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Coupled measure of noncompactness and functional integral equations

Coupled measure of noncompactness and functional integral equations 1IntroductionIn nonlinear analysis, one of the most important tools is the concept of measure of noncompactness (MNC) to address the problems in functional operator equations. This important concept in mathematical sciences has been defined by many authors in various ways (see [1,2,3, 4,5,6, 7,8]). In [9], Aghajani et al. established some generalizations of Darbo’s fixed-point theorem and presented an application in functional integral equations.In this paper, we investigate the fixed-point results that generalize Darbo’s fixed-point theorem and many existing results in the literature by introducing the notion of coupled MNC. As an application, we prove the existence of solutions of a functional integral equation in Banach space BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}). Finally, an example is supplied to illustrate the results.Throughout this study, we consider EEas a Banach space and briefly represent a measure of noncompactness with MNC, B(υ,r)B\left(\upsilon ,r)represents a closed ball in Banach space EEto center υ\upsilon and radius rr. Also, we use Br{B}_{r}to represent B(θ,r)B\left(\theta ,r), where θ\theta is the zero element, the family of all nonempty bounded subsets of EEis represented with ℬE{{\mathcal{ {\mathcal B} }}}_{E}. To begin, we have the following preliminaries from [6,10,11].Definition 1.1[6]. Let μ:ℬE→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. The family ℬE{{\mathcal{ {\mathcal B} }}}_{E}is called MNC on Banach space EEif the following conditions hold: (1)For each U1∈ℬE{{\mathcal{U}}}^{1}\in {{\mathcal{ {\mathcal B} }}}_{E}, μ(U1)=θ\mu \left({{\mathcal{U}}}^{1})=\theta iff U1{{\mathcal{U}}}^{1}is a precompact set;(2)For each pair (U1,U2)∈ℬE×ℬE\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}\times {{\mathcal{ {\mathcal B} }}}_{E}, we have U1⊆U2impliesμ(U1)≤μ(U2);{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{2}\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1})\le \mu \left({{\mathcal{U}}}^{2});(3)For each U1∈ℬE{{\mathcal{U}}}^{1}\in {{\mathcal{ {\mathcal B} }}}_{E}, μ(U1)=μ(U1¯)=μ(convU1),\mu \left({{\mathcal{U}}}^{1})=\mu \left(\overline{{{\mathcal{U}}}^{1}})=\mu \left({\rm{conv}}\hspace{0.33em}{{\mathcal{U}}}^{1}),where U1¯\overline{{{\mathcal{U}}}^{1}}represents the closure of U1{{\mathcal{U}}}^{1}and convU1{\rm{conv}}\hspace{0.33em}{{\mathcal{U}}}^{1}represents the convex hull of U1{{\mathcal{U}}}^{1};(4)μ(λU1+(1−λ)U2)≤λμ(U1)+(1−λ)μ(U2)\mu \left(\lambda {{\mathcal{U}}}^{1}+\left(1-\lambda ){{\mathcal{U}}}^{2})\le \lambda \mu \left({{\mathcal{U}}}^{1})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{2})for λ∈[0,1]\lambda \in \left[0,1];(5)If {υn}0+∞∈ℬE{\left\{{\upsilon }_{n}\right\}}_{0}^{+\infty }\in {{\mathcal{ {\mathcal B} }}}_{E}is a decreasing sequence of closed sets and limn→+∞μ(υn)=0{\mathrm{lim}}_{n\to +\infty }\mu \left({\upsilon }_{n})=0, then U+∞1=⋂n=0+∞Un1≠∅{{\mathcal{U}}}_{+\infty }^{1}={\bigcap }_{n=0}^{+\infty }{{\mathcal{U}}}_{n}^{1}\ne \varnothing .In this part, we have the following theorems from [10,11,12].Theorem 1.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a compact and continuous operator. Then, FFhas at least one fixed-point.Theorem 1.3(Schauder) Let GGbe a nonempty, closed, and convex subset of a normed space and FFbe a continuous operator from GGinto a compact subset of GG. Then, FFhas a fixed-point.Theorem 1.4(Darbo) Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous operator. Suppose there is λ∈[0,1)\lambda \in {[}0,1)such that μ(FU)≤λμ(U)\mu \left(F{\mathcal{U}})\le \lambda \mu \left({\mathcal{U}})for each U∈G{\mathcal{U}}\in G. Then, FFhas a fixed-point.Theorem 1.5(Brouwer) Let GGbe a nonempty, compact, and convex subset of a finite-dimensional normed space and F:G→GF:G\to Gbe a continuous operator. Then, FFhas a fixed-point.Lemma 1.6[9] Let φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}be an upper semicontinuous and nondecreasing function. In this case, the following conditions are equivalent: (1)limn→+∞φ1n(r)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(r)=0for every r>0r\gt 0;(2)φ1(r)<r{\varphi }_{1}\left(r)\lt rfor every r>0r\gt 0.2Coupled MNCWe start this section with the following concept, and then, we turn to the main subject.Definition 2.1Let EEbe a Banach space and μ:ℬE2→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{2}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. We say that μ\mu is a coupled MNC on EE, if it has the following conditions: (1)kerμ={(U1,U2)∈ℬE2:μ(U1,U2)=θ}\ker \mu =\left\{\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}:\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\theta \right\}is nonempty;(2)For every (U1,U2)∈ℬE2,μ(U1,U2)=θ⇔(U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2},\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\theta \iff \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})is a precompact set;(3)For each ((U1,U2),(U′1,U′2))∈ℬE2×ℬE2\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}))\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}\times {{\mathcal{ {\mathcal B} }}}_{E}^{2}and (U1,U2)⊆(U′1,U′2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}), where U1⊆U′1{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{^{\prime} 1}and U2⊆U′2{{\mathcal{U}}}^{2}\subseteq {{\mathcal{U}}}^{^{\prime} 2}, we have (U1,U2)⊆(U′1,U′2)impliesμ(U1,U2)≤μ(U′1,U′2);\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2})\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\le \mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2});(4)For every (U1,U2)∈ℬE2\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}, μ(U1¯,U2¯)=μ(U1,U2)=μ(conv(U1,U2)),\mu \left(\overline{{{\mathcal{U}}}^{1}},\overline{{{\mathcal{U}}}^{2}})=\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\mu \left({\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),where conv(U1,U2){\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})denotes the convex hull of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2});(5)μ(λ(U1,U2)+(1−λ)(U′1,U′2))≤λμ(U1,U2)+(1−λ)μ(U′1,U′2)\mu \left(\lambda \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\left(1-\lambda )\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}))\le \lambda \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2})for λ∈[0,1]\lambda \in \left[0,1];(6)If {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }in ℬE{{\mathcal{ {\mathcal B} }}}_{E}are decreasing sequences of closed sets and limn→+∞μ{(Un1,Un2)}0+∞=0,then(U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)≠∅.\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu {\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{0}^{+\infty }=0,\hspace{1em}\hspace{0.1em}\text{then}\hspace{0.1em}\hspace{0.33em}\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})=\mathop{\bigcap }\limits_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\ne \varnothing .Theorem 2.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(2.1)φ2(μ(FU1,FU2))≤φ2(μ(U1,U2))−φ1(μ(U1,U2)),{\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2}))\le {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),for each ∅≠U1⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G, ∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where μ\mu is an arbitrary coupled MNC and φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.ProofTaking U01,U02=G{{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}=G, Un+11=conv(FUn1)¯,Un+12=conv(FUn2)¯{{\mathcal{U}}}_{n+1}^{1}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},{{\mathcal{U}}}_{n+1}^{2}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}, for n=0,1,2,…,n=0,1,2,\ldots ,we obtain Un+11⊆Un1,Un+12⊆Un2{{\mathcal{U}}}_{n+1}^{1}\hspace{0.25em}\subseteq {{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n+1}^{2}\subseteq {{\mathcal{U}}}_{n}^{2}for n=0,1,…n=0,1,\ldots . Therefore, {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }are decreasing sequences of closed and convex sets. Moreover, from (2.1), we have (2.2)φ2(μ(Un+11,Un+12))=φ2(μ(conv(FUn1)¯,conv(FUn2)¯))=φ2(μ(FUn1,FUn2))≤φ2(μ(Un1,Un2))−φ1(μ(Un1,Un2)),\begin{array}{rcl}{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))& =& {\varphi }_{2}\left(\mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}))\\ & =& {\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})),\end{array}for n=0,1,2,…n=0,1,2,\ldots . Since the sequence {μ(Un1,Un2)}\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nonnegative and nonincreasing, we deduce that μ(Un1,Un2)→m\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to mwhen nntends to infinity, where m≥0m\ge 0is a real number. On the other hand, considering equation (2.2), we obtain (2.3)lim supn→+∞φ2(μ(Un+11,Un+12))≤lim supn→+∞φ2(μ(Un1,Un2))−liminfn→+∞φ1(μ(Un1,Un2)).\mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))\le \mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))-\mathop{\mathrm{liminf}}\limits_{n\to +\infty }{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})).This yields φ2(m)≤φ2(m)−φ1(m){\varphi }_{2}\left(m)\le {\varphi }_{2}\left(m)-{\varphi }_{1}\left(m). Consequently, φ1(m)=0{\varphi }_{1}\left(m)=0and so m=0m=0. Therefore, we infer μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\hspace{-0.08em}\to \hspace{-0.08em}+\infty . Now, considering that (Un+11,Un+12)⊆(Un1,Un2)\left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})\hspace{-0.08em}\subseteq \hspace{-0.08em}\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}), by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is nonempty, closed, and convex. Furthermore, the set (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})under the operator FFis invariant and (U+∞1,U+∞2)∈kerμ\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})\in \ker \mu . So, by applying Theorem 1.2, the proof is complete.□Theorem 2.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Ga continuous map such that(2.4)μ(FU1,FU2)≤φ1(μ(U1,U2)),\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),for each ∅≠U1⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G, ∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where μ\mu is an arbitrary coupled MNC and φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for every s≥0s\ge 0. Then, FFhas at least one fixed-point.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction, where U01,U02=G{{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}=G, Un+11=conv(FUn1)¯,Un+12=conv(FUn2)¯{{\mathcal{U}}}_{n+1}^{1}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},{{\mathcal{U}}}_{n+1}^{2}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}, for n=0,1,…n=0,1,\ldots . Moreover, in the same as the previous method, we can assume μ(Un1,Un2)>0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\gt 0for all n=1,2,…n=1,2,\ldots . In addition, by given assumptions, we obtain (2.5)μ(Un+11,Un+12)=μ(conv(FUn1)¯,conv(FUn2)¯)=μ(FUn1,FUn2)≤φ1(μ(Un1,Un2))≤φ12(μ(Un−11,Un−12))⋮≤φ1n+1(μ(U01,U02)).\begin{array}{rcl}\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})& =& \mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})})\\ & =& \mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2})\\ & \le & {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{1}^{2}\left(\mu \left({{\mathcal{U}}}_{n-1}^{1},{{\mathcal{U}}}_{n-1}^{2}))\\ & \vdots & \\ & \le & {\varphi }_{1}^{n+1}\left(\mu \left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2})).\end{array}This shows that μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\to +\infty . Since the sequence {(Un1,Un2)}\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nested, by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is a nonempty, closed, and convex subset of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}). Therefore, we obtain that (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is a member of kerμ\ker \mu . So, (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is compact. Next, note that FFmaps (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})into itself, and considering Theorem 1.2, we deduce that FFhas fixed-point in (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2}). So the proof is complete.□Now, from the aforementioned theorem, we have the following.Corollary 2.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe an operator such that(2.6)‖(Fυ1,Fυ2)−(Fυ′1,Fυ′2)‖≤φ1(‖υ1υ2−υ′1υ′2‖),forallυ1,υ′1,υ2,υ′2∈G,\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert ),\hspace{0.33em}{for}\hspace{0.33em}{all}\hspace{0.33em}{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in G,where φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for any s≥0s\ge 0. Then, FFhas a fixed-point in GG.ProofLet μ:ℬE2→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{2}\to {{\mathcal{ {\mathcal R} }}}_{+}and μ(U1,U2)≔diam(U1,U2),\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}):= {\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),where diam(U1,U2)=sup{‖υ1υ2−υ′1υ′2‖:υ1,υ′1∈U1,υ2,υ′2∈U2}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\sup \left\{\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert :{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}\right\}. It can be easily seen that μ\mu is coupled MNC in EEby Definition 2.1. Furthermore, since φ1{\varphi }_{1}is nondecreasing, then in view of (2.6), we have supυ1,υ′1∈U1,υ2,υ′2∈U2‖(Fυ1,Fυ2)−(Fυ′1,Fυ′2)‖≤supυ1,υ′1∈U1,υ2,υ′2∈U2φ1‖υ1υ2−υ′1υ′2‖≤φ1(supυ1,υ′1∈U1,υ2,υ′2∈U2‖υ1υ2−υ′1υ′2‖),\begin{array}{rcl}\mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\Vert & \le & \mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}{\varphi }_{1}\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \\ & \le & {\varphi }_{1}\left(\mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \right),\end{array}which yields that μ(FU1,FU2)≤φ1(μ(U1,U2)).\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})).By using Theorem 2.3, the proof is complete.□3Tripled MNCIn this section, as a result of Section 2, we define the notion of tripled MNC as follows.Definition 3.1Let EEbe a Banach space and μ:ℬE3→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{3}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. We say that μ\mu is a tripled MNC on E,E,if it has the following conditions: (1)kerμ={(U1,U2,U3)∈ℬE3:μ(U1,U2,U3)=θ}\ker \mu =\left\{\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}:\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\theta \right\}is nonempty;(2)For every (U1,U2,U3)∈ℬE3,μ(U1,U2,U3)=θ⇔(U1,U2,U3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3},\hspace{1em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\theta \iff \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})is a precompact set;(3)For each ((U1,U2,U3),(U′1,U′2,U′3))∈ℬE3×ℬE3\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}),\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3}))\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}\times {{\mathcal{ {\mathcal B} }}}_{E}^{3}, ((U1,U2,U3)⊆(U′1,U′2,U′3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})yields U1⊆U′1,U2⊆U′2{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{2}\subseteq {{\mathcal{U}}}^{^{\prime} 2}and U3⊆U′3{{\mathcal{U}}}^{3}\subseteq {{\mathcal{U}}}^{^{\prime} 3}), we have (U1,U2,U3)⊆(U′1,U′2,U′3)impliesμ(U1,U2,U3)≤μ(U′1,U′2,U′3);\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\le \mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3});(4)For every (U1,U2,U3)∈ℬE3\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}, one has μ(U1¯,U2¯,U3¯)=μ(U1,U2,U3)=μ(conv(U1,U2,U3)),\mu \left(\overline{{{\mathcal{U}}}^{1}},\overline{{{\mathcal{U}}}^{2}},\overline{{{\mathcal{U}}}^{3}})=\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\mu \left({\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),where conv(U1,U2,U3){\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})denotes the convex hull of (U1,U2,U3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3});(5)μ(λ(U1,U2,U3)+(1−λ)(U′1,U′2,U′3))≤λμ(U1,U2,U3)+(1−λ)μ(U′1,U′2,U′3)\mu \left(\lambda \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})+\left(1-\lambda )\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3}))\le \lambda \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})for λ∈[0,1]\lambda \in \left[0,1];(6)If {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }and {Un3}0+∞{\left\{{{\mathcal{U}}}_{n}^{3}\right\}}_{0}^{+\infty }in ℬE{{\mathcal{ {\mathcal B} }}}_{E}are decreasing sequences of closed sets and limn→+∞μ{(Un1,Un2,Un3)}0+∞=0{\mathrm{lim}}_{n\to +\infty }\mu {\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2},{{\mathcal{U}}}_{n}^{3})\right\}}_{0}^{+\infty }=0, then (U+∞1,U+∞2,U+∞3)=⋂n=0+∞(Un1,Un2,Un3)≠∅\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2},{{\mathcal{U}}}_{+\infty }^{3})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2},{{\mathcal{U}}}_{n}^{3})\ne \varnothing .Theorem 3.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(3.1)φ2(μ(FU1,FU2,FU3))≤φ2(μ(U1,U2,U3))−φ1(μ(U1,U2,U3)),{\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3}))\le {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where μ\mu is an arbitrary tripled MNC and φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.Theorem 3.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(3.2)μ(FU1,FU2,FU3)≤φ1(μ(U1,U2,U3)),\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where μ\mu is an arbitrary tripled MNC and φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(s)=0for every s≥0s\ge 0. Then, FFhas at least one fixed-point.Corollary 3.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe an operator such that(3.3)‖(Fυ1,Fυ2,Fυ3)−(Fυ′1,Fυ′2,Fυ′3)‖≤φ1(‖υ1υ2υ3−υ′1υ′2υ′3‖),\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2},F{\upsilon }^{3})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2},F{\upsilon }^{^{\prime} 3})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}{\upsilon }^{3}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}{\upsilon }^{^{\prime} 3}\Vert ),for all υ1,υ′1,υ2,υ′2,υ3,υ′3∈G{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2},{\upsilon }^{3},{\upsilon }^{^{\prime} 3}\in G, where φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(s)=0for any s≥0s\ge 0. Then, FFhas a fixed-point in GG.4MNC and JS-contractionIn this section, we tend to prove some results of MNC for the family of JS-contractive-type mappings. Also, we generalize Darbo’s fixed-point theorem to coupled and tripled MNC through JS-contraction-type mappings.Denote by Θ\Theta the set of all functions θ:(0,+∞)→(1,+∞)\theta :\left(0,+\infty )\to \left(1,+\infty )so that: (θ1)θ\theta is continuous and increasing;(θ2)limn→+∞tn=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{t}_{n}=0iff limn→+∞θ(tn)=1{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}\theta \left({t}_{n})=1for all {tn}⊆(0,+∞)\left\{{t}_{n}\right\}\subseteq \left(0,+\infty ).Theorem 4.1[13] Let (G,ϱ)\left(G,\varrho )be a complete metric space and F:G→GF:G\to Gbe a given mapping. Suppose that there exist θ∈Θ\theta \in \Theta and ν∈(0,1)\nu \in \left(0,1)such that for all ι,ς∈G\iota ,\varsigma \in G, (4.1)ϱ(Fι,Fς)≠0⇒θ(ϱ(Fι,Fς))≤(θ(ϱ(ι,ς)))ν.\varrho \left(F\iota ,F\varsigma )\ne 0\Rightarrow \theta \left(\varrho \left(F\iota ,F\varsigma ))\le {\left(\theta \left(\varrho \left(\iota ,\varsigma )))}^{\nu }.Then, FFhas a unique fixed-point.Theorem 4.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to G, be a continuous map such that(4.2)θ(φ2(μ(FU1,FU2)))≤θ(φ2(μ(U1,U2)))θ(φ2(φ1(μ(U1,U2)))),\hspace{-24.4em}\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))))},for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where θ∈Θ\theta \in \Theta , and μ\mu is an arbitrary coupled MNC and functions φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}, such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction. Moreover, from (4.2), we obtain (4.3)θ(φ2(μ(Un+11,Un+12)))=θ(φ2(μ(conv(FUn1)¯,conv(FUn2)¯)))=θ(φ2(μ(FUn1,FUn2)))≤θ(φ2(μ(Un1,Un2)))θ(φ2(φ1(μ(Un1,Un2)))),\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})))=\theta \left({\varphi }_{2}\left(\mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})})))=\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))))},for n=0,1,2,…n=0,1,2,\ldots . Since the sequence {μ(Un1,Un2)}\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nonnegative and nonincreasing, we deduce that μ(Un1,Un2)→m\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to mwhen nntends to infinity, where m≥0m\ge 0is a real number. On the other hand, considering equation (4.3), we obtain (4.4)lim supn→+∞θ(φ2(μ(Un+11,Un+12)))≤lim supn→+∞θ(φ2(μ(Un1,Un2)))θ(φ2(φ1(μ(Un1,Un2)))),\mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})))\le \mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }\frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))))},which yields that θ(φ2(m))≤θ(φ2(m))θ(φ2(φ1(m)))\theta \left({\varphi }_{2}\left(m))\le \frac{\theta \left({\varphi }_{2}\left(m))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(m)))}. Consequently, θ(φ2(φ1(m)))=1\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(m)))=1, then φ2(φ1(m))=0{\varphi }_{2}\left({\varphi }_{1}\left(m))=0and φ1(m)=0{\varphi }_{1}\left(m)=0so m=0m=0. Therefore, we infer μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\to +\infty . Now, considering that (Un+11,Un+12)⊆(Un1,Un2)\left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})\subseteq \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}), by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is nonempty, closed, and convex. Furthermore, the set (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})under the operator FFis invariant and (U+∞1,U+∞2)∈kerμ\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})\in ker\mu . So, by applying Theorem 1.2, the proof is complete.□Denote by Ψ\Psi the set of all functions φ2:(1,+∞)→(1,+∞){\varphi }_{2}:\left(1,+\infty )\to \left(1,+\infty )so that: (φ21)φ2{\varphi }_{2}is continuous and increasing;(φ22)limn→+∞φ2n(s)=1{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{2}^{n}\left(s)=1for all s∈(1,+∞)s\in \left(1,+\infty ).Theorem 4.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Ga continuous map such that(4.5)θ(μ(FU1,FU2))≤φ2(θ(μ(U1,U2))),\theta \left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2}))\le {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where θ∈Θ\theta \in \Theta , φ2∈Ψ{\varphi }_{2}\in \Psi , and μ\mu is an arbitrary coupled MNC. Then, FFhas at least one fixed-point.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction.If for an integer N∈NN\in {\mathbb{N}}one has μ(UN1,UN2)=0\mu \left({{\mathcal{U}}}_{N}^{1},{{\mathcal{U}}}_{N}^{2})=0, then (UN1,UN2)\left({{\mathcal{U}}}_{N}^{1},{{\mathcal{U}}}_{N}^{2})is a precompact set. So the Schauder theorem ensures the existence of a fixed-point for FF. Therefore, we can assume μ(Un1,Un2)>0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\gt 0for all n∈N∪{0}n\in {\mathbb{N}}\cup \left\{0\right\}.Obviously, {(Un1,Un2)}n∈N{\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{n\in {\mathbb{N}}}is a sequence of nonempty, bounded, closed, and convex subsets such that (U01,U02)⊇(U11,U12)⊇⋯⊇(Un1,Un2)⊇(Un+11,Un+12).\left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2})\supseteq \left({{\mathcal{U}}}_{1}^{1},{{\mathcal{U}}}_{1}^{2})\supseteq \cdots \supseteq \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\supseteq \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}).On the other hand, (4.6)θ(μ(Un+11,Un+12))=θ(μ(FUn1,FUn2))≤φ2(θ(μ(Un1,Un2)))≤⋮≤φ2n+1(θ(μ(U01,U02))).\begin{array}{rcl}\theta \left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))& =& \theta \left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))\\ & \le & \\ & \vdots & \\ & \le & {\varphi }_{2}^{n+1}\left(\theta \left(\mu \left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}))).\end{array}Thus, {μ(Un1,Un2)}n∈N{\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{n\in {\mathbb{N}}}is a convergent sequence. Assume that limn→+∞μ(Un1,Un2)=η.\hspace{-10.5em}\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})=\eta .By taking the limit from (4.6), limn→+∞θ(μ(Un+11,Un+12))=1{\mathrm{lim}}_{n\to +\infty }\theta \left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))=1. By (θ2),\left({\theta }_{2}),we obtain limn→+∞μ(Un+11,Un+12)=0.\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})=0.From Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is a nonempty, closed, and convex subset of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}). Therefore, we obtain (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is a member of kerμ\ker \mu . So, (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is compact. Note that FFmaps (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})into itself, and considering Theorem 1.2, we deduce that FFhas a fixed-point in (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2}). So the proof is complete.□Theorem 4.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(4.7)θ(φ2(μ(FU1,FU2,FU3)))≤θ(φ2(μ(U1,U2,U3)))θ(φ2(φ1(μ(U1,U2,U3)))),\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))))},for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where θ∈Θ\theta \in \Theta and μ\mu is an arbitrary tripled MNC and functions φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{ {\mathcal R} }_{+}\to { {\mathcal R} }_{+}, such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{ {\mathcal R} }_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.Theorem 4.5Let G be a nonempty, bounded, closed, and convex subset of a Banach space E and F:G→GF:G\to Gbe a continuous map such that(4.8)θ(μ(FU1,FU2,FU3))≤φ2(θ(μ(U1,U2,U3))),\theta \left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3}))\le {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where θ∈Θ\theta \in \Theta , φ2Ψ∈{\varphi }_{2}\Psi \in , and μ\mu is an arbitrary tripled MNC. Then, FFhas at least one fixed-point.5ApplicationWe offer the applications of Theorem 2.3 to prove the existence of solutions of a functional integral equation in the Banach space BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})consisting of all real functions that are bounded and continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. This space is endowed with the supremum norm ‖υ1‖=sup{‖υ1(r)‖:r∈ℛ+}.\Vert {\upsilon }^{1}\Vert =\sup \left\{\Vert {\upsilon }^{1}\left(r)\Vert :r\in {{\mathcal{ {\mathcal R} }}}_{+}\right\}.We choose nonempty bounded subsets U1,U2{{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}of BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})and a number L>0L\gt 0. For ε>0\varepsilon \gt 0, υ1∈U1{\upsilon }^{1}\in {{\mathcal{U}}}^{1}, and υ2∈U2{\upsilon }^{2}\in {{\mathcal{U}}}^{2}, we show the modulus of continuity of function (υ1,υ2)\left({\upsilon }^{1},{\upsilon }^{2})on the interval [0,L]\left[0,L]by ωL((υ1,υ2),ε){\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ): ωL((υ1,υ2),ε)=sup{∣υ1υ2(r)−υ1υ2(s)∣:r,s∈[0,L],∣r−s∣≤ε}.{\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon )=\sup \left\{| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{1}{\upsilon }^{2}\left(s)| :r,s\in \left[0,L],| r-s| \le \varepsilon \right\}.Also, ωL((U1,U2),ε)=sup{ωL((υ1,υ2),ε):υ1∈U1,υ2∈U2},ω0L(U1,U2)=limε→0ωL((U1,U2),ε),ω0(U1,U2)=limL→+∞ω0L(U1,U2).\begin{array}{rcl}{\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon )& =& \sup \left\{{\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ):{\upsilon }^{1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2}\in {{\mathcal{U}}}^{2}\right\},\\ {\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon ),\\ {\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& \mathop{\mathrm{lim}}\limits_{L\to +\infty }{\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}).\end{array}Moreover, for r∈ℛ+r\in {{\mathcal{ {\mathcal R} }}}_{+}, we define (U1,U2)(r)={(υ1,υ2)(r):υ1∈U1,υ2∈U2},μ(U1,U2)=ω0(U1,U2)+lim supr→+∞diam(U1,U2)(r),diam(U1,U2)(r)=sup{∣υ1υ2(r)−υ′1υ′2(r)∣:υ1,υ′1∈U1,υ2,υ′2∈U2}.\begin{array}{rcl}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)& =& \left\{\left({\upsilon }^{1},{\upsilon }^{2})\left(r):{\upsilon }^{1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2}\in {{\mathcal{U}}}^{2}\right\},\\ \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& {\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r),\\ {\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)& =& \sup \left\{| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| :{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}\right\}.\end{array}Now, we consider the following hypotheses: (i)Let f1:ℛ+×ℛ2→ℛ{f}_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}be a continuous function. Furthermore, the function r→f1(r,0,0)r\to {f}_{1}\left(r,0,0)is a member of BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}).(ii)There is an upper semicontinuous function φ1∈ϕ{\varphi }_{1}\in \phi , where ϕ\phi is family of all functions φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}, which φ1{\varphi }_{1}is a nondecreasing function such that limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for each s≥0s\ge 0, such that ∣f1(r,υ1,υ2)−f1(r,υ′1,υ′2)∣≤φ1(∣υ1υ2−υ′1υ′2∣),r∈ℛ+υ1,υ′1,υ2,υ′2∈ℛ.| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| \le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ),\hspace{1em}r\in {{\mathcal{ {\mathcal R} }}}_{+}\hspace{1em}{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {\mathcal{ {\mathcal R} }}.In addition, we presume that φ1(s)+φ1(s′)≤φ1(s+s′){\varphi }_{1}\left(s)+{\varphi }_{1}\left(s^{\prime} )\le {\varphi }_{1}\left(s+s^{\prime} )for all s,s′∈ℛ+s,s^{\prime} \in {{\mathcal{ {\mathcal R} }}}_{+}.(iii)Let f2:ℛ+2×ℛ2→ℛ{f}_{2}:{{\mathcal{ {\mathcal R} }}}_{+}^{2}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}be a continuous function, and there are continuous functions u,v:ℛ+→ℛ+u,v:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that limr→+∞u(r)∫0rv(s)ds=0\mathop{\mathrm{lim}}\limits_{r\to +\infty }u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s=0and ∣f2(r,s,υ1,υ2)∣≤u(r)v(s)| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})| \le u\left(r)v\left(s)for r,s∈ℛ+r,s\in {{\mathcal{ {\mathcal R} }}}_{+}such that s≤rs\le rand for every υ1,υ2∈ℛ{\upsilon }^{1},{\upsilon }^{2}\in {\mathcal{ {\mathcal R} }}.(iv)There is a positive solution r01{r}_{0}^{1}from φ1(r1)+p≤r1,{\varphi }_{1}\left({r}^{1})+p\le {r}^{1},where p=sup{∣f1(r,0,0)+u(r)∫0rv(s)ds:r≥0}.p=\sup \left\{| {f}_{1}\left(r,0,0)+u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s:r\ge 0\right\}.Now, we consider the integral equation (5.1)(υ1,υ2)(r)=f1(r,υ1(r),υ2(r))+∫0rf2(r,s,υ1(s),υ2(s))ds.\left({\upsilon }^{1},{\upsilon }^{2})\left(r)={f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))+\underset{0}{\overset{r}{\int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s.We define operator FFon BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})by (5.2)(Fυ1,Fυ2)(r)=f1(r,υ1(r),υ2(r))+∫0rf2(r,s,υ1(s),υ2(s))ds,forr∈ℛ+,\left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)={f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))+\underset{0}{\overset{r}{\int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s,\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}r\in {{\mathcal{ {\mathcal R} }}}_{+},where the function (Fυ1,Fυ2)\left(F{\upsilon }^{1},F{\upsilon }^{2})is continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}.Theorem 5.1According to hypotheses (i)–(iv), relation (5.1) has at least one solution in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}).ProofFor arbitrary υ1,υ2∈BC(ℛ+){\upsilon }^{1},{\upsilon }^{2}\in BC\left({{\mathcal{ {\mathcal R} }}}_{+}), using the aforementioned hypotheses, we obtain ∣(Fυ1,Fυ2)(r)∣≤∣f1(r,υ1(r),υ2(r))−f1(r,0,0)∣+∣f1(r,0,0)∣+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds≤φ1(∣υ1υ2(r)∣)+∣f1(r,0,0)∣+u(r)∫0rv(s)ds=φ1(∣υ1υ2(r)∣)+∣f1(r,0,0)∣+a(r),\begin{array}{rcl}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)| & \le & | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r,0,0)| +| {f}_{1}\left(r,0,0)| +\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ & \le & {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+| {f}_{1}\left(r,0,0)| +u\left(r)\underset{0}{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ & =& {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+| {f}_{1}\left(r,0,0)| +a\left(r),\end{array}where a(r)≔u(r)∫0rv(s)ds.\hspace{-12.25em}a\left(r):= u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s.Since φ1{\varphi }_{1}is nondecreasing, in accordance with the fourth condition, we obtain ‖(Fυ1,Fυ2)‖≤φ1(‖υ1υ2‖)+p.\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}\Vert )+p.Thus, FFis a self-mapping of BC(ℛ+)BC\left({{\mathcal{ {\mathcal R} }}}_{+}). On the other hand, applying assumption (iv), we deduce that FFis a self-mapping of the ball Br01{B}_{{r}_{0}^{1}}. To show that FFis continuous on Br01{B}_{{r}_{0}^{1}}, take ε>0\varepsilon \gt 0and υ1,υ′1,υ2,υ′2∈Br01{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {B}_{{r}_{0}^{1}}such that ‖υ1υ2−υ′1υ′2‖<ε\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \lt \varepsilon , we obtain (5.3)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))−f2(r,s,υ′1(s),υ′2(s))∣ds\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \\ \hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))-{f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\end{array}≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds+∫0r∣f2(r,s,υ′1(s),υ′2(s))∣ds≤φ1(ε)+2a(r),\begin{array}{l}\hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\varphi }_{1}\left(\varepsilon )+2a\left(r),\end{array}for any r∈ℛ+r\in {{\mathcal{ {\mathcal R} }}}_{+}. By assumption (iii), there is a number L>0L\gt 0such that (5.4)2u(r)∫0rv(s)ds≤ε,for everyL≤r.2u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s\le \varepsilon ,\hspace{0.33em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}L\le r.So, considering Lemma 1.6 and the similar evaluation mentioned earlier, for an arbitrary L≤r,L\le r,we have (5.5)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤2ε.| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \le 2\varepsilon .Now, we define ωL(f2,ε)≔sup{∣f2(r,s,υ1,υ2)−f2(r,s,υ′1,υ′2)∣:r,s∈[0,L],υ1,υ′1,υ2,υ′2∈[−r01,r01],∣υ1υ2−υ′1υ′2∣≤ε}.{\omega }^{L}({f}_{2},\varepsilon ):= \sup \left\{| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})-{f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| :r,s\in \left[0,L],{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| \le \varepsilon \right\}.Due to the uniform continuity of f2(r,s,υ1,υ2){f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})on [0,L]2×[−r01,r01]2,{\left[0,L]}^{2}\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2},we infer that ωL(f2,ε)→0{\omega }^{L}({f}_{2},\varepsilon )\to 0as ε→0\varepsilon \to 0. Now, considering the first part of equation (5.3), for the arbitrary constant r∈[0,L],r\in \left[0,L],we obtain (5.6)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤φ1(ε)+∫0LωL(f2,ε)ds=φ1(ε)+LωL(f2,ε).| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \le {\varphi }_{1}\left(\varepsilon )+\underset{0}{\overset{L}{\int }}{\omega }^{L}({f}_{2},\varepsilon ){\rm{d}}s={\varphi }_{1}\left(\varepsilon )+L{\omega }^{L}({f}_{2},\varepsilon ).By combining (5.5) and (5.6) and based on the aforementioned fact about ωL(f2,ε){\omega }^{L}({f}_{2},\varepsilon ), the operator FFon the ball Br01{B}_{{r}_{0}^{1}}, is continuous. Next, we can choose arbitrary nonempty subsets U1,U2{{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}of the ball Br01{B}_{{r}_{0}^{1}}. To do this, we consider constant numbers L>0L\gt 0and ε>0\varepsilon \gt 0. Also, take arbitrary numbers r,r′∈[0,L]r,r^{\prime} \in \left[0,L]with ∣r−r′∣≤ε| r-r^{\prime} | \le \varepsilon . Without loss of generality, it can be assumed that r′<rr^{\prime} \lt r. So, for υ1∈U1{\upsilon }^{1}\in {{\mathcal{U}}}^{1}and υ2∈U2,{\upsilon }^{2}\in {{\mathcal{U}}}^{2},we obtain (5.7)∣(Fυ1,Fυ2)(r)−(Fυ1,Fυ2)(r′)∣≤∣f1(r,υ1(r),υ2(r))−f1(r′,υ1(r′),υ2(r′))∣+∫0rf2(r,s,υ1(s),υ2(s))ds−∫0r′f2(r′,s,υ1(s),υ2(s))ds≤∣f1(r,υ1(r),υ2(r))−f1(r′,υ1(r),υ2(r))∣+∣f1(r′,υ1(r),υ2(r))−f1(r′,υ1(r′),υ2(r′))∣+∫0rf2(r,s,υ1(s),υ2(s))ds−∫0rf2(r′,s,υ1(s),υ2(s))ds+∫0rf2(r′,s,υ1(s),υ2(s))ds−∫0r′f2(r′,s,υ1(s),υ2(s))ds≤ω1L(f1,ε)+φ1(∣υ1υ2(r)−υ1υ2(r′)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))ds−f2(r′,s,υ1(s),υ2(s))∣ds+∫r′r∣f2(r′,s,υ1(s),υ2(s))∣ds≤ω1L(f1,ε)+φ1(ωL((υ1,υ2),ε))+∫0rω1L(f2,ε)ds+u(r′)∫r′rv(s)ds≤ω1L(f1,ε)+φ1(ωL((υ1,υ2),ε))+Lω1L(f2,ε)+εsup{u(r′)v(r):r,r′∈[0,L]},\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r^{\prime} )| \\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r^{\prime} ),{\upsilon }^{2}\left(r^{\prime} ))| +\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-\underset{0}{\overset{r^{\prime} }{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|\\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))| +| {f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r^{\prime} ),{\upsilon }^{2}\left(r^{\prime} ))| \\ \hspace{1.0em}\hspace{1.0em}+\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|+\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\right.\\ \hspace{1.0em}\hspace{1.0em}\left.-\underset{0}{\overset{r^{\prime} }{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{1}{\upsilon }^{2}\left(r^{\prime} )| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\hspace{1.0em}+\underset{r^{\prime} }{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left({\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ))+\underset{0}{\overset{r}{\displaystyle \int }}{\omega }_{1}^{L}({f}_{2},\varepsilon ){\rm{d}}s+u\left(r^{\prime} )\underset{r^{\prime} }{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left({\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ))+L{\omega }_{1}^{L}({f}_{2},\varepsilon )+\varepsilon \sup \left\{u\left(r^{\prime} )v\left(r):r,r^{\prime} \in \left[0,L]\right\},\end{array}where ω1L(f1,ε)≔sup{∣f1(r,υ1,υ2)−f1(r′,υ1,υ2)∣:r,r′∈[0,L],υ1,υ2∈[−r01,r01],∣r−r′∣≤ε},ω1L(f2,ε)≔sup{∣f2(r,s,υ1,υ2)−f2(r′,s,υ1,υ2)∣:r,r′,s∈[0,L],υ1,υ2∈[−r01,r01],∣r−r′∣≤ε}.\begin{array}{rcl}{\omega }_{1}^{L}({f}_{1},\varepsilon )& := & \sup \left\{| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1},{\upsilon }^{2})| :\hspace{1em}r,r^{\prime} \in \left[0,L],{\upsilon }^{1},{\upsilon }^{2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| r-r^{\prime} | \le \varepsilon \right\},\\ {\omega }_{1}^{L}({f}_{2},\varepsilon )& := & \sup \left\{| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})-{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1},{\upsilon }^{2})| :\hspace{1em}r,r^{\prime} ,s\in \left[0,L],{\upsilon }^{1},{\upsilon }^{2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| r-r^{\prime} | \le \varepsilon \right\}.\end{array}In addition, due to the uniform continuity of f1{f}_{1}on [0,L]×[−r01,r01]2\left[0,L]\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2}and f2{f}_{2}on [0,L]2×[−r01,r01]2{\left[0,L]}^{2}\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2}, we deduce that ω1L(f1,ε)→0{\omega }_{1}^{L}({f}_{1},\varepsilon )\to 0and ω1L(f2,ε)→0{\omega }_{1}^{L}({f}_{2},\varepsilon )\to 0as ε→0\varepsilon \to 0. Also, since u=u(r)u=u\left(r)and v=v(r)v=v\left(r)are continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}, we have sup{u(r′)v(r):r,r′∈[0,L]}<+∞.\sup \left\{u\left(r^{\prime} )v\left(r):r,r^{\prime} \in \left[0,L]\right\}\lt +\infty .Therefore, from (5.7), we obtain ω0L(FU1,FU2)≤limε→0φ1(ωL((U1,U2),ε)).{\omega }_{0}^{L}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\varphi }_{1}\left({\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon )).As a result, given the upper semicontinuity of the function φ1{\varphi }_{1}, we obtain ω0L(FU1,FU2)≤φ1(ω0L(U1,U2)),\hspace{-17.95em}{\omega }_{0}^{L}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left({\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),and eventually (5.8)ω0(FU1,FU2)≤φ1(ω0(U1,U2)).\hspace{-17.7em}{\omega }_{0}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left({\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})).Now, take arbitrary functions υ1,υ′1∈U1{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1}and υ2,υ′2∈U2{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}. Then, for r∈ℛ,r\in {\mathcal{ {\mathcal R} }},we obtain ∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤∣f1(r,υ1(r),υ2(r))−f1(r,υ′1(r),υ′2(r))∣+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds+∫0r∣f2(r,s,υ′1(s),υ′2(s))∣ds≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+2u(r)∫0rv(s)ds=φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+2a(r).\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1}\left(r),{\upsilon }^{^{\prime} 2}\left(r))| +\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+2u\left(r)\underset{0}{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ \hspace{1.0em}={\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+2a\left(r).\end{array}Hence, from the aforementioned inequality, we have diam(FU1,FU2)(r)≤φ1(diam(U1,U2)(r))+2a(r).\hspace{4.65em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}\left({\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r))+2a\left(r).As a result, given the upper semicontinuity of φ1{\varphi }_{1}, we obtain (5.9)lim supr→+∞diam(FU1,FU2)(r)≤φ1(lim supr→+∞diam(U1,U2)(r)).\hspace{1.25em}\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}(\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)).By combining (5.8) and (5.9) and considering the superadditivity of φ1{\varphi }_{1}, we obtain ω0(FU1,FU2)+lim supr→+∞diam(FU1,FU2)(r)≤φ1(ω0(U1,U2)+lim supr→+∞diam(U1,U2)(r)),{\omega }_{0}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}\left({\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)),or equivalently, (5.10)μ(FU1,FU2)≤φ1(μ(U1,U2)),\hspace{1.25em}\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),where μ\mu is coupled MNC in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}). So, from (5.10) and using Theorem 2.3, the result is obtained.□Example 5.2Let us define the functional integral equation as follows, which is a special mode of equation (5.1), (5.11)(υ1,υ2)(r)≔rr+1ln(1+∣υ1υ2(r)∣)+∫0res−1−rcosυ1υ2(s)1+∣sinυ1υ2(s)∣ds,(forr∈ℛ+).\left({\upsilon }^{1},{\upsilon }^{2})\left(r):= \frac{r}{r+1}\mathrm{ln}\left(1+| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+\underset{0}{\overset{r}{\int }}\frac{{e}^{s-1-r}\cos {\upsilon }^{1}{\upsilon }^{2}\left(s)}{1+| \sin {\upsilon }^{1}{\upsilon }^{2}\left(s)| }{\rm{d}}s,\hspace{0.33em}\left(\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}r\in {{\mathcal{ {\mathcal R} }}}_{+}).Here, f1(r,υ1,υ2)=rr+1ln(1+∣υ1υ2∣),f2(r,s,υ1,υ2)=es−1−rcosυ1υ21+∣sinυ1υ2∣.\begin{array}{rcl}{f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})& =& \frac{r}{r+1}\mathrm{ln}\left(1+| {\upsilon }^{1}{\upsilon }^{2}| ),\phantom{\rule[-1.25em]{}{0ex}}\\ {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})& =& \frac{{e}^{s-1-r}\cos {\upsilon }^{1}{\upsilon }^{2}}{1+| \sin {\upsilon }^{1}{\upsilon }^{2}| }.\end{array}In fact, if we take φ1(s)=ln(1+s),{\varphi }_{1}\left(s)=ln\left(1+s),we see that φ1(s)<s{\varphi }_{1}\left(s)\lt sfor s>0s\gt 0. Evidently, φ1{\varphi }_{1}is concave and increasing on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Moreover, for υ1,υ′1,υ2,υ′2∈ℛ{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {\mathcal{ {\mathcal R} }}with ∣υ1υ2∣≥∣υ′1υ′2∣| {\upsilon }^{1}{\upsilon }^{2}| \ge | {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| and for r>0,r\gt 0,we obtain ∣f1(r,υ1,υ2)−f1(r,υ′1,υ′2)∣=rr+1ln1+∣υ1υ2∣1+∣υ′1υ′2∣≤ln1+∣υ1υ2∣−∣υ′1υ′2∣1+∣υ′1υ′2∣<ln(1+(∣υ1υ2−υ′1υ′2∣))=φ1(∣υ1υ2−υ′1υ′2∣).\begin{array}{rcl}| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| & =& \frac{r}{r+1}\mathrm{ln}\frac{1+| {\upsilon }^{1}{\upsilon }^{2}| }{1+| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }\\ & \le & \mathrm{ln}\left(1+\frac{| {\upsilon }^{1}{\upsilon }^{2}| -| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }{1+| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }\right)\\ & \lt & \mathrm{ln}\left(1+\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ))\\ & =& {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ).\end{array}In the case ∣υ′1υ′2∣≥∣υ1υ2∣| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| \ge | {\upsilon }^{1}{\upsilon }^{2}| , the same can be done. Therefore, we conclude that the function f1{f}_{1}gives hypothesis (ii) and also (i). In addition, note that the function f2{f}_{2}operates continuously from ℛ+2×ℛ2→ℛ{{\mathcal{ {\mathcal R} }}}_{+}^{2}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}. Furthermore, ∣f2(r,s,υ1,υ2)∣≤es−1−r,r,s∈ℛ+,υ1,υ2∈ℛ.| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})| \le {e}^{s-1-r},\hspace{1.0em}r,s\in {{\mathcal{ {\mathcal R} }}}_{+},\hspace{0.33em}{\upsilon }^{1},{\upsilon }^{2}\in {\mathcal{ {\mathcal R} }}.Then, if u(r)≔e−1−ru\left(r):= {e}^{-1-r}, v(s)≔esv\left(s):= {e}^{s}, we see that hypothesis (iii) holds. In fact, limr→+∞u(r)∫0rv(s)ds=limr→+∞e−1−r∫0resds=0.\mathop{\mathrm{lim}}\limits_{r\to +\infty }u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s=\mathop{\mathrm{lim}}\limits_{r\to +\infty }{e}^{-1-r}\underset{0}{\overset{r}{\int }}{e}^{s}{\rm{d}}s=0.Now, we calculate ppaccording to assumption (iv). Then, p=sup{∣f1(r,0,0)∣+u(r)∫0rv(s)ds:r≥0}=sup{e−1:r≥0}=e−1.p=\sup \left\{| {f}_{1}\left(r,0,0)| +u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s:r\ge 0\right\}=\sup \left\{{e}^{-1}:r\ge 0\right\}={e}^{-1}.In addition, we consider the hypothesis inequality (iv), we have ln(1+r1)+p≤r1.\mathrm{ln}\left(1+{r}^{1})+p\le {r}^{1}.It can be easily seen that every r1≥1{r}^{1}\ge 1holds in the aforementioned inequality. Thus, as a number r01,{r}_{0}^{1},we can catch r01=1{r}_{0}^{1}=1. Therefore, we conclude that according to Theorem 5.1, equation (5.11) has at least one solution that is on the ball Br01=B1{B}_{{r}_{0}^{1}}={B}_{1}, in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Coupled measure of noncompactness and functional integral equations

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de Gruyter
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© 2022 Hasan Hosseinzadeh et al., published by De Gruyter
ISSN
2391-5455
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2391-5455
DOI
10.1515/math-2022-0015
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Abstract

1IntroductionIn nonlinear analysis, one of the most important tools is the concept of measure of noncompactness (MNC) to address the problems in functional operator equations. This important concept in mathematical sciences has been defined by many authors in various ways (see [1,2,3, 4,5,6, 7,8]). In [9], Aghajani et al. established some generalizations of Darbo’s fixed-point theorem and presented an application in functional integral equations.In this paper, we investigate the fixed-point results that generalize Darbo’s fixed-point theorem and many existing results in the literature by introducing the notion of coupled MNC. As an application, we prove the existence of solutions of a functional integral equation in Banach space BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}). Finally, an example is supplied to illustrate the results.Throughout this study, we consider EEas a Banach space and briefly represent a measure of noncompactness with MNC, B(υ,r)B\left(\upsilon ,r)represents a closed ball in Banach space EEto center υ\upsilon and radius rr. Also, we use Br{B}_{r}to represent B(θ,r)B\left(\theta ,r), where θ\theta is the zero element, the family of all nonempty bounded subsets of EEis represented with ℬE{{\mathcal{ {\mathcal B} }}}_{E}. To begin, we have the following preliminaries from [6,10,11].Definition 1.1[6]. Let μ:ℬE→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. The family ℬE{{\mathcal{ {\mathcal B} }}}_{E}is called MNC on Banach space EEif the following conditions hold: (1)For each U1∈ℬE{{\mathcal{U}}}^{1}\in {{\mathcal{ {\mathcal B} }}}_{E}, μ(U1)=θ\mu \left({{\mathcal{U}}}^{1})=\theta iff U1{{\mathcal{U}}}^{1}is a precompact set;(2)For each pair (U1,U2)∈ℬE×ℬE\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}\times {{\mathcal{ {\mathcal B} }}}_{E}, we have U1⊆U2impliesμ(U1)≤μ(U2);{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{2}\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1})\le \mu \left({{\mathcal{U}}}^{2});(3)For each U1∈ℬE{{\mathcal{U}}}^{1}\in {{\mathcal{ {\mathcal B} }}}_{E}, μ(U1)=μ(U1¯)=μ(convU1),\mu \left({{\mathcal{U}}}^{1})=\mu \left(\overline{{{\mathcal{U}}}^{1}})=\mu \left({\rm{conv}}\hspace{0.33em}{{\mathcal{U}}}^{1}),where U1¯\overline{{{\mathcal{U}}}^{1}}represents the closure of U1{{\mathcal{U}}}^{1}and convU1{\rm{conv}}\hspace{0.33em}{{\mathcal{U}}}^{1}represents the convex hull of U1{{\mathcal{U}}}^{1};(4)μ(λU1+(1−λ)U2)≤λμ(U1)+(1−λ)μ(U2)\mu \left(\lambda {{\mathcal{U}}}^{1}+\left(1-\lambda ){{\mathcal{U}}}^{2})\le \lambda \mu \left({{\mathcal{U}}}^{1})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{2})for λ∈[0,1]\lambda \in \left[0,1];(5)If {υn}0+∞∈ℬE{\left\{{\upsilon }_{n}\right\}}_{0}^{+\infty }\in {{\mathcal{ {\mathcal B} }}}_{E}is a decreasing sequence of closed sets and limn→+∞μ(υn)=0{\mathrm{lim}}_{n\to +\infty }\mu \left({\upsilon }_{n})=0, then U+∞1=⋂n=0+∞Un1≠∅{{\mathcal{U}}}_{+\infty }^{1}={\bigcap }_{n=0}^{+\infty }{{\mathcal{U}}}_{n}^{1}\ne \varnothing .In this part, we have the following theorems from [10,11,12].Theorem 1.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a compact and continuous operator. Then, FFhas at least one fixed-point.Theorem 1.3(Schauder) Let GGbe a nonempty, closed, and convex subset of a normed space and FFbe a continuous operator from GGinto a compact subset of GG. Then, FFhas a fixed-point.Theorem 1.4(Darbo) Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous operator. Suppose there is λ∈[0,1)\lambda \in {[}0,1)such that μ(FU)≤λμ(U)\mu \left(F{\mathcal{U}})\le \lambda \mu \left({\mathcal{U}})for each U∈G{\mathcal{U}}\in G. Then, FFhas a fixed-point.Theorem 1.5(Brouwer) Let GGbe a nonempty, compact, and convex subset of a finite-dimensional normed space and F:G→GF:G\to Gbe a continuous operator. Then, FFhas a fixed-point.Lemma 1.6[9] Let φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}be an upper semicontinuous and nondecreasing function. In this case, the following conditions are equivalent: (1)limn→+∞φ1n(r)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(r)=0for every r>0r\gt 0;(2)φ1(r)<r{\varphi }_{1}\left(r)\lt rfor every r>0r\gt 0.2Coupled MNCWe start this section with the following concept, and then, we turn to the main subject.Definition 2.1Let EEbe a Banach space and μ:ℬE2→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{2}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. We say that μ\mu is a coupled MNC on EE, if it has the following conditions: (1)kerμ={(U1,U2)∈ℬE2:μ(U1,U2)=θ}\ker \mu =\left\{\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}:\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\theta \right\}is nonempty;(2)For every (U1,U2)∈ℬE2,μ(U1,U2)=θ⇔(U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2},\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\theta \iff \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})is a precompact set;(3)For each ((U1,U2),(U′1,U′2))∈ℬE2×ℬE2\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}))\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}\times {{\mathcal{ {\mathcal B} }}}_{E}^{2}and (U1,U2)⊆(U′1,U′2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}), where U1⊆U′1{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{^{\prime} 1}and U2⊆U′2{{\mathcal{U}}}^{2}\subseteq {{\mathcal{U}}}^{^{\prime} 2}, we have (U1,U2)⊆(U′1,U′2)impliesμ(U1,U2)≤μ(U′1,U′2);\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2})\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\le \mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2});(4)For every (U1,U2)∈ℬE2\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\in {{\mathcal{ {\mathcal B} }}}_{E}^{2}, μ(U1¯,U2¯)=μ(U1,U2)=μ(conv(U1,U2)),\mu \left(\overline{{{\mathcal{U}}}^{1}},\overline{{{\mathcal{U}}}^{2}})=\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\mu \left({\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),where conv(U1,U2){\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})denotes the convex hull of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2});(5)μ(λ(U1,U2)+(1−λ)(U′1,U′2))≤λμ(U1,U2)+(1−λ)μ(U′1,U′2)\mu \left(\lambda \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\left(1-\lambda )\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2}))\le \lambda \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2})for λ∈[0,1]\lambda \in \left[0,1];(6)If {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }in ℬE{{\mathcal{ {\mathcal B} }}}_{E}are decreasing sequences of closed sets and limn→+∞μ{(Un1,Un2)}0+∞=0,then(U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)≠∅.\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu {\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{0}^{+\infty }=0,\hspace{1em}\hspace{0.1em}\text{then}\hspace{0.1em}\hspace{0.33em}\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})=\mathop{\bigcap }\limits_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\ne \varnothing .Theorem 2.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(2.1)φ2(μ(FU1,FU2))≤φ2(μ(U1,U2))−φ1(μ(U1,U2)),{\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2}))\le {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),for each ∅≠U1⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G, ∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where μ\mu is an arbitrary coupled MNC and φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.ProofTaking U01,U02=G{{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}=G, Un+11=conv(FUn1)¯,Un+12=conv(FUn2)¯{{\mathcal{U}}}_{n+1}^{1}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},{{\mathcal{U}}}_{n+1}^{2}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}, for n=0,1,2,…,n=0,1,2,\ldots ,we obtain Un+11⊆Un1,Un+12⊆Un2{{\mathcal{U}}}_{n+1}^{1}\hspace{0.25em}\subseteq {{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n+1}^{2}\subseteq {{\mathcal{U}}}_{n}^{2}for n=0,1,…n=0,1,\ldots . Therefore, {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }are decreasing sequences of closed and convex sets. Moreover, from (2.1), we have (2.2)φ2(μ(Un+11,Un+12))=φ2(μ(conv(FUn1)¯,conv(FUn2)¯))=φ2(μ(FUn1,FUn2))≤φ2(μ(Un1,Un2))−φ1(μ(Un1,Un2)),\begin{array}{rcl}{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))& =& {\varphi }_{2}\left(\mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}))\\ & =& {\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})),\end{array}for n=0,1,2,…n=0,1,2,\ldots . Since the sequence {μ(Un1,Un2)}\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nonnegative and nonincreasing, we deduce that μ(Un1,Un2)→m\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to mwhen nntends to infinity, where m≥0m\ge 0is a real number. On the other hand, considering equation (2.2), we obtain (2.3)lim supn→+∞φ2(μ(Un+11,Un+12))≤lim supn→+∞φ2(μ(Un1,Un2))−liminfn→+∞φ1(μ(Un1,Un2)).\mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))\le \mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }{\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))-\mathop{\mathrm{liminf}}\limits_{n\to +\infty }{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})).This yields φ2(m)≤φ2(m)−φ1(m){\varphi }_{2}\left(m)\le {\varphi }_{2}\left(m)-{\varphi }_{1}\left(m). Consequently, φ1(m)=0{\varphi }_{1}\left(m)=0and so m=0m=0. Therefore, we infer μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\hspace{-0.08em}\to \hspace{-0.08em}+\infty . Now, considering that (Un+11,Un+12)⊆(Un1,Un2)\left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})\hspace{-0.08em}\subseteq \hspace{-0.08em}\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}), by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is nonempty, closed, and convex. Furthermore, the set (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})under the operator FFis invariant and (U+∞1,U+∞2)∈kerμ\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})\in \ker \mu . So, by applying Theorem 1.2, the proof is complete.□Theorem 2.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Ga continuous map such that(2.4)μ(FU1,FU2)≤φ1(μ(U1,U2)),\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),for each ∅≠U1⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G, ∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where μ\mu is an arbitrary coupled MNC and φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for every s≥0s\ge 0. Then, FFhas at least one fixed-point.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction, where U01,U02=G{{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}=G, Un+11=conv(FUn1)¯,Un+12=conv(FUn2)¯{{\mathcal{U}}}_{n+1}^{1}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},{{\mathcal{U}}}_{n+1}^{2}=\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})}, for n=0,1,…n=0,1,\ldots . Moreover, in the same as the previous method, we can assume μ(Un1,Un2)>0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\gt 0for all n=1,2,…n=1,2,\ldots . In addition, by given assumptions, we obtain (2.5)μ(Un+11,Un+12)=μ(conv(FUn1)¯,conv(FUn2)¯)=μ(FUn1,FUn2)≤φ1(μ(Un1,Un2))≤φ12(μ(Un−11,Un−12))⋮≤φ1n+1(μ(U01,U02)).\begin{array}{rcl}\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})& =& \mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})})\\ & =& \mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2})\\ & \le & {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{1}^{2}\left(\mu \left({{\mathcal{U}}}_{n-1}^{1},{{\mathcal{U}}}_{n-1}^{2}))\\ & \vdots & \\ & \le & {\varphi }_{1}^{n+1}\left(\mu \left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2})).\end{array}This shows that μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\to +\infty . Since the sequence {(Un1,Un2)}\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nested, by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is a nonempty, closed, and convex subset of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}). Therefore, we obtain that (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is a member of kerμ\ker \mu . So, (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is compact. Next, note that FFmaps (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})into itself, and considering Theorem 1.2, we deduce that FFhas fixed-point in (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2}). So the proof is complete.□Now, from the aforementioned theorem, we have the following.Corollary 2.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe an operator such that(2.6)‖(Fυ1,Fυ2)−(Fυ′1,Fυ′2)‖≤φ1(‖υ1υ2−υ′1υ′2‖),forallυ1,υ′1,υ2,υ′2∈G,\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert ),\hspace{0.33em}{for}\hspace{0.33em}{all}\hspace{0.33em}{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in G,where φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for any s≥0s\ge 0. Then, FFhas a fixed-point in GG.ProofLet μ:ℬE2→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{2}\to {{\mathcal{ {\mathcal R} }}}_{+}and μ(U1,U2)≔diam(U1,U2),\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}):= {\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),where diam(U1,U2)=sup{‖υ1υ2−υ′1υ′2‖:υ1,υ′1∈U1,υ2,υ′2∈U2}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})=\sup \left\{\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert :{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}\right\}. It can be easily seen that μ\mu is coupled MNC in EEby Definition 2.1. Furthermore, since φ1{\varphi }_{1}is nondecreasing, then in view of (2.6), we have supυ1,υ′1∈U1,υ2,υ′2∈U2‖(Fυ1,Fυ2)−(Fυ′1,Fυ′2)‖≤supυ1,υ′1∈U1,υ2,υ′2∈U2φ1‖υ1υ2−υ′1υ′2‖≤φ1(supυ1,υ′1∈U1,υ2,υ′2∈U2‖υ1υ2−υ′1υ′2‖),\begin{array}{rcl}\mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\Vert & \le & \mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}{\varphi }_{1}\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \\ & \le & {\varphi }_{1}\left(\mathop{\sup }\limits_{{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}}\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \right),\end{array}which yields that μ(FU1,FU2)≤φ1(μ(U1,U2)).\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})).By using Theorem 2.3, the proof is complete.□3Tripled MNCIn this section, as a result of Section 2, we define the notion of tripled MNC as follows.Definition 3.1Let EEbe a Banach space and μ:ℬE3→ℛ+\mu :{{\mathcal{ {\mathcal B} }}}_{E}^{3}\to {{\mathcal{ {\mathcal R} }}}_{+}be a mapping. We say that μ\mu is a tripled MNC on E,E,if it has the following conditions: (1)kerμ={(U1,U2,U3)∈ℬE3:μ(U1,U2,U3)=θ}\ker \mu =\left\{\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}:\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\theta \right\}is nonempty;(2)For every (U1,U2,U3)∈ℬE3,μ(U1,U2,U3)=θ⇔(U1,U2,U3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3},\hspace{1em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\theta \iff \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})is a precompact set;(3)For each ((U1,U2,U3),(U′1,U′2,U′3))∈ℬE3×ℬE3\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}),\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3}))\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}\times {{\mathcal{ {\mathcal B} }}}_{E}^{3}, ((U1,U2,U3)⊆(U′1,U′2,U′3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})yields U1⊆U′1,U2⊆U′2{{\mathcal{U}}}^{1}\subseteq {{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{2}\subseteq {{\mathcal{U}}}^{^{\prime} 2}and U3⊆U′3{{\mathcal{U}}}^{3}\subseteq {{\mathcal{U}}}^{^{\prime} 3}), we have (U1,U2,U3)⊆(U′1,U′2,U′3)impliesμ(U1,U2,U3)≤μ(U′1,U′2,U′3);\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\subseteq \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})\hspace{0.33em}\hspace{0.1em}\text{implies}\hspace{0.1em}\hspace{0.33em}\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\le \mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3});(4)For every (U1,U2,U3)∈ℬE3\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})\in {{\mathcal{ {\mathcal B} }}}_{E}^{3}, one has μ(U1¯,U2¯,U3¯)=μ(U1,U2,U3)=μ(conv(U1,U2,U3)),\mu \left(\overline{{{\mathcal{U}}}^{1}},\overline{{{\mathcal{U}}}^{2}},\overline{{{\mathcal{U}}}^{3}})=\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})=\mu \left({\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),where conv(U1,U2,U3){\rm{conv}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})denotes the convex hull of (U1,U2,U3)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3});(5)μ(λ(U1,U2,U3)+(1−λ)(U′1,U′2,U′3))≤λμ(U1,U2,U3)+(1−λ)μ(U′1,U′2,U′3)\mu \left(\lambda \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})+\left(1-\lambda )\left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3}))\le \lambda \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})+\left(1-\lambda )\mu \left({{\mathcal{U}}}^{^{\prime} 1},{{\mathcal{U}}}^{^{\prime} 2},{{\mathcal{U}}}^{^{\prime} 3})for λ∈[0,1]\lambda \in \left[0,1];(6)If {Un1}0+∞,{Un2}0+∞{\left\{{{\mathcal{U}}}_{n}^{1}\right\}}_{0}^{+\infty },{\left\{{{\mathcal{U}}}_{n}^{2}\right\}}_{0}^{+\infty }and {Un3}0+∞{\left\{{{\mathcal{U}}}_{n}^{3}\right\}}_{0}^{+\infty }in ℬE{{\mathcal{ {\mathcal B} }}}_{E}are decreasing sequences of closed sets and limn→+∞μ{(Un1,Un2,Un3)}0+∞=0{\mathrm{lim}}_{n\to +\infty }\mu {\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2},{{\mathcal{U}}}_{n}^{3})\right\}}_{0}^{+\infty }=0, then (U+∞1,U+∞2,U+∞3)=⋂n=0+∞(Un1,Un2,Un3)≠∅\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2},{{\mathcal{U}}}_{+\infty }^{3})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2},{{\mathcal{U}}}_{n}^{3})\ne \varnothing .Theorem 3.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(3.1)φ2(μ(FU1,FU2,FU3))≤φ2(μ(U1,U2,U3))−φ1(μ(U1,U2,U3)),{\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3}))\le {\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))-{\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where μ\mu is an arbitrary tripled MNC and φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.Theorem 3.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(3.2)μ(FU1,FU2,FU3)≤φ1(μ(U1,U2,U3)),\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where μ\mu is an arbitrary tripled MNC and φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(s)=0for every s≥0s\ge 0. Then, FFhas at least one fixed-point.Corollary 3.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe an operator such that(3.3)‖(Fυ1,Fυ2,Fυ3)−(Fυ′1,Fυ′2,Fυ′3)‖≤φ1(‖υ1υ2υ3−υ′1υ′2υ′3‖),\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2},F{\upsilon }^{3})-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2},F{\upsilon }^{^{\prime} 3})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}{\upsilon }^{3}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}{\upsilon }^{^{\prime} 3}\Vert ),for all υ1,υ′1,υ2,υ′2,υ3,υ′3∈G{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2},{\upsilon }^{3},{\upsilon }^{^{\prime} 3}\in G, where φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}is a nondecreasing function with limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{1}^{n}\left(s)=0for any s≥0s\ge 0. Then, FFhas a fixed-point in GG.4MNC and JS-contractionIn this section, we tend to prove some results of MNC for the family of JS-contractive-type mappings. Also, we generalize Darbo’s fixed-point theorem to coupled and tripled MNC through JS-contraction-type mappings.Denote by Θ\Theta the set of all functions θ:(0,+∞)→(1,+∞)\theta :\left(0,+\infty )\to \left(1,+\infty )so that: (θ1)θ\theta is continuous and increasing;(θ2)limn→+∞tn=0{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{t}_{n}=0iff limn→+∞θ(tn)=1{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}\theta \left({t}_{n})=1for all {tn}⊆(0,+∞)\left\{{t}_{n}\right\}\subseteq \left(0,+\infty ).Theorem 4.1[13] Let (G,ϱ)\left(G,\varrho )be a complete metric space and F:G→GF:G\to Gbe a given mapping. Suppose that there exist θ∈Θ\theta \in \Theta and ν∈(0,1)\nu \in \left(0,1)such that for all ι,ς∈G\iota ,\varsigma \in G, (4.1)ϱ(Fι,Fς)≠0⇒θ(ϱ(Fι,Fς))≤(θ(ϱ(ι,ς)))ν.\varrho \left(F\iota ,F\varsigma )\ne 0\Rightarrow \theta \left(\varrho \left(F\iota ,F\varsigma ))\le {\left(\theta \left(\varrho \left(\iota ,\varsigma )))}^{\nu }.Then, FFhas a unique fixed-point.Theorem 4.2Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to G, be a continuous map such that(4.2)θ(φ2(μ(FU1,FU2)))≤θ(φ2(μ(U1,U2)))θ(φ2(φ1(μ(U1,U2)))),\hspace{-24.4em}\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))))},for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where θ∈Θ\theta \in \Theta , and μ\mu is an arbitrary coupled MNC and functions φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}, such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction. Moreover, from (4.2), we obtain (4.3)θ(φ2(μ(Un+11,Un+12)))=θ(φ2(μ(conv(FUn1)¯,conv(FUn2)¯)))=θ(φ2(μ(FUn1,FUn2)))≤θ(φ2(μ(Un1,Un2)))θ(φ2(φ1(μ(Un1,Un2)))),\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})))=\theta \left({\varphi }_{2}\left(\mu \left(\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{1})},\overline{{\rm{conv}}\left(F{{\mathcal{U}}}_{n}^{2})})))=\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))))},for n=0,1,2,…n=0,1,2,\ldots . Since the sequence {μ(Un1,Un2)}\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}is nonnegative and nonincreasing, we deduce that μ(Un1,Un2)→m\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to mwhen nntends to infinity, where m≥0m\ge 0is a real number. On the other hand, considering equation (4.3), we obtain (4.4)lim supn→+∞θ(φ2(μ(Un+11,Un+12)))≤lim supn→+∞θ(φ2(μ(Un1,Un2)))θ(φ2(φ1(μ(Un1,Un2)))),\mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})))\le \mathop{\mathrm{lim\; sup}}\limits_{n\to +\infty }\frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}))))},which yields that θ(φ2(m))≤θ(φ2(m))θ(φ2(φ1(m)))\theta \left({\varphi }_{2}\left(m))\le \frac{\theta \left({\varphi }_{2}\left(m))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(m)))}. Consequently, θ(φ2(φ1(m)))=1\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(m)))=1, then φ2(φ1(m))=0{\varphi }_{2}\left({\varphi }_{1}\left(m))=0and φ1(m)=0{\varphi }_{1}\left(m)=0so m=0m=0. Therefore, we infer μ(Un1,Un2)→0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\to 0as n→+∞n\to +\infty . Now, considering that (Un+11,Un+12)⊆(Un1,Un2)\left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})\subseteq \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2}), by Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is nonempty, closed, and convex. Furthermore, the set (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})under the operator FFis invariant and (U+∞1,U+∞2)∈kerμ\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})\in ker\mu . So, by applying Theorem 1.2, the proof is complete.□Denote by Ψ\Psi the set of all functions φ2:(1,+∞)→(1,+∞){\varphi }_{2}:\left(1,+\infty )\to \left(1,+\infty )so that: (φ21)φ2{\varphi }_{2}is continuous and increasing;(φ22)limn→+∞φ2n(s)=1{\mathrm{lim}}_{n\to +\infty }\hspace{0.25em}{\varphi }_{2}^{n}\left(s)=1for all s∈(1,+∞)s\in \left(1,+\infty ).Theorem 4.3Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Ga continuous map such that(4.5)θ(μ(FU1,FU2))≤φ2(θ(μ(U1,U2))),\theta \left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2}))\le {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}))),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, where θ∈Θ\theta \in \Theta , φ2∈Ψ{\varphi }_{2}\in \Psi , and μ\mu is an arbitrary coupled MNC. Then, FFhas at least one fixed-point.ProofAccording to the proof of Theorem 2.2, we define the sequences {Un1},{Un2}\left\{{{\mathcal{U}}}_{n}^{1}\right\},\left\{{{\mathcal{U}}}_{n}^{2}\right\}by induction.If for an integer N∈NN\in {\mathbb{N}}one has μ(UN1,UN2)=0\mu \left({{\mathcal{U}}}_{N}^{1},{{\mathcal{U}}}_{N}^{2})=0, then (UN1,UN2)\left({{\mathcal{U}}}_{N}^{1},{{\mathcal{U}}}_{N}^{2})is a precompact set. So the Schauder theorem ensures the existence of a fixed-point for FF. Therefore, we can assume μ(Un1,Un2)>0\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\gt 0for all n∈N∪{0}n\in {\mathbb{N}}\cup \left\{0\right\}.Obviously, {(Un1,Un2)}n∈N{\left\{\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{n\in {\mathbb{N}}}is a sequence of nonempty, bounded, closed, and convex subsets such that (U01,U02)⊇(U11,U12)⊇⋯⊇(Un1,Un2)⊇(Un+11,Un+12).\left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2})\supseteq \left({{\mathcal{U}}}_{1}^{1},{{\mathcal{U}}}_{1}^{2})\supseteq \cdots \supseteq \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\supseteq \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}).On the other hand, (4.6)θ(μ(Un+11,Un+12))=θ(μ(FUn1,FUn2))≤φ2(θ(μ(Un1,Un2)))≤⋮≤φ2n+1(θ(μ(U01,U02))).\begin{array}{rcl}\theta \left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))& =& \theta \left(\mu \left(F{{\mathcal{U}}}_{n}^{1},F{{\mathcal{U}}}_{n}^{2}))\\ & \le & {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})))\\ & \le & \\ & \vdots & \\ & \le & {\varphi }_{2}^{n+1}\left(\theta \left(\mu \left({{\mathcal{U}}}_{0}^{1},{{\mathcal{U}}}_{0}^{2}))).\end{array}Thus, {μ(Un1,Un2)}n∈N{\left\{\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})\right\}}_{n\in {\mathbb{N}}}is a convergent sequence. Assume that limn→+∞μ(Un1,Un2)=η.\hspace{-10.5em}\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu \left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})=\eta .By taking the limit from (4.6), limn→+∞θ(μ(Un+11,Un+12))=1{\mathrm{lim}}_{n\to +\infty }\theta \left(\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2}))=1. By (θ2),\left({\theta }_{2}),we obtain limn→+∞μ(Un+11,Un+12)=0.\mathop{\mathrm{lim}}\limits_{n\to +\infty }\mu \left({{\mathcal{U}}}_{n+1}^{1},{{\mathcal{U}}}_{n+1}^{2})=0.From Definition 2.1 (6), (U+∞1,U+∞2)=⋂n=0+∞(Un1,Un2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})={\bigcap }_{n=0}^{+\infty }\left({{\mathcal{U}}}_{n}^{1},{{\mathcal{U}}}_{n}^{2})is a nonempty, closed, and convex subset of (U1,U2)\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}). Therefore, we obtain (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is a member of kerμ\ker \mu . So, (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})is compact. Note that FFmaps (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2})into itself, and considering Theorem 1.2, we deduce that FFhas a fixed-point in (U+∞1,U+∞2)\left({{\mathcal{U}}}_{+\infty }^{1},{{\mathcal{U}}}_{+\infty }^{2}). So the proof is complete.□Theorem 4.4Let GGbe a nonempty, bounded, closed, and convex subset of a Banach space EEand F:G→GF:G\to Gbe a continuous map such that(4.7)θ(φ2(μ(FU1,FU2,FU3)))≤θ(φ2(μ(U1,U2,U3)))θ(φ2(φ1(μ(U1,U2,U3)))),\theta \left({\varphi }_{2}\left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3})))\le \frac{\theta \left({\varphi }_{2}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3})))}{\theta \left({\varphi }_{2}\left({\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))))},for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where θ∈Θ\theta \in \Theta and μ\mu is an arbitrary tripled MNC and functions φ1,φ2:ℛ+→ℛ+{\varphi }_{1},{\varphi }_{2}:{ {\mathcal R} }_{+}\to { {\mathcal R} }_{+}, such that φ2{\varphi }_{2}is continuous and φ1{\varphi }_{1}is lower semicontinuous on ℛ+{ {\mathcal R} }_{+}. Furthermore, φ1(0)=0{\varphi }_{1}\left(0)=0and φ1(s)>0{\varphi }_{1}\left(s)\gt 0for s>0s\gt 0. Then, FFhas at least one fixed-point in GG.Theorem 4.5Let G be a nonempty, bounded, closed, and convex subset of a Banach space E and F:G→GF:G\to Gbe a continuous map such that(4.8)θ(μ(FU1,FU2,FU3))≤φ2(θ(μ(U1,U2,U3))),\theta \left(\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2},F{{\mathcal{U}}}^{3}))\le {\varphi }_{2}\left(\theta \left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2},{{\mathcal{U}}}^{3}))),for each ∅≠U1⊆G,∅≠U2⊆G\varnothing \ne {{\mathcal{U}}}^{1}\subseteq G,\varnothing \ne {{\mathcal{U}}}^{2}\subseteq G, and ∅≠U3⊆G\varnothing \ne {{\mathcal{U}}}^{3}\subseteq G, where θ∈Θ\theta \in \Theta , φ2Ψ∈{\varphi }_{2}\Psi \in , and μ\mu is an arbitrary tripled MNC. Then, FFhas at least one fixed-point.5ApplicationWe offer the applications of Theorem 2.3 to prove the existence of solutions of a functional integral equation in the Banach space BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})consisting of all real functions that are bounded and continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. This space is endowed with the supremum norm ‖υ1‖=sup{‖υ1(r)‖:r∈ℛ+}.\Vert {\upsilon }^{1}\Vert =\sup \left\{\Vert {\upsilon }^{1}\left(r)\Vert :r\in {{\mathcal{ {\mathcal R} }}}_{+}\right\}.We choose nonempty bounded subsets U1,U2{{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}of BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})and a number L>0L\gt 0. For ε>0\varepsilon \gt 0, υ1∈U1{\upsilon }^{1}\in {{\mathcal{U}}}^{1}, and υ2∈U2{\upsilon }^{2}\in {{\mathcal{U}}}^{2}, we show the modulus of continuity of function (υ1,υ2)\left({\upsilon }^{1},{\upsilon }^{2})on the interval [0,L]\left[0,L]by ωL((υ1,υ2),ε){\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ): ωL((υ1,υ2),ε)=sup{∣υ1υ2(r)−υ1υ2(s)∣:r,s∈[0,L],∣r−s∣≤ε}.{\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon )=\sup \left\{| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{1}{\upsilon }^{2}\left(s)| :r,s\in \left[0,L],| r-s| \le \varepsilon \right\}.Also, ωL((U1,U2),ε)=sup{ωL((υ1,υ2),ε):υ1∈U1,υ2∈U2},ω0L(U1,U2)=limε→0ωL((U1,U2),ε),ω0(U1,U2)=limL→+∞ω0L(U1,U2).\begin{array}{rcl}{\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon )& =& \sup \left\{{\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ):{\upsilon }^{1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2}\in {{\mathcal{U}}}^{2}\right\},\\ {\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon ),\\ {\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& \mathop{\mathrm{lim}}\limits_{L\to +\infty }{\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}).\end{array}Moreover, for r∈ℛ+r\in {{\mathcal{ {\mathcal R} }}}_{+}, we define (U1,U2)(r)={(υ1,υ2)(r):υ1∈U1,υ2∈U2},μ(U1,U2)=ω0(U1,U2)+lim supr→+∞diam(U1,U2)(r),diam(U1,U2)(r)=sup{∣υ1υ2(r)−υ′1υ′2(r)∣:υ1,υ′1∈U1,υ2,υ′2∈U2}.\begin{array}{rcl}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)& =& \left\{\left({\upsilon }^{1},{\upsilon }^{2})\left(r):{\upsilon }^{1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2}\in {{\mathcal{U}}}^{2}\right\},\\ \mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})& =& {\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r),\\ {\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)& =& \sup \left\{| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| :{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}\right\}.\end{array}Now, we consider the following hypotheses: (i)Let f1:ℛ+×ℛ2→ℛ{f}_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}be a continuous function. Furthermore, the function r→f1(r,0,0)r\to {f}_{1}\left(r,0,0)is a member of BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}).(ii)There is an upper semicontinuous function φ1∈ϕ{\varphi }_{1}\in \phi , where ϕ\phi is family of all functions φ1:ℛ+→ℛ+{\varphi }_{1}:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}, which φ1{\varphi }_{1}is a nondecreasing function such that limn→+∞φ1n(s)=0{\mathrm{lim}}_{n\to +\infty }{\varphi }_{1}^{n}\left(s)=0for each s≥0s\ge 0, such that ∣f1(r,υ1,υ2)−f1(r,υ′1,υ′2)∣≤φ1(∣υ1υ2−υ′1υ′2∣),r∈ℛ+υ1,υ′1,υ2,υ′2∈ℛ.| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| \le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ),\hspace{1em}r\in {{\mathcal{ {\mathcal R} }}}_{+}\hspace{1em}{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {\mathcal{ {\mathcal R} }}.In addition, we presume that φ1(s)+φ1(s′)≤φ1(s+s′){\varphi }_{1}\left(s)+{\varphi }_{1}\left(s^{\prime} )\le {\varphi }_{1}\left(s+s^{\prime} )for all s,s′∈ℛ+s,s^{\prime} \in {{\mathcal{ {\mathcal R} }}}_{+}.(iii)Let f2:ℛ+2×ℛ2→ℛ{f}_{2}:{{\mathcal{ {\mathcal R} }}}_{+}^{2}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}be a continuous function, and there are continuous functions u,v:ℛ+→ℛ+u,v:{{\mathcal{ {\mathcal R} }}}_{+}\to {{\mathcal{ {\mathcal R} }}}_{+}such that limr→+∞u(r)∫0rv(s)ds=0\mathop{\mathrm{lim}}\limits_{r\to +\infty }u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s=0and ∣f2(r,s,υ1,υ2)∣≤u(r)v(s)| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})| \le u\left(r)v\left(s)for r,s∈ℛ+r,s\in {{\mathcal{ {\mathcal R} }}}_{+}such that s≤rs\le rand for every υ1,υ2∈ℛ{\upsilon }^{1},{\upsilon }^{2}\in {\mathcal{ {\mathcal R} }}.(iv)There is a positive solution r01{r}_{0}^{1}from φ1(r1)+p≤r1,{\varphi }_{1}\left({r}^{1})+p\le {r}^{1},where p=sup{∣f1(r,0,0)+u(r)∫0rv(s)ds:r≥0}.p=\sup \left\{| {f}_{1}\left(r,0,0)+u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s:r\ge 0\right\}.Now, we consider the integral equation (5.1)(υ1,υ2)(r)=f1(r,υ1(r),υ2(r))+∫0rf2(r,s,υ1(s),υ2(s))ds.\left({\upsilon }^{1},{\upsilon }^{2})\left(r)={f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))+\underset{0}{\overset{r}{\int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s.We define operator FFon BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+})by (5.2)(Fυ1,Fυ2)(r)=f1(r,υ1(r),υ2(r))+∫0rf2(r,s,υ1(s),υ2(s))ds,forr∈ℛ+,\left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)={f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))+\underset{0}{\overset{r}{\int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s,\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}r\in {{\mathcal{ {\mathcal R} }}}_{+},where the function (Fυ1,Fυ2)\left(F{\upsilon }^{1},F{\upsilon }^{2})is continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}.Theorem 5.1According to hypotheses (i)–(iv), relation (5.1) has at least one solution in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}).ProofFor arbitrary υ1,υ2∈BC(ℛ+){\upsilon }^{1},{\upsilon }^{2}\in BC\left({{\mathcal{ {\mathcal R} }}}_{+}), using the aforementioned hypotheses, we obtain ∣(Fυ1,Fυ2)(r)∣≤∣f1(r,υ1(r),υ2(r))−f1(r,0,0)∣+∣f1(r,0,0)∣+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds≤φ1(∣υ1υ2(r)∣)+∣f1(r,0,0)∣+u(r)∫0rv(s)ds=φ1(∣υ1υ2(r)∣)+∣f1(r,0,0)∣+a(r),\begin{array}{rcl}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)| & \le & | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r,0,0)| +| {f}_{1}\left(r,0,0)| +\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ & \le & {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+| {f}_{1}\left(r,0,0)| +u\left(r)\underset{0}{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ & =& {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+| {f}_{1}\left(r,0,0)| +a\left(r),\end{array}where a(r)≔u(r)∫0rv(s)ds.\hspace{-12.25em}a\left(r):= u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s.Since φ1{\varphi }_{1}is nondecreasing, in accordance with the fourth condition, we obtain ‖(Fυ1,Fυ2)‖≤φ1(‖υ1υ2‖)+p.\Vert \left(F{\upsilon }^{1},F{\upsilon }^{2})\Vert \le {\varphi }_{1}\left(\Vert {\upsilon }^{1}{\upsilon }^{2}\Vert )+p.Thus, FFis a self-mapping of BC(ℛ+)BC\left({{\mathcal{ {\mathcal R} }}}_{+}). On the other hand, applying assumption (iv), we deduce that FFis a self-mapping of the ball Br01{B}_{{r}_{0}^{1}}. To show that FFis continuous on Br01{B}_{{r}_{0}^{1}}, take ε>0\varepsilon \gt 0and υ1,υ′1,υ2,υ′2∈Br01{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {B}_{{r}_{0}^{1}}such that ‖υ1υ2−υ′1υ′2‖<ε\Vert {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\Vert \lt \varepsilon , we obtain (5.3)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))−f2(r,s,υ′1(s),υ′2(s))∣ds\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \\ \hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))-{f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\end{array}≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds+∫0r∣f2(r,s,υ′1(s),υ′2(s))∣ds≤φ1(ε)+2a(r),\begin{array}{l}\hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\varphi }_{1}\left(\varepsilon )+2a\left(r),\end{array}for any r∈ℛ+r\in {{\mathcal{ {\mathcal R} }}}_{+}. By assumption (iii), there is a number L>0L\gt 0such that (5.4)2u(r)∫0rv(s)ds≤ε,for everyL≤r.2u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s\le \varepsilon ,\hspace{0.33em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}L\le r.So, considering Lemma 1.6 and the similar evaluation mentioned earlier, for an arbitrary L≤r,L\le r,we have (5.5)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤2ε.| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \le 2\varepsilon .Now, we define ωL(f2,ε)≔sup{∣f2(r,s,υ1,υ2)−f2(r,s,υ′1,υ′2)∣:r,s∈[0,L],υ1,υ′1,υ2,υ′2∈[−r01,r01],∣υ1υ2−υ′1υ′2∣≤ε}.{\omega }^{L}({f}_{2},\varepsilon ):= \sup \left\{| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})-{f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| :r,s\in \left[0,L],{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| \le \varepsilon \right\}.Due to the uniform continuity of f2(r,s,υ1,υ2){f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})on [0,L]2×[−r01,r01]2,{\left[0,L]}^{2}\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2},we infer that ωL(f2,ε)→0{\omega }^{L}({f}_{2},\varepsilon )\to 0as ε→0\varepsilon \to 0. Now, considering the first part of equation (5.3), for the arbitrary constant r∈[0,L],r\in \left[0,L],we obtain (5.6)∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤φ1(ε)+∫0LωL(f2,ε)ds=φ1(ε)+LωL(f2,ε).| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \le {\varphi }_{1}\left(\varepsilon )+\underset{0}{\overset{L}{\int }}{\omega }^{L}({f}_{2},\varepsilon ){\rm{d}}s={\varphi }_{1}\left(\varepsilon )+L{\omega }^{L}({f}_{2},\varepsilon ).By combining (5.5) and (5.6) and based on the aforementioned fact about ωL(f2,ε){\omega }^{L}({f}_{2},\varepsilon ), the operator FFon the ball Br01{B}_{{r}_{0}^{1}}, is continuous. Next, we can choose arbitrary nonempty subsets U1,U2{{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}of the ball Br01{B}_{{r}_{0}^{1}}. To do this, we consider constant numbers L>0L\gt 0and ε>0\varepsilon \gt 0. Also, take arbitrary numbers r,r′∈[0,L]r,r^{\prime} \in \left[0,L]with ∣r−r′∣≤ε| r-r^{\prime} | \le \varepsilon . Without loss of generality, it can be assumed that r′<rr^{\prime} \lt r. So, for υ1∈U1{\upsilon }^{1}\in {{\mathcal{U}}}^{1}and υ2∈U2,{\upsilon }^{2}\in {{\mathcal{U}}}^{2},we obtain (5.7)∣(Fυ1,Fυ2)(r)−(Fυ1,Fυ2)(r′)∣≤∣f1(r,υ1(r),υ2(r))−f1(r′,υ1(r′),υ2(r′))∣+∫0rf2(r,s,υ1(s),υ2(s))ds−∫0r′f2(r′,s,υ1(s),υ2(s))ds≤∣f1(r,υ1(r),υ2(r))−f1(r′,υ1(r),υ2(r))∣+∣f1(r′,υ1(r),υ2(r))−f1(r′,υ1(r′),υ2(r′))∣+∫0rf2(r,s,υ1(s),υ2(s))ds−∫0rf2(r′,s,υ1(s),υ2(s))ds+∫0rf2(r′,s,υ1(s),υ2(s))ds−∫0r′f2(r′,s,υ1(s),υ2(s))ds≤ω1L(f1,ε)+φ1(∣υ1υ2(r)−υ1υ2(r′)∣)+∫0r∣f2(r,s,υ1(s),υ2(s))ds−f2(r′,s,υ1(s),υ2(s))∣ds+∫r′r∣f2(r′,s,υ1(s),υ2(s))∣ds≤ω1L(f1,ε)+φ1(ωL((υ1,υ2),ε))+∫0rω1L(f2,ε)ds+u(r′)∫r′rv(s)ds≤ω1L(f1,ε)+φ1(ωL((υ1,υ2),ε))+Lω1L(f2,ε)+εsup{u(r′)v(r):r,r′∈[0,L]},\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r^{\prime} )| \\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r^{\prime} ),{\upsilon }^{2}\left(r^{\prime} ))| +\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-\underset{0}{\overset{r^{\prime} }{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|\\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))| +| {f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1}\left(r^{\prime} ),{\upsilon }^{2}\left(r^{\prime} ))| \\ \hspace{1.0em}\hspace{1.0em}+\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|+\left|\hspace{-0.25em}\underset{0}{\overset{r}{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\right.\\ \hspace{1.0em}\hspace{1.0em}\left.-\underset{0}{\overset{r^{\prime} }{\displaystyle \int }}{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s\hspace{-0.25em}\right|\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{1}{\upsilon }^{2}\left(r^{\prime} )| )+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s)){\rm{d}}s-{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\hspace{1.0em}+\underset{r^{\prime} }{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left({\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ))+\underset{0}{\overset{r}{\displaystyle \int }}{\omega }_{1}^{L}({f}_{2},\varepsilon ){\rm{d}}s+u\left(r^{\prime} )\underset{r^{\prime} }{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ \hspace{1.0em}\le {\omega }_{1}^{L}({f}_{1},\varepsilon )+{\varphi }_{1}\left({\omega }^{L}\left(\left({\upsilon }^{1},{\upsilon }^{2}),\varepsilon ))+L{\omega }_{1}^{L}({f}_{2},\varepsilon )+\varepsilon \sup \left\{u\left(r^{\prime} )v\left(r):r,r^{\prime} \in \left[0,L]\right\},\end{array}where ω1L(f1,ε)≔sup{∣f1(r,υ1,υ2)−f1(r′,υ1,υ2)∣:r,r′∈[0,L],υ1,υ2∈[−r01,r01],∣r−r′∣≤ε},ω1L(f2,ε)≔sup{∣f2(r,s,υ1,υ2)−f2(r′,s,υ1,υ2)∣:r,r′,s∈[0,L],υ1,υ2∈[−r01,r01],∣r−r′∣≤ε}.\begin{array}{rcl}{\omega }_{1}^{L}({f}_{1},\varepsilon )& := & \sup \left\{| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r^{\prime} ,{\upsilon }^{1},{\upsilon }^{2})| :\hspace{1em}r,r^{\prime} \in \left[0,L],{\upsilon }^{1},{\upsilon }^{2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| r-r^{\prime} | \le \varepsilon \right\},\\ {\omega }_{1}^{L}({f}_{2},\varepsilon )& := & \sup \left\{| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})-{f}_{2}\left(r^{\prime} ,s,{\upsilon }^{1},{\upsilon }^{2})| :\hspace{1em}r,r^{\prime} ,s\in \left[0,L],{\upsilon }^{1},{\upsilon }^{2}\in \left[-{r}_{0}^{1},{r}_{0}^{1}],| r-r^{\prime} | \le \varepsilon \right\}.\end{array}In addition, due to the uniform continuity of f1{f}_{1}on [0,L]×[−r01,r01]2\left[0,L]\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2}and f2{f}_{2}on [0,L]2×[−r01,r01]2{\left[0,L]}^{2}\times {\left[-{r}_{0}^{1},{r}_{0}^{1}]}^{2}, we deduce that ω1L(f1,ε)→0{\omega }_{1}^{L}({f}_{1},\varepsilon )\to 0and ω1L(f2,ε)→0{\omega }_{1}^{L}({f}_{2},\varepsilon )\to 0as ε→0\varepsilon \to 0. Also, since u=u(r)u=u\left(r)and v=v(r)v=v\left(r)are continuous on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}, we have sup{u(r′)v(r):r,r′∈[0,L]}<+∞.\sup \left\{u\left(r^{\prime} )v\left(r):r,r^{\prime} \in \left[0,L]\right\}\lt +\infty .Therefore, from (5.7), we obtain ω0L(FU1,FU2)≤limε→0φ1(ωL((U1,U2),ε)).{\omega }_{0}^{L}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\varphi }_{1}\left({\omega }^{L}\left(\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2}),\varepsilon )).As a result, given the upper semicontinuity of the function φ1{\varphi }_{1}, we obtain ω0L(FU1,FU2)≤φ1(ω0L(U1,U2)),\hspace{-17.95em}{\omega }_{0}^{L}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left({\omega }_{0}^{L}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),and eventually (5.8)ω0(FU1,FU2)≤φ1(ω0(U1,U2)).\hspace{-17.7em}{\omega }_{0}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left({\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})).Now, take arbitrary functions υ1,υ′1∈U1{\upsilon }^{1},{\upsilon }^{^{\prime} 1}\in {{\mathcal{U}}}^{1}and υ2,υ′2∈U2{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {{\mathcal{U}}}^{2}. Then, for r∈ℛ,r\in {\mathcal{ {\mathcal R} }},we obtain ∣(Fυ1,Fυ2)(r)−(Fυ′1,Fυ′2)(r)∣≤∣f1(r,υ1(r),υ2(r))−f1(r,υ′1(r),υ′2(r))∣+∫0r∣f2(r,s,υ1(s),υ2(s))∣ds+∫0r∣f2(r,s,υ′1(s),υ′2(s))∣ds≤φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+2u(r)∫0rv(s)ds=φ1(∣υ1υ2(r)−υ′1υ′2(r)∣)+2a(r).\begin{array}{l}| \left(F{\upsilon }^{1},F{\upsilon }^{2})\left(r)-\left(F{\upsilon }^{^{\prime} 1},F{\upsilon }^{^{\prime} 2})\left(r)| \\ \hspace{1.0em}\le | {f}_{1}\left(r,{\upsilon }^{1}\left(r),{\upsilon }^{2}\left(r))-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1}\left(r),{\upsilon }^{^{\prime} 2}\left(r))| +\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{1}\left(s),{\upsilon }^{2}\left(s))| {\rm{d}}s+\underset{0}{\overset{r}{\displaystyle \int }}| {f}_{2}\left(r,s,{\upsilon }^{^{\prime} 1}\left(s),{\upsilon }^{^{\prime} 2}\left(s))| {\rm{d}}s\\ \hspace{1.0em}\le {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+2u\left(r)\underset{0}{\overset{r}{\displaystyle \int }}v\left(s){\rm{d}}s\\ \hspace{1.0em}={\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}\left(r)-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}\left(r)| )+2a\left(r).\end{array}Hence, from the aforementioned inequality, we have diam(FU1,FU2)(r)≤φ1(diam(U1,U2)(r))+2a(r).\hspace{4.65em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}\left({\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r))+2a\left(r).As a result, given the upper semicontinuity of φ1{\varphi }_{1}, we obtain (5.9)lim supr→+∞diam(FU1,FU2)(r)≤φ1(lim supr→+∞diam(U1,U2)(r)).\hspace{1.25em}\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}(\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)).By combining (5.8) and (5.9) and considering the superadditivity of φ1{\varphi }_{1}, we obtain ω0(FU1,FU2)+lim supr→+∞diam(FU1,FU2)(r)≤φ1(ω0(U1,U2)+lim supr→+∞diam(U1,U2)(r)),{\omega }_{0}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\left(r)\le {\varphi }_{1}\left({\omega }_{0}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})+\mathop{\mathrm{lim\; sup}}\limits_{r\to +\infty }\hspace{0.25em}{\rm{diam}}\left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})\left(r)),or equivalently, (5.10)μ(FU1,FU2)≤φ1(μ(U1,U2)),\hspace{1.25em}\mu \left(F{{\mathcal{U}}}^{1},F{{\mathcal{U}}}^{2})\le {\varphi }_{1}\left(\mu \left({{\mathcal{U}}}^{1},{{\mathcal{U}}}^{2})),where μ\mu is coupled MNC in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}). So, from (5.10) and using Theorem 2.3, the result is obtained.□Example 5.2Let us define the functional integral equation as follows, which is a special mode of equation (5.1), (5.11)(υ1,υ2)(r)≔rr+1ln(1+∣υ1υ2(r)∣)+∫0res−1−rcosυ1υ2(s)1+∣sinυ1υ2(s)∣ds,(forr∈ℛ+).\left({\upsilon }^{1},{\upsilon }^{2})\left(r):= \frac{r}{r+1}\mathrm{ln}\left(1+| {\upsilon }^{1}{\upsilon }^{2}\left(r)| )+\underset{0}{\overset{r}{\int }}\frac{{e}^{s-1-r}\cos {\upsilon }^{1}{\upsilon }^{2}\left(s)}{1+| \sin {\upsilon }^{1}{\upsilon }^{2}\left(s)| }{\rm{d}}s,\hspace{0.33em}\left(\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}r\in {{\mathcal{ {\mathcal R} }}}_{+}).Here, f1(r,υ1,υ2)=rr+1ln(1+∣υ1υ2∣),f2(r,s,υ1,υ2)=es−1−rcosυ1υ21+∣sinυ1υ2∣.\begin{array}{rcl}{f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})& =& \frac{r}{r+1}\mathrm{ln}\left(1+| {\upsilon }^{1}{\upsilon }^{2}| ),\phantom{\rule[-1.25em]{}{0ex}}\\ {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})& =& \frac{{e}^{s-1-r}\cos {\upsilon }^{1}{\upsilon }^{2}}{1+| \sin {\upsilon }^{1}{\upsilon }^{2}| }.\end{array}In fact, if we take φ1(s)=ln(1+s),{\varphi }_{1}\left(s)=ln\left(1+s),we see that φ1(s)<s{\varphi }_{1}\left(s)\lt sfor s>0s\gt 0. Evidently, φ1{\varphi }_{1}is concave and increasing on ℛ+{{\mathcal{ {\mathcal R} }}}_{+}. Moreover, for υ1,υ′1,υ2,υ′2∈ℛ{\upsilon }^{1},{\upsilon }^{^{\prime} 1},{\upsilon }^{2},{\upsilon }^{^{\prime} 2}\in {\mathcal{ {\mathcal R} }}with ∣υ1υ2∣≥∣υ′1υ′2∣| {\upsilon }^{1}{\upsilon }^{2}| \ge | {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| and for r>0,r\gt 0,we obtain ∣f1(r,υ1,υ2)−f1(r,υ′1,υ′2)∣=rr+1ln1+∣υ1υ2∣1+∣υ′1υ′2∣≤ln1+∣υ1υ2∣−∣υ′1υ′2∣1+∣υ′1υ′2∣<ln(1+(∣υ1υ2−υ′1υ′2∣))=φ1(∣υ1υ2−υ′1υ′2∣).\begin{array}{rcl}| {f}_{1}\left(r,{\upsilon }^{1},{\upsilon }^{2})-{f}_{1}\left(r,{\upsilon }^{^{\prime} 1},{\upsilon }^{^{\prime} 2})| & =& \frac{r}{r+1}\mathrm{ln}\frac{1+| {\upsilon }^{1}{\upsilon }^{2}| }{1+| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }\\ & \le & \mathrm{ln}\left(1+\frac{| {\upsilon }^{1}{\upsilon }^{2}| -| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }{1+| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| }\right)\\ & \lt & \mathrm{ln}\left(1+\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ))\\ & =& {\varphi }_{1}\left(| {\upsilon }^{1}{\upsilon }^{2}-{\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| ).\end{array}In the case ∣υ′1υ′2∣≥∣υ1υ2∣| {\upsilon }^{^{\prime} 1}{\upsilon }^{^{\prime} 2}| \ge | {\upsilon }^{1}{\upsilon }^{2}| , the same can be done. Therefore, we conclude that the function f1{f}_{1}gives hypothesis (ii) and also (i). In addition, note that the function f2{f}_{2}operates continuously from ℛ+2×ℛ2→ℛ{{\mathcal{ {\mathcal R} }}}_{+}^{2}\times {{\mathcal{ {\mathcal R} }}}^{2}\to {\mathcal{ {\mathcal R} }}. Furthermore, ∣f2(r,s,υ1,υ2)∣≤es−1−r,r,s∈ℛ+,υ1,υ2∈ℛ.| {f}_{2}\left(r,s,{\upsilon }^{1},{\upsilon }^{2})| \le {e}^{s-1-r},\hspace{1.0em}r,s\in {{\mathcal{ {\mathcal R} }}}_{+},\hspace{0.33em}{\upsilon }^{1},{\upsilon }^{2}\in {\mathcal{ {\mathcal R} }}.Then, if u(r)≔e−1−ru\left(r):= {e}^{-1-r}, v(s)≔esv\left(s):= {e}^{s}, we see that hypothesis (iii) holds. In fact, limr→+∞u(r)∫0rv(s)ds=limr→+∞e−1−r∫0resds=0.\mathop{\mathrm{lim}}\limits_{r\to +\infty }u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s=\mathop{\mathrm{lim}}\limits_{r\to +\infty }{e}^{-1-r}\underset{0}{\overset{r}{\int }}{e}^{s}{\rm{d}}s=0.Now, we calculate ppaccording to assumption (iv). Then, p=sup{∣f1(r,0,0)∣+u(r)∫0rv(s)ds:r≥0}=sup{e−1:r≥0}=e−1.p=\sup \left\{| {f}_{1}\left(r,0,0)| +u\left(r)\underset{0}{\overset{r}{\int }}v\left(s){\rm{d}}s:r\ge 0\right\}=\sup \left\{{e}^{-1}:r\ge 0\right\}={e}^{-1}.In addition, we consider the hypothesis inequality (iv), we have ln(1+r1)+p≤r1.\mathrm{ln}\left(1+{r}^{1})+p\le {r}^{1}.It can be easily seen that every r1≥1{r}^{1}\ge 1holds in the aforementioned inequality. Thus, as a number r01,{r}_{0}^{1},we can catch r01=1{r}_{0}^{1}=1. Therefore, we conclude that according to Theorem 5.1, equation (5.11) has at least one solution that is on the ball Br01=B1{B}_{{r}_{0}^{1}}={B}_{1}, in BC(ℛ+){\rm{BC}}\left({{\mathcal{ {\mathcal R} }}}_{+}).

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: fixed-point; JS-contraction-type mapping; Darbo’s fixed-point theorem; measure of noncompactness; coupled measure of noncompactness; functional integral equation; 47H09; 47H10; 34A12

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