now give a numerical example of the studied method. Let H = H =(R , · ). Define 1 2 2 2 2 2 2 2 2 the mappings S : R → R , U : R → R ,and T : R → R by i i i –3i –2i S (x , x )= (x , x ), U (x , x )= x , x , i ∈ N, i 1 2 1 2 i 1 2 1 2 i +1 i +1 and x 1 (x , sin )if x =0, 1 2 3i x T (x , x )= i ∈ N i 1 2 (x ,0) if x =0, 1 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 18 of 21 for all x , x ∈ R.Then S are -demicontractive mappings for all i ∈ N and F(S )= 1 2 i i i=1 {(0, 0)}, U are -demicontractive mappings for all i ∈ N and F(U )=0 × R,and T i i i 4 i=1 are 0-demicontractive mappings for all i ∈ N and F(T )= R × 0. Next, we define the i=1 2 2 2 2 2 2 2 2 mappings f : R → R , B : R → R , A : R → R ,and A : R → R by 1 2 x x x 1 2 2 f (x , x )= , , B(x , x )= x , , A (x , x )=(x ,2x ), 1 2 1 2 1 1 1 2 1 1 8 8 2 and A (x , x )=(x – x ,2x ) 2 1 2 2 1 1 for all x , x ∈ R.Then f is a -contraction, B is a self-adjoint strongly positive bounded 1 2 linear operator with coefficient ξ = ,and A , A are bounded linear operators. Define the 1 2 real sequence {α }, {β },and {γ } as follows: n,i n,i n,i 1if n = i =1, 1 n ( )if n > i, n+1 α = n,i n–1 1 n 1– ( )if n = i >1, ⎪ i i=1 n+1 0otherwise, 1if n = i =1, ⎨ 1 n ( )if n > i, 3 n+1 β = n,i n–1 1 n 1– ( )if n = i >1, i=1 i n+1 ⎪ 3 0otherwise, and 1if n = i =1, 1 n ( )if n > i, i+1 2n+1 γ = n,i n–1 1 n 1– ( )if n = i >1, ⎪ i+1 i=1 4 2n+1 0otherwise, that is, ⎛ ⎞ 100 0 0 0 0 0 ... ⎜ ⎟ 1/3 2/3 0 0 0 0 0 0 ... ⎜ ⎟ ⎜ ⎟ 3/8 3/16 7/16 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 2/5 1/5 1/10 3/10 0 0 0 0 . . . ⎜ ⎟ α = ⎜ ⎟ , n,i 5/12 5/24 5/48 5/96 7/32 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/7 3/14 3/28 3/56 3/112 19/112 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/16 7/32 7/64 7/128 7/256 7/512 71/512 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 19 of 21 Figure 1 Graph for errors ⎛ ⎞ 1 0 0 0 0 0 0 0 ... ⎜ ⎟ 2/9 7/9 0 0 0 0 0 0 ... ⎜ ⎟ ⎜ ⎟ 1/4 1/12 2/3 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 4/15 4/45 4/135 83/135 0 0 0 0 . . . ⎜ ⎟ β = , ⎜ ⎟ n,i 5/18 5/54 5/162 5/486 143/243 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 2/7 2/21 2/63 2/189 1/284 325/567 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/24 7/72 7/216 7/648 1/278 1/833 58/103 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . and ⎛ ⎞ 1 0 00 00 0 0... ⎜ ⎟ 1/40 39/40 0 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/112 3/448 433/448 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 1/36 1/144 1/576 185/192 0 0 0 0 . . . ⎜ ⎟ γ = ⎜ ⎟ . n,i 5/176 5/704 1/563 1/2253 51/53 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/104 3/416 1/555 1/2219 1/8875 976/1015 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/240 7/960 1/549 1/2194 1/8777 1/35,109 618/643 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 We see that lim α = , lim β = ,and lim γ = for i ∈ N.Now,we n→∞ n,i n→∞ n,i n→∞ n,i i i 2i+3 2 3 2 start with the initial point x = (1, 1) and let {x } be the sequence generated by (12). Sup- 1 n pose that x is of the form x =(a , b ). where a , b ∈ R. The criterion for stopping our n n n n n n –6 n n n testingmethodistaken as x – x  <10 .Choose δ = , θ = , τ = ,and n–1 n 2 n n n 11n–1 30n–1 2n–1 σ = for all n ∈ N.Figure 1 shows the errors x – x  of our proposed method. n n–1 n 2 0.01 The values of x and x – x  are shown in Table 1. n n–1 n 2 We observe from Table 1 that x → (0, 0) ∈ . We also note that the error is bounded –6 by x – x  <10 ,and we canuse x = (0.00000003, 0.00000117) to approximate the 30 31 2 31 solution of (7)withaccuracyatleast 6D.P. 6Conclusion We introduce a new algorithm for solving the split common fixed point problem (7)of the infinite families of demicontractive mappings in Hilbert spaces. Strong convergence of the proposed algorithm is obtained under some suitable control conditions. The main Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 20 of 21 Table 1 Numerical experiment for x na b x – x n n n–1 n 2 1 1.00000000 1.00000000 – 2 0.12500000 0.62500000 0.95197164 3 0.01751567 0.39224395 0.25637524 4 0.00414010 0.24675959 0.14609793 5 0.00202951 0.15549870 0.09128529 6 0.00140947 0.09811767 0.05738438 7 0.00107109 0.06197693 0.03614232 8 0.00063002 0.03918347 0.02279773 9 0.00047832 0.02479206 0.01439221 10 0.00030270 0.01569709 0.00909667 11 0.00022553 0.00994467 0.00575293 12 0.00014616 0.00630378 0.00364176 13 0.00008740 0.00399788 0.00230665 14 0.00005861 0.00253664 0.00146152 . . . . . . . . . . . . 28 0.00000009 0.00000450 0.00000257 29 0.00000007 0.00000287 0.00000163 30 0.00000007 0.00000183 0.00000104 31 0.00000003 0.00000117 0.00000066 results of this paper can be considered as an extension of work by Eslamian [12] by provid- ing an algorithm for finding a solution of problem (7), which is a generalization of prob- lem (5). Acknowledgements The authors would like to thank Chiang Mai University and Center of Excellence in Mathematics, CHE, Bangkok 10400, Thailand, for the financial support. Abbreviations SFP, The split feasibility problem; SCFP, The split common fixed point problem; MSSFP, The multiple set split feasibility problem; SCNPP, The split common null point problem; SVIP, The split variational inequality problem; SEP, The split equilibrium problem. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript. Author details 1 2 Ph.D. Degree Program in Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. Center of Excellence in Mathematics, CHE, Bangkok, Thailand. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 5 September 2017 Accepted: 7 January 2018 References 1. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) 2. Byrne, C.: Iterative oblique projection onto convex subsets ant the split feasibility problem. Inverse Probl. 18, 441–453 (2002) 3. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) 4. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple set split feasibility problem and its applications. Inverse Probl. 21, 2071–2084 (2005) Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 21 of 21 5. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009) 6. Moudafi, A.: A note on the split common fixed point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083–4087 (2011) 7. Qin, L.J., Wang, L., Chang, S.S.: Multiple-set split feasibility problem for a finite family of asymptotically quasi-nonexpansive mappings. Panam. Math. J. 22(1), 37–45 (2012) 8. Wang, F., Xu, H.K.: Approximation curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. (2010). https://doi.org/10.1155/2010/102085 9. Xu, H.K.: A variable Krasnosekel’skii–Mann algorithm and the multiple-sets split feasibility problem. Inverse Probl. 22, 2021–2034 (2006) 10. Yang, Q.: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004) 11. Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 587–600 (2010) 12. Eslamian, M., Eslamian, P.: Strong convergence of a split common fixed point problem. Numer. Funct. Anal. Optim. 37, 1248–1266 (2016) 13. Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) 14. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009) 15. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) 16. Mainge, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007) 17. Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65(2), 443–465 (2016) 18. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012) 19. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problem. Math. Stud. 63, 123–145 (1994) 20. Combettes, P.L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fixed Point Theory and Applications Springer Journals

The split common fixed point problem for infinite families of demicontractive mappings

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Abstract

Center of Excellence in In this paper, we propose a new algorithm for solving the split common fixed point Mathematics and Applied Mathematics, Department of problem for infinite families of demicontractive mappings. Strong convergence of the Mathematics, Faculty of Science, proposed method is established under suitable control conditions. We apply our Chiang Mai University, Chiang Mai, main results to study the split common null point problem, the split variational Thailand Center of Excellence in inequality problem, and the split equilibrium problem in the framework of a real Mathematics, CHE, Bangkok, Hilbert space. A numerical example supporting our main result is also given. Thailand Full list of author information is MSC: 47H09; 47J05; 47J25; 47N10 available at the end of the article Keywords: Fixed point problem; Demicontractive mappings; Null point problem; Variational inequality problem; Equilibrium problem 1 Introduction Let H be a real Hilbert space with inner product ·, · and norm ·.Let I denote the identity mapping. Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H ,respectively.Let A : H → H be a bounded linear operator with adjoint oper- 1 2 1 2 ator A . The split feasibility problem (SFP), which was first introduced by Censor and Elfving [1], is to find ∗ ∗ v ∈ C such that Av ∈ Q.(1) Let P and P be the orthogonal projections onto the sets C and Q,respectively. Assume C Q that (1) has a solution. It known that v ∈ H solves (1) if and only if it solves the fixed point equation ∗ ∗ ∗ v = P I + γ A (P – I)A v , C Q where γ > 0 is any positive constant. SFP has been used to model significant real-world inverse problems in sensor networks, radiation therapy treatment planning, antenna design, immaterial science, computerized tomography, etc. (see [2–4]). © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 2 of 21 The split common fixed point problem (SCFP) for mappings T and S,which wasfirst introduced by Censor and Segal [5], is to find ∗ ∗ v ∈ F(T)suchthat Av ∈ F(S), (2) where T : H → H and S : H → H are two mappings satisfying F(T)= {x ∈ H : 1 1 2 2 1 Tx = x}= ∅ and F(S)= {x ∈ H : Sx = x}= ∅, respectively. Since each closed and convex subset may be considered as a fixed point set of a projection onto the subset, the SCFP is a generalization of the SFP. Recently, the SFP and SCFP have been studied by many authors; see, for example, [6–11]. In 2010, Moudafi [11] introduced the following algorithm for solving (2)for twodemi- contractive mappings: x ∈ H choose arbitrarily, 1 1 (3) u = x + γαA (S – I)Ax , n n n x =(1 – β )u + β Tu , n ∈ N. n+1 n n n n He proved that {x } converges weakly to some solution of SCFP. The multiple set split feasibility problem (MSSFP), which was first introduced by Censor et al. [4], is to find m r ∗ ∗ v ∈ C such that Av ∈ Q,(4) i i i=1 i=1 m r where {C } and {Q } are families of nonempty closed convex subsets of real Hilbert i i i=1 i=1 spaces H and H ,respectively. We seethatif m = r =1, then problem (4)reduces to prob- 1 2 lem (1). Recently, Eslamian [12] considered the problem of finding a point m m m ∗ ∗ ∗ v ∈ F(U)suchthat A v ∈ F(S)and A v ∈ F(T ), (5) i 1 i 2 i i=1 i=1 i=1 where A , A : H → H are bounded linear operators, and U : H → H , T : H → H 1 2 1 2 i 1 1 i 2 2 and S : H → H , i = 1,2,..., m.Healsopresented anew algorithmtosolve (5)for finite i 2 2 families of quasi-nonexpansive mappings: x ∈ H choose arbitrarily, 1 1 1 ∗ u = x + ηβA (S – I)A x , n n i 1 n ⎨ 1 i=1 m y = u + η β A (T – I)A u , (6) n n i 2 n i=1 m 2 ⎪ m z = α y + α U y , n n,0 n n,i i n i=1 x = θ γ f (x )+(I – θ B)z , n ∈ N. n+1 n n n n He proved that {x } converges strongly to some solution of (5)under some controlcon- ditions. Question. Can we modify algorithm (6) to a simple one for solving the problem of finding ∞ ∞ ∞ ∗ ∗ ∗ v ∈ F(U)suchthat A v ∈ F(S)and A v ∈ F(T ), (7) i 1 i 2 i i=1 i=1 i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 3 of 21 where A , A : H → H are bounded linear operators, and {U : H → H : i ∈ N}, 1 2 1 2 i 1 1 {T : H → H : i ∈ N} and {S : H → H : i ∈ N} are infinite families of k -, k -, and k - i 2 2 i 2 2 3 2 1 demicontractive mappings, respectively. In this work, we introduce a new algorithm for solving problem (7) for infinite families of demicontractive mappings and prove its strong convergence to a solution of problem (7). 2Preliminaries Throughout this paper, we adopt the following notations. (i) “→”and “” denote the strong and weak convergence, respectively. (ii) ω (x ) denotes the set of the cluster points of {x } in the weak topology, that is, ω n n ∃{x } of {x } such that x  x. n n n i i (iii) is the solution set of problem (7), that is, ∞ ∞ ∞ ∗ ∗ ∗ = v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T ) . i 1 i 2 i i=1 i=1 i=1 A mapping P is said to be a metric projection of H onto C if for every x ∈ H,there exists auniquenearest pointin C,denoted by P x,suchthat x – P x≤x – z, ∀z ∈ C. It is known that P is a firmly nonexpansive mapping. Moreover, P is characterized by the C C following property: x – P x, y – P x≤ 0 for all x ∈ H, y ∈ C. A bounded linear operator C C B : H → H is said to be strongly positive if there is a constant ξ >0 such that Bx, x≥ ξ x for all x ∈ H. Definition 2.1 The mapping T : H → H is said to be (i) L-Lipschitzian if there exists L >0 such that Tu – Tv≤ Lu – v for all u, v ∈ H; (ii) α-contraction if T is α-Lipschitzian with α ∈ [0, 1),thatis, Tu – Tv≤ αu – v for all u, v ∈ H; (iii) nonexpansive if T is 1-Lipschitzian; (iv) quasi-nonexpansive if F(T) = ∅ and Tu – v≤u – v for all u ∈ H, v ∈ F(T); (v) firmly nonexpansive if 2 2 Tu – Tv ≤u – v – (u – v)–(Tu – Tv) for all u, v ∈ H; or equivalently, for all u, v ∈ H, Tu – Tv ≤Tu – Tv, u – v; Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 4 of 21 (vi) λ-inverse strongly monotone if there exists λ >0 such that u – v, Tu – Tv≥ λTu – Tv for all u, v ∈ H; (vii) k-demicontractive if F(T) = ∅ and there exists k ∈ [0, 1) such that 2 2 2 Tu – v ≤u – v + ku – Tu for all u ∈ H, v ∈ F(T). The following example is an infinite family of k-demicontractive mappings in R . 2 2 Example 2.2 For i ∈ N,let U : R → R be defined for all x , x ∈ R by i 1 2 –2i U (x , x )= x , x , i 1 2 1 2 i +1 and · is the Euclidean norm on R .Observe that F(U )=0 × R for all i ∈ N,thatis, if x =(x , x ) ∈ R × R and p =(0, p ) ∈ F(U ), then 1 2 2 i –2i U x – p = x , x –(0, p ) i 1 2 2 i +1 –2i 2 2 = |x | + |x – p | 1 2 2 i +1 2 2 ≤ 4|x | + |x – p | 1 2 2 2 2 2 2 = |x | + (1 + 1) |x | + |x – p | 1 1 2 2 3 2i 2 2 ≤x – p + 1+ |x | 4 i +1 2 2 = x – p + U x – x . So, U are -demicontractive mappings for all i ∈ N. Definition 2.3 The mapping T : H → H is said to be demiclosed at zero if for any sequence {u }⊂ H with u  u and Tu → 0, we have Tu =0. n n n Lemma 2.4 ([13]) Assume that B is a self-adjoint strongly positive bounded linear operator –1 on a Hilbert space H with coefficient ξ >0 and 0< μ ≤B . Then I – μB≤ 1– ξμ. Lemma 2.5 ([14]) Let H be a real Hilbert space. Then the following results hold: 2 2 2 (i) u + v = u +2u, v + v ∀u, v ∈ H; 2 2 (ii) u + v ≤u +2v, u + v∀u, v ∈ H. Lemma 2.6 ([15]) Let {a } be a sequence of nonnegative real numbers satisfying the fol- lowing relation: a ≤ (1 – γ )a + δ , n ∈ N, n+1 n n n where Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 5 of 21 (i) {γ }⊂ (0, 1), γ = ∞; n n n=1 δ ∞ (ii) lim sup ≤ 0 or |δ | < ∞. n→∞ γ n=1 Then lim a =0. n→∞ n Lemma 2.7 ([16]) Let {κ } be a sequence of real numbers that does not decrease at infinity, that is, there exists at a subsequence {κ } of {κ } that satisfies κ < κ for all i ∈ N. For n n n n +1 i i i every n ≥ n , define the integer sequence {τ(n)} as follows: τ(n)= max{l ∈ N : l ≤ n, κ < κ }, l l+1 where n ∈ N is such that {l ≤ n : κ < κ }= ∅. Then: o o l l+1 (i) τ(n ) ≤ τ(n +1) ≤ ··· , and τ(n) →∞; o o (ii) for all n ≥ n , max{κ , κ }≤ κ . o n τ(n) τ(n)+1 3 Results and discussion In this section, we propose a new algorithm, which is a modification of (6)and proveits strong convergence under some suitable conditions. We start with the following important lemma. Lemma 3.1 For two real Hilbert spaces H and H , let A : H → H be a bounded linear 1 2 1 2 operator with adjoint operator A . If T : H → H is a k-demicontractive mapping, then 2 2 2 2 2 ∗ ∗ ∗ 2 x + δA (T – I)Ax – x ≤ x – x – δ 1– k – δA (T – I)Ax ∗ ∗ for all x ∈ H such that Ax ∈ F(T). Proof Suppose that T : H → H is a k-demicontractive mapping and let x ∈ H be such 2 2 1 that Ax ∈ F(T). Then we have 2 2 ∗ ∗ ∗ ∗ ∗ x – x + δA (T – I)Ax ≤ x – x +2δ x – x , A (T – I)Ax 2 2 + δ A (T – I)Ax.(8) Since A is a bounded linear operator with adjoint operator A and T is a k-demicontractive mapping, by Lemma 2.5(ii) we deduce that ∗ ∗ ∗ x – x , A (T – I)Ax = Ax – Ax ,(T – I)Ax = TAx – Ax , TAx – Ax – (T – I)Ax 2 2 ∗ 2 ∗ = TAx – Ax + TAx – Ax – Ax – Ax – (T – I)Ax ∗ 2 ≤ Ax – Ax + kTAx – Ax 2 2 2 ∗ + TAx – Ax – Ax – Ax – (T – I)Ax k –1 = (T – I)Ax.(9) 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 6 of 21 From (8)and (9)weget 2 2 2 ∗ ∗ ∗ 2 x + δA (T – I)Ax – x ≤ x – x – δ 1– k – δA (T – I)Ax . This completes the proof. Lemma 3.2 For two real Hilbert spaces H and H , let A : H → H be a bounded linear 1 2 1 2 operator with adjoint operator A , and let {T : H → H : i ∈ N} be an infinite family of i 2 2 k-demicontractive mappings. Let {x } be sequence in H , and let n 1 u = x + α δ A (T – I)Ax , ∀n ∈ N, (10) n n n,i n i n i=1 where {α } is a real sequence in [0, 1] satisfying α =1. Then we have n,i n,i i=1 2 2 2 ∗ ∗ 2 u – x ≤ x – x – α δ 1– k – δ A (T – I)Ax n n n,i n n i n i=1 ∗ ∗ for all x ∈ H such that Ax ∈ F(T ). 1 i i=1 ∗ ∗ Proof Let x ∈ H be such that Ax ∈ F(T ). From (10) and Lemma 3.1 we obtain 1 i i=1 2 2 ∗ ∗ ∗ u – x ≤ α x – x + δ A (T – I)Ax n n,i n n i n i=1 2 2 ∗ 2 ≤ α x – x – δ 1– k – δ A (T – I)Ax n,i n n n i n i=1 2 2 ∗ 2 = x – x – α δ 1– k – δ A (T – I)Ax . n n,i n n i n i=1 This completes the proof. Lemma 3.3 Let {T : H → H : i ∈ N} be an infinite family of k-demicontractive mappings i 1 1 from a Hilbert space H to itself. Let {x } be sequence in H , and let 1 n 1 u = x + α δ (T – I)x , ∀n ∈ N, (11) n n n,i n i n i=1 where {α } is a real sequence in [0, 1] satisfying α =1. Then we have n,i n,i i=1 2 2 2 ∗ ∗ u – x ≤ x – x – α δ (1 – k – δ ) (T – I)x n n n,i n n i n i=1 for all x ∈ F(T ). i=1 Proof The statement directly follows from Lemma 3.2 by putting H = H and A = I. 1 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 7 of 21 Now, we introduce a new algorithm for solving problem (7) for an infinite family of demicontractive mappings and then prove its strong convergence. Theorem 3.4 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let f : H → H 1 1 1 2 be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear op- erator on H with coefficient ξ >2ρ and B =1. Let {S : H → H : i ∈ N}, {T : H → H : 1 i 2 2 i 2 2 i ∈ N}, and {U : H → H : i ∈ N} be infinite families of k -, k -, and k -demicontractive i 1 1 1 2 3 mappings such that S –I, T –I, and U –I are demiclosed at zero, respectively. Suppose that i i i ∞ ∞ ∞ ∗ ∗ ∗ = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= ∅. For arbitrary x ∈ H , i 1 i 2 i 1 1 i=1 i=1 i=1 let {u }, {v }, {y }, and {x } be generated by n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (12) y = v + γ τ (U – I)v , ⎪ n n n,i n i n i=1 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 1–k (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Proof For any u, v ∈ H , by Lemma 2.4 we have P (f + I – B)u – P (f + I – B)v ≤ (f + I – B)u –(f + I – B)v ≤ f (u)– f (v) + I – Bu – v ≤ ρu – v +(1– ξ)u – v ≤ (1 – ρ)u – v, that is, the mapping P (f + I – B) is a contraction. So, by the Banach contraction principle ∗ ∗ ∗ there is a unique element x ∈ H such that x = P (f + I – B)x . ∞ ∞ ∗ ∗ ∗ ∗ ∗ Let x = P (f + I – B)x ,thatis, x ∈ F(U )is such that A x ∈ F(S )and A x ∈ i 1 i 2 i=1 i=1 F(T ). From Lemmas 3.2 and 3.3 and from (12)weobtain i=1 2 2 2 ∗ ∗ 2 u – x ≤ x – x – α δ 1– k – δ A  (S – I)A x , (13) n n n,i n 1 n 1 i 1 n i=1 2 2 2 ∗ ∗ 2 v – x ≤ u – x – β θ 1– k – θ A  (T – I)A u , (14) n n n,i n 2 n 2 i 2 n i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 8 of 21 and 2 2 2 ∗ ∗ y – x ≤ v – x – λ τ (1 – k – τ ) (U – I)v . (15) n n n,i n 3 n i n i=1 Therefore 2 2 2 ∗ ∗ 2 y – x ≤ x – x – α δ 1– k – δ A  (S – I)A x n n n,i n 1 n 1 i 1 n i=1 – β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 – λ τ (1 – k – τ ) (U – I)v . (16) n,i n 3 n i n i=1 By conditions (C4), (C5),and (C6) we have ∗ ∗ y – x ≤ x – x . (17) n n –1 By condition (C3) we may assume that σ ∈ (0, B ) for all n ∈ N. By Lemma 2.4 we get I – σ B≤ 1– σ ξ.From(12)and (17)weget n n ∗ ∗ x – x = σ f (y )+(I – σ B)y – x n+1 n n n n ∗ ∗ = σ f (y )– Bx +(I – σ B) y – x n n n n ∗ ∗ ∗ ∗ ≤ σ f (y )– f x + f x – Bx + I – σ B y – x n n n n ∗ ∗ ∗ ∗ ≤ σ ρ y – x + σ f x – Bx +(1– σ ξ) y – x n n n n n ∗ ∗ f (x )– Bx ≤ 1– σ (ξ – ρ) x – x + σ (ξ – ρ) n n n ξ – ρ ∗ ∗ f (x )– Bx ≤ max x – x , ξ – ρ ∗ ∗ f (x )– Bx ≤ max x – x , . (18) ξ – ρ Therefore {x } is bounded, and we also have that {y } and {f (y )} are bounded. To this end, n n n we consider the following two cases. ∗ ∞ Case 1. Suppose that {x – x } is nonincreasing for some n ∈ N.Thenweget that n o n=n lim x – x  exists. By (16), (17), and Lemma 2.5(i) we get n→∞ n 2 2 ∗ ∗ ∗ x – x = σ f (y )– Bx +(I – σ B) y – x n+1 n n n n 2 2 ∗ ∗ ≤ σ f (y )– Bx +(1– σ ξ) y – x n n n n ∗ ∗ +2σ (1 – σ ξ) f (y )– Bx y – x n n n n ≤ σ M + x – x n n Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 9 of 21 – α δ 1– k – δ A  (S – I)A x n,i n 1 n 1 i 1 n i=1 – β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 – λ τ (1 – k – τ ) (U – I)v , n,i n 3 n i n i=1 where ∗ ∗ ∗ M = sup f (y )– Bx +2 f (y )– Bx x – x . n n n This implies, for j = 1,2,..., n, α δ 1– k – δ A  (S – I)A x n,j n 1 n 1 j 1 n ≤ α δ 1– k – δ A  (S – I)A x n,i n 1 n 1 i 1 n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x , (19) n n n+1 β θ 1– k – θ A  (T – I)A u n,j n 2 n 2 j 2 n ≤ β θ 1– k – θ A  (T – I)A u n,i n 2 n 2 i 2 n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x , (20) n n n+1 and 2 2 λ τ (1 – k – τ ) (U – I)v ≤ λ τ (1 – k – τ ) (U – I)v n,j n 3 n j n n,i n 3 n i n i=1 2 2 ∗ ∗ ≤ σ M + x – x – x – x . (21) n n n+1 From (19), (20), (21), and conditions (C2)–(C6) we obtain lim (S – I)A x = lim α (S – I)A x = 0, (22) j 1 n n,i i 1 n n→∞ n→∞ i=1 lim (T – I)A u = lim β (T – I)A u = 0, (23) j 2 n n,i i 2 n n→∞ n→∞ i=1 and lim (U – I)v = lim λ (U – I)v = 0. (24) j n n,i i n n→∞ n→∞ i=1 Next, we show that ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x ≤ 0, where x = P (f + I – B)x . n→∞ Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 10 of 21 To see this, choose a subsequence {x } of {x } such that n n ∗ ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x = lim f x – Bx , x – x . n n p→∞ n→∞ Since the sequence {x } is bounded, there exists a subsequence {x } of {x } such that n n n p p p x  z ∈ H . Without loss of generality, we may assume that x  z ∈ H .Since A is a n 1 n 1 1 p p bounded linear operator, this yields that A x  A z. By the demiclosedness principle of 1 n 1 S – I at zero and (22)weget A z ∈ F(S ). By (12)and (22)wehave i 1 i i=1 2 ∗ u – x  = x + α δ A (S – I)A x – x n n n n,i n i 1 n n i=1 ≤ α δ A  (S – I)A x →0as n →∞. n,i n 1 i 1 n i=1 Similarly, we also have v – u → 0as n →∞.Using thefactthat x  z and n n n u – x → 0, we conclude that u  z.Since A is a bounded linear operator, we n n n 2 get that A u  A z. By the demiclosedness principle of T – I at zero and (23)weget 2 n 2 i A z ∈ F(T ). Again, since u  z and v –u → 0, we conclude that v  z.Bythe 2 i n n n n i=1 p p demiclosedness principle of U – I at zero and (24)wealsohave z ∈ F(U ). Therefore i i i=1 z ∈ . ∗ ∗ Since x = P (f + I – B)x and z ∈ ,weget ∗ ∗ ∗ ∗ ∗ ∗ lim sup f x – Bx , x – x = lim f x – Bx , x – x n n p→∞ n→∞ ∗ ∗ ∗ = f x – Bx , z – x ≤ 0. (25) Using Lemma 2.5 and (17), we have 2 2 ∗ ∗ ∗ x – x = σ f (y )– Bx +(I – σ B) y – x n+1 n n n n ∗ ∗ ∗ ≤ (1 – σ ξ) y – x +2σ f (y )– Bx , x – x n n n n n+1 ∗ ∗ ∗ ≤ (1 – σ ξ) x – x +2ρσ y – x x – x n n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x n n+1 2 2 2 ∗ ∗ ∗ ≤ (1 – σ ξ) x – x + ρσ x – x + x – x n n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x n n+1 2 2 ∗ ∗ = 1– σ (ξ – ρ) x – x + ρσ x – x n n n n+1 ∗ ∗ ∗ +2σ f x – Bx , x – x . n n+1 This implies that σ (ξ – ρ) 2σ 2 2 n n ∗ ∗ ∗ ∗ ∗ x – x ≤ 1– x – x + f x – Bx , x – x . (26) n+1 n n+1 1– σ ρ 1– σ ρ n n By (25), (26), and Lemma 2.6 we conclude that x → x as n →∞. n Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 11 of 21 Case 2. Suppose that there exists an integer m such that ∗ ∗ x – x ≤ x – x . m m +1 o o Put κ = x – x  for all n ≥ m .Thenwehave κ ≤ κ .Let {μ(n)} be the sequence n n o m m +1 o o defined by μ(n)= max{l ∈ N : l ≤ n, κ ≤ κ } l l+1 for all n ≥ m . By Lemma 2.7 we obtain that {μ(n)} is a nondecreasing sequence such that lim μ(n)= ∞ and κ ≤ κ for all n ≥ m . μ(n) μ(n)+1 o n→∞ By thesameargumentasincase1we obtain lim (S – I)A x =0, lim (T – I)A u =0, i 1 μ(n) i 2 μ(n) n→∞ n→∞ and lim (U – I)v =0. i μ(n) n→∞ By the demiclosedness principle of S – I, T – I,and U – I at zero, we have ω (x ) ⊂ . i i i ω μ(n) This implies that ∗ ∗ ∗ lim sup f x – Bx , x – x ≤ 0. μ(n) n→∞ By a similar argument from (26)wealsohave σ (ξ – ρ) 2σ μ(n) μ(n) 2 2 ∗ ∗ ∗ κ ≤ 1– κ + f x – Bx , x – x . μ(n)+1 μ(n)+1 μ(n) 1– σ ρ 1– σ ρ μ(n) μ(n) So, we get lim κ =0 and also have lim κ = 0. By Lemma 2.7 we have n→∞ μ(n) n→∞ μ(n)+1 0 ≤ κ ≤ max{κ , κ }≤ κ . n n μ(n) μ(n)+1 Therefore x → x as n →∞. This completes the proof. By setting T = I for all i ∈ N in Theorem 3.4 we obtain the following result. Corollary 3.5 Let H and H be two real Hilbert spaces, let A : H → H be a bounded 1 2 1 1 2 linear operator with adjoint operator A . Let f : H → H be a ρ-contraction mapping, 1 1 and let B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {S : H → H : i ∈ N} and {U : H → H : i ∈ N} be infinite fami- i 2 2 i 1 1 lies of k -and k -demicontractive mappings such that S – Iand U – Iare demiclosed at 1 3 i i ∞ ∞ ∗ ∗ zero, respectively. Suppose that  = {v ∈ F(U ): A v ∈ F(S )}= ∅. For arbitrary i 1 i i=1 i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 12 of 21 x ∈ H , let {u }, {y }, and {x } be generated by 1 1 n n n ⎪ u = x + α δ A (S – I)A x , n n n,i n i 1 n ⎨ i=1 1 y = u + γ τ (U – I)u , (27) n n n,i n i n i=1 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {τ }, {σ }, {α }, and {γ } are sequences in [0, 1] satisfying the following condi- n n n n,i n,i tions: n n (C1) α = γ =1 for all n ∈ N; n,i n,i i=1 i=1 (C2) lim inf α >0 and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Remark 3.6 By the same setting as in Corollary 3.5,Eslamian[17] used another algorithm for solving the same problem as in Corollary 3.5;see [17], Theorem 3.3. Note that each step of our algorithm is much easier for computation than that of Eslamian [17]because our algorithm concerns only the finite sum. By setting f (y)= v for all y ∈ H and B = I in Theorem 3.4 we obtain the following result. Corollary 3.7 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let {S : H → H : i 2 2 1 2 i ∈ N}, {T : H → H : i ∈ N}, and {U : H → H : i ∈ N} be infinite families of k -, k -, and i 2 2 i 1 1 1 2 k -demicontractive mappings such that S – I, T – I, and U – I are demiclosed at zero, re- 3 i i i ∞ ∞ ∞ ∗ ∗ ∗ spectively. Suppose that = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= i 1 i 2 i i=1 i=1 i=1 ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (28) y = v + γ τ (U – I)v , ⎪ n n n,i n i n i=1 x = σ v +(1– σ )y , n ∈ N, n+1 n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 1–k (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 1–k (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1 – k . 1 n 2 3 Then the sequence {x } converges strongly to x = P (v). It is known that every quasi-nonexpansive mapping is 0-demicontractive mapping, so the following result is directly obtained by Theorem 3.2. Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 13 of 21 Corollary 3.8 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let {S : H → i 2 1 2 H : i ∈ N}, {T : H → H : i ∈ N}, and {U : H → H : i ∈ N} be infinite families of quasi- 2 i 2 2 i 1 1 nonexpansive mappings such that S – I, T – I, and U – I are demiclosed at zero, respec- i i i ∞ ∞ ∞ ∗ ∗ ∗ tively. Suppose that = {v ∈ F(U ): A v ∈ F(S ) and A v ∈ F(T )}= ∅. i 1 i 2 i i=1 i=1 i=1 For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n u = x + α δ A (S – I)A x , n n n,i n i 1 n i=1 1 ⎨ n v = u + β θ A (T – I)A u , n n n,i n i 2 n i=1 2 (29) y = v + γ τ (U – I)v , n n n,i n i n ⎪ i=1 x = σ v +(1– σ )y , n ∈ N, n+1 n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, and {γ } are sequences in [0, 1] satisfying the fol- n n n n n,i n,i n,i lowing conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 Then the sequence {x } converges strongly to x = P (v). 4 Applications 4.1 The split common null point problem Let M be the set-valued mapping of H into 2 .The effective domain of M is denoted by D(M), that is, D(M)= {x ∈ H : Mx = ∅}. The mapping M is said to be monotone if x – y, u – v≥ 0, ∀x, y ∈ D(M), u ∈ Mx, v ∈ My. A monotone mapping M is said to be maximal if the graph G(M) is not properly contained in the graph of any other monotone map, where G(M)= {(x, y) ∈ H × H : y ∈ Mx}.Itis known that M is maximal if and only if for (x, u) ∈ H × H, x – y, u – v≥ 0for every (y, v) ∈ G(M)implies u ∈ Mx. For the maximal monotone operator M,wecan associateits resolvent J defined by M –1 J ≡ (I + δM) : H → D(M), where δ >0. It is known that if M is a maximal monotone operator, then the resolvent J is firmly M –1 nonexpansive, and F(J )= M 0 ≡{x ∈ H :0 ∈ Mx} for every δ >0. H H 1 2 Let H and H be two real Hilbert spaces. Let M : H → 2 , O : H → 2 ,and P : 1 2 i 1 i 2 i H → 2 be multivalued mappings. The split common null point problem (SCNPP) [18] is to find a point u ∈ H such that 0 ∈ M u (30) i=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 14 of 21 ∗ ∗ and the points v = A u ∈ H satisfy j 2 0 ∈ O v , (31) j=1 where A : H → H (1 ≤ j ≤ q) are bounded linear operators. j 1 2 Now, we apply Theorem 3.4 to solve the problem of finding a point u ∈ H such that 0 ∈ M u (32) i=1 ∗ ∗ ∗ ∗ and the points v = A u ∈ H and s = A u ∈ H satisfy 1 2 2 2 ∞ ∞ ∗ ∗ 0 ∈ O v and 0 ∈ P s , (33) i i i=1 i=1 where A , A : H → H are bounded linear operators. 1 2 1 2 Since every firmly nonexpansive mapping is a 0-demicontractive mapping, we obtain the following theorem for problem (32)–(33). Theorem 4.1 Let H and H be two real Hilbert spaces, and let A , A : H → H be two 1 2 1 2 1 2 ∗ ∗ bounded linear operators with adjoint operators A and A , respectively. Let f : H → H 1 1 1 2 be a ρ-contraction mapping, and let B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {M : H → 2 : i ∈ N}, {O : 1 i 1 i H H 2 2 H → 2 : i ∈ N}, and {P : H → 2 : i ∈ N} be maximal monotone mappings. Sup- 2 i 2 ∞ ∞ ∞ ∗ –1 ∗ –1 ∗ –1 pose that  = {v ∈ M 0: A v ∈ O 0 and A v ∈ P 0}= ∅. For arbitrary 1 2 i=1 i i=1 i i=1 i x ∈ H , let {u }, {v }, {y }, and {x } be generated by 1 1 n n n n n O ∗ i u = x + α δ A (J – I)A x , ⎪ n n n,i n r 1 n i=1 1 1 n P ⎨ ∗ i v = u + β θ A (J – I)A u , n n n,i n 2 n i=1 2 2 (34) y = v + γ τ (J – I)v , ⎪ n n n,i n r n i=1 3 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where r , r , r >0 and {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satis- 1 2 3 n n n n n,i n,i n,i fying the following conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . 4.2 The split variational inequality problem Let C and Q be nonempty closed convex subsets of two real Hilbert spaces H and H , 1 2 respectively. Let A : H → H be a bounded linear operator, g : H → H ,and h : H → H . 1 2 1 1 2 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 15 of 21 The split variational inequality problem (SVIP) is to find a point u ∈ C such that ∗ ∗ g u , x – u ≥ 0, ∀x ∈ C, (35) ∗ ∗ and the point v = Au ∈ Q satisfy ∗ ∗ h v , y – v ≥ 0, ∀y ∈ Q. (36) We denote the solution set of the SVIP by Ω = SVIP(C, Q, g, h, A). The set of all solu- tions of variational inequality problem (35)isdenoted by VIP(C, g), and it is known that VIP(C, g)= F(P (I – λg)) for all λ >0. Let A , A : H → H be two bounded linear operators, g : H → H ,and h , l : H → H . 1 2 1 2 i 1 1 i i 2 2 In this section, we apply Theorem 3.4 to solve the problem of finding a point u ∈ C i=1 such that ∗ ∗ g u , x – u ≥ 0, ∀x ∈ C , (37) i i i=1 ∞ ∞ ∗ ∗ ∗ ∗ and the point v = A u ∈ Q , s = A u ∈ K satisfy 1 i 2 i i=1 i=1 ∞ ∞ ∗ ∗ ∗ ∗ h v , y – v ≥ 0, ∀y ∈ Q,and l s , z – s ≥ 0, ∀z ∈ K , (38) i i i i i=1 i=1 where {C } is a family of nonempty closed convex subsets of a real Hilbert space H ,and i i∈N 1 {Q } and {K } are two families of nonempty closed convex subsets of a real Hilbert i i∈N i i∈N space H . We now prove a strong convergence theorem for problem (37)–(38). Theorem 4.2 Let {C } be the family of nonempty closed convex subsets of a real Hilbert i i∈N space H , let {Q } and {K } be two families of nonempty closed convex subsets of a real 1 i i∈N i i∈N Hilbert space H , and let A , A : H → H be two bounded linear operators with adjoint 2 1 2 1 2 ∗ ∗ operators A and A , respectively. Let f : H → H be a ρ-contraction mapping, and let 1 1 1 2 B be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let {g : H → H : i ∈ N}, {h : H → H ; i ∈ N}, and {l : H → H ; i ∈ N} be i 1 1 i 2 2 i 2 2 r -, r -, and r -inverse strongly monotone mappings, respectively. Let r = min{r , r , r } and 1 2 3 1 2 3 ∞ ∞ ∗ ∗ ∗ μ ∈ (0, 2r). Suppose that  = {v ∈ VIP(C , g ): A v ∈ VIP(Q , h ) and A v ∈ i i 1 i i 2 i=1 i=1 VIP(K , l )}= ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by i i 1 1 n n n n i=1 u = x + α δ A (P (I – μh )– I)A x , n n n,i n Q i 1 n i=1 1 i ⎨ n v = u + β θ A (P (I – μl )– I)A u , n n n,i n K i 2 n i=1 2 i (39) y = v + γ τ (P (I – μg )– I)v , n n n,i n C i n i=1 i x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satisfying the following n n n n n,i n,i n,i conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 16 of 21 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . Proof It is known that S := P (I – μh ), T =: P (I – μl ), and U := P (I – μg ) are nonex- i Q i i K i i C i i i i pensive mappings for all μ ∈ (0, 2r), and hence they are 0-demicontractive mappings. We obtain the desired result from Theorem 3.4. 4.3 The split equilibrium problem Let H and H be two real Hilbert spaces, and let C and Q be nonempty closed convex 1 2 subsets of H and H ,respectively. Let A : H → H be a bounded linear operator, and let 1 2 1 2 g : C × C → R and h : Q × Q → R be two bifunctions. The split equilibrium problem (SEP) is to find a point u ∈ C such that g u , x ≥ 0, ∀x ∈ C, (40) and Au ∈ Q satisfy h Au , y ≥ 0, ∀y ∈ Q. (41) The set of all solutions of equilibrium problem (40)isdenoted by EP(g). Lemma 4.3 ([19]) Let C be a nonempty closed convex subset of H, and let g be a bifunction of C ×Cinto R satisfying the following conditions: (A1) g(x, x)=0 for all x ∈ C; (A2) g is monotone, that is, g(x, y)+ g(y, x) ≤ 0 for all x, y ∈ C; (A3) for all x, y, z ∈ C, lim sup g tz +(1– t)x, y ≤ g(x, y); t↓0 (A4) g(x, ·) is convex and lower semicontinuous for all x ∈ C. Let g : C × C → R be a bifunction satisfying conditions (A1)–(A4), and let r >0 and x ∈ H. Then there exists z ∈ Csuch that g(z, y)+ y – z, z – x≥ 0 for all y ∈ C. Lemma 4.4 ([20]) Let C be a nonempty closed convex subset of H, and let g be a bifunction of C × Cinto R satisfying conditions (A1)–(A4). For r >0 and x ∈ H, define the mapping T : H → Cof g by T x = z ∈ C : g(z, y)+ y – z, z – x≥ 0, ∀y ∈ C , ∀x ∈ H. Then the following hold: (i) T is single-valued; (ii) T is firmly nonexpansive; r Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 17 of 21 (iii) F(T )= EP(g); (iv) EP(g) is closed and convex. Let A , A : H → H be two bounded linear operators, and let g : C × C → R and 1 2 1 2 i i i h , l : Q × Q → R be bifunctions for all i ∈ N. In this section, we apply Theorem 3.4 to i i i i solve the problem of finding a point ∞ ∞ ∞ ∗ ∗ ∗ u ∈ EP(g)suchthat A v ∈ EP(h)and A v ∈ EP(l ). (42) i 1 i 2 i i=1 i=1 i=1 h l g i i i By Lemma 4.4(iii) we have that T , T ,and T are firmly nonexpansive mappings, and r r r 1 2 3 hence they are 0-demicontractive mappings. We obtain the following result from Theo- rem 3.4. Theorem 4.5 Let {C } be a family of nonempty closed convex subsets of a real Hilbert i i∈N space H , let {Q } and {K } be two families of nonempty closed convex subsets of a real 1 i i∈N i i∈N Hilbert space H , and let A , A : H → H be two bounded linear operators with adjoint 2 1 2 1 2 ∗ ∗ operators A and A , respectively. Let f : H → H be a ρ-contraction mapping, and let B 1 1 1 2 be a self-adjoint strongly positive bounded linear operator on H with coefficient ξ >2ρ and B =1. Let g : C × C → R and h , l : Q × Q → R be bifunctions satisfying conditions i i i i i i i ∞ ∞ ∗ ∗ ∗ (A1)–(A4) for all i ∈ N. Suppose that  = {v ∈ EP(g ): A v ∈ EP(h ) and A v ∈ i 1 i 2 i=1 i=1 EP(l )}= ∅. For arbitrary x ∈ H , let {u }, {v }, {y }, and {x } be generated by i 1 1 n n n n i=1 ∗ i u = x + α δ A (T – I)A x , n n n,i n r 1 n i=1 1 1 ⎨ n ∗ i v = u + β θ A (T – I)A u , n n n,i n r 2 n i=1 2 2 (43) y = v + γ τ (T – I)v , ⎪ n n n,i n r n i=1 3 x = σ f (y )+(I – σ B)y , n ∈ N, n+1 n n n n where r , r , r >0 and {δ }, {θ }, {τ }, {σ }, {α }, {β }, {γ } are sequences in [0, 1] satis- 1 2 3 n n n n n,i n,i n,i fying the following conditions: n n n (C1) α = β = γ =1 for all n ∈ N; n,i n,i n,i i=1 i=1 i=1 (C2) lim inf α >0, lim inf β >0, and lim inf γ >0 for all i ∈ N; n→∞ n,i n→∞ n,i n→∞ n,i (C3) lim σ =0 and σ = ∞; n→∞ n n n=1 (C4) 0< a ≤ δ ≤ a < ; 1 n 2 2 (C5) 0< b ≤ θ ≤ b < ; 1 n 2 2 (C6) 0< c ≤ τ ≤ c <1. 1 n 2 ∗ ∗ Then the sequence {x } converges strongly to x = P (f + I – B)x . 5 Numerical example for the main result We now give a numerical example of the studied method. Let H = H =(R , · ). Define 1 2 2 2 2 2 2 2 2 the mappings S : R → R , U : R → R ,and T : R → R by i i i –3i –2i S (x , x )= (x , x ), U (x , x )= x , x , i ∈ N, i 1 2 1 2 i 1 2 1 2 i +1 i +1 and x 1 (x , sin )if x =0, 1 2 3i x T (x , x )= i ∈ N i 1 2 (x ,0) if x =0, 1 2 Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 18 of 21 for all x , x ∈ R.Then S are -demicontractive mappings for all i ∈ N and F(S )= 1 2 i i i=1 {(0, 0)}, U are -demicontractive mappings for all i ∈ N and F(U )=0 × R,and T i i i 4 i=1 are 0-demicontractive mappings for all i ∈ N and F(T )= R × 0. Next, we define the i=1 2 2 2 2 2 2 2 2 mappings f : R → R , B : R → R , A : R → R ,and A : R → R by 1 2 x x x 1 2 2 f (x , x )= , , B(x , x )= x , , A (x , x )=(x ,2x ), 1 2 1 2 1 1 1 2 1 1 8 8 2 and A (x , x )=(x – x ,2x ) 2 1 2 2 1 1 for all x , x ∈ R.Then f is a -contraction, B is a self-adjoint strongly positive bounded 1 2 linear operator with coefficient ξ = ,and A , A are bounded linear operators. Define the 1 2 real sequence {α }, {β },and {γ } as follows: n,i n,i n,i 1if n = i =1, 1 n ( )if n > i, n+1 α = n,i n–1 1 n 1– ( )if n = i >1, ⎪ i i=1 n+1 0otherwise, 1if n = i =1, ⎨ 1 n ( )if n > i, 3 n+1 β = n,i n–1 1 n 1– ( )if n = i >1, i=1 i n+1 ⎪ 3 0otherwise, and 1if n = i =1, 1 n ( )if n > i, i+1 2n+1 γ = n,i n–1 1 n 1– ( )if n = i >1, ⎪ i+1 i=1 4 2n+1 0otherwise, that is, ⎛ ⎞ 100 0 0 0 0 0 ... ⎜ ⎟ 1/3 2/3 0 0 0 0 0 0 ... ⎜ ⎟ ⎜ ⎟ 3/8 3/16 7/16 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 2/5 1/5 1/10 3/10 0 0 0 0 . . . ⎜ ⎟ α = ⎜ ⎟ , n,i 5/12 5/24 5/48 5/96 7/32 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/7 3/14 3/28 3/56 3/112 19/112 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/16 7/32 7/64 7/128 7/256 7/512 71/512 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 19 of 21 Figure 1 Graph for errors ⎛ ⎞ 1 0 0 0 0 0 0 0 ... ⎜ ⎟ 2/9 7/9 0 0 0 0 0 0 ... ⎜ ⎟ ⎜ ⎟ 1/4 1/12 2/3 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 4/15 4/45 4/135 83/135 0 0 0 0 . . . ⎜ ⎟ β = , ⎜ ⎟ n,i 5/18 5/54 5/162 5/486 143/243 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 2/7 2/21 2/63 2/189 1/284 325/567 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/24 7/72 7/216 7/648 1/278 1/833 58/103 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . and ⎛ ⎞ 1 0 00 00 0 0... ⎜ ⎟ 1/40 39/40 0 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/112 3/448 433/448 0 0 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 1/36 1/144 1/576 185/192 0 0 0 0 . . . ⎜ ⎟ γ = ⎜ ⎟ . n,i 5/176 5/704 1/563 1/2253 51/53 0 0 0 . . . ⎜ ⎟ ⎜ ⎟ 3/104 3/416 1/555 1/2219 1/8875 976/1015 0 0 . . . ⎜ ⎟ ⎜ ⎟ 7/240 7/960 1/549 1/2194 1/8777 1/35,109 618/643 0 . . . ⎝ ⎠ . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 We see that lim α = , lim β = ,and lim γ = for i ∈ N.Now,we n→∞ n,i n→∞ n,i n→∞ n,i i i 2i+3 2 3 2 start with the initial point x = (1, 1) and let {x } be the sequence generated by (12). Sup- 1 n pose that x is of the form x =(a , b ). where a , b ∈ R. The criterion for stopping our n n n n n n –6 n n n testingmethodistaken as x – x  <10 .Choose δ = , θ = , τ = ,and n–1 n 2 n n n 11n–1 30n–1 2n–1 σ = for all n ∈ N.Figure 1 shows the errors x – x  of our proposed method. n n–1 n 2 0.01 The values of x and x – x  are shown in Table 1. n n–1 n 2 We observe from Table 1 that x → (0, 0) ∈ . We also note that the error is bounded –6 by x – x  <10 ,and we canuse x = (0.00000003, 0.00000117) to approximate the 30 31 2 31 solution of (7)withaccuracyatleast 6D.P. 6Conclusion We introduce a new algorithm for solving the split common fixed point problem (7)of the infinite families of demicontractive mappings in Hilbert spaces. Strong convergence of the proposed algorithm is obtained under some suitable control conditions. The main Hanjing and Suantai Fixed Point Theory and Applications (2018) 2018:14 Page 20 of 21 Table 1 Numerical experiment for x na b x – x n n n–1 n 2 1 1.00000000 1.00000000 – 2 0.12500000 0.62500000 0.95197164 3 0.01751567 0.39224395 0.25637524 4 0.00414010 0.24675959 0.14609793 5 0.00202951 0.15549870 0.09128529 6 0.00140947 0.09811767 0.05738438 7 0.00107109 0.06197693 0.03614232 8 0.00063002 0.03918347 0.02279773 9 0.00047832 0.02479206 0.01439221 10 0.00030270 0.01569709 0.00909667 11 0.00022553 0.00994467 0.00575293 12 0.00014616 0.00630378 0.00364176 13 0.00008740 0.00399788 0.00230665 14 0.00005861 0.00253664 0.00146152 . . . . . . . . . . . . 28 0.00000009 0.00000450 0.00000257 29 0.00000007 0.00000287 0.00000163 30 0.00000007 0.00000183 0.00000104 31 0.00000003 0.00000117 0.00000066 results of this paper can be considered as an extension of work by Eslamian [12] by provid- ing an algorithm for finding a solution of problem (7), which is a generalization of prob- lem (5). Acknowledgements The authors would like to thank Chiang Mai University and Center of Excellence in Mathematics, CHE, Bangkok 10400, Thailand, for the financial support. Abbreviations SFP, The split feasibility problem; SCFP, The split common fixed point problem; MSSFP, The multiple set split feasibility problem; SCNPP, The split common null point problem; SVIP, The split variational inequality problem; SEP, The split equilibrium problem. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript. Author details 1 2 Ph.D. Degree Program in Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. Center of Excellence in Mathematics, CHE, Bangkok, Thailand. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 5 September 2017 Accepted: 7 January 2018 References 1. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) 2. Byrne, C.: Iterative oblique projection onto convex subsets ant the split feasibility problem. Inverse Probl. 18, 441–453 (2002) 3. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006) 4. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple set split feasibility problem and its applications. 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Inverse Probl. 20, 1261–1266 (2004) 11. Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Probl. 26, 587–600 (2010) 12. Eslamian, M., Eslamian, P.: Strong convergence of a split common fixed point problem. Numer. Funct. Anal. Optim. 37, 1248–1266 (2016) 13. Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006) 14. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009) 15. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) 16. Mainge, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469–479 (2007) 17. Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65(2), 443–465 (2016) 18. 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Fixed Point Theory and ApplicationsSpringer Journals

Published: May 28, 2018

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