# Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ i -strictly pseudocontractive mappings

## Abstract

In this article, by using the S-mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.

## 1 Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T of H into itself is called nonexpansive if Tx - Tyx - y for all x, y H. We denote by F(T) the set of fixed points of T (i.e., F(T) = {x H : Tx = x}) Goebel and Kirk  showed that F(T) is always closed convex, and also nonempty provided T has a bounded trajectory.

Recall the mapping T is said to be κ-strict pseudo-contration if there exist κ [0, 1) such that

${∥Tx-Ty∥}^{2}\le {∥x-y∥}^{2}+\kappa {∥\left(I-T\right)x-\left(I-T\right)y∥}^{2}\forall x,y\in D\left(T\right).$
(1.1)

Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is T is nonexpansive if and only if T is 0-strict pseudo-contractive. If κ = 1, T is said to be pseudo-contraction mapping. T is strong pseudo-contraction if there exists a positive constant λ (0, 1) such that T + λI is pseudo-contraction. In a real Hilbert space H (1.1) is equivalent to

$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^{2}-\frac{1-\kappa }{2}{∥\left(I-T\right)x-\left(I-T\right)y∥}^{2}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in D\left(T\right).$
(1.2)

T is pseudo-contraction if and only if

$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^{2}\phantom{\rule{1em}{0ex}}\forall x,y\in D\left(T\right).$

T is strong pseudo-contraction if there exists a positive constant λ (0, 1)

$⟨Tx-Ty,x-y⟩\le \left(1-\lambda \right){∥x-y∥}^{2}\phantom{\rule{1em}{0ex}}\forall x,y\in D\left(T\right)$

The class of κ-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings and class of strong pseudo-contraction mappings is independent of the class of κ-strict pseudo-contraction.

A mapping A of C into H is called inverse-strongly monotone, see  if there exists a positive real number α such that

$⟨x-y,Ax-Ay⟩\ge \alpha {∥Ax-Ay∥}^{2}$

for all x, y C.

The equilibrium problem for G is to determine its equilibrium points, i.e., the set

$EP\left(G\right)=\left\{x\in G:G\left(x,y\right)\ge 0,\phantom{\rule{1em}{0ex}}\forall y\in C\right\}.$
(1.3)

Given a mapping T : CH, let G(x, y) = 〈Tx, y - x〉 for all x, y C. Then, z EP(F) if and only if 〈Tz, y - z〉 ≥ 0 for all y C, i.e., z is a solution of the variational inequality. Let A : CH be a nonlinear mapping. The variational inequality problem is to find a u C such that

$⟨v-u,Au⟩\ge 0$
(1.4)

for all v C. The set of solutions of the variational inequality is denoted by VI(C, A).

In 2005, Combettes and Hirstoaga  introduced some iterative schemes of finding the best approximation to the initial data when EP(G) is nonempty and proved strong convergence theorem.

Also in  Combettes and Hiratoaga, following  define S r : HC by

${S}_{r}\left(x\right)=\left\{z\in C:G\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0\forall y\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\in C\right\}.$
(1.5)

hey proved that under suitable hypotheses G, S r is single-valued and firmly nonexpansive with F(S r ) = EP(G).

Numerous problems in physics, optimization, and economics reduce to find a element of EP(G) (see, e.g., )

Let CB(H) be the family of all nonempty closed bounded subsets of H and $ℋ\left(.,.\right)$ be the Hausdorff metric on CB(H) defined as

$ℋ\left(U,V\right)=\text{max}\left\{\underset{u\in U}{\text{sup}}d\left(u,V\right),\underset{v\in V}{\text{sup}}d\left(U,v\right)\right\},\phantom{\rule{1em}{0ex}}\forall U,V\in CB\left(H\right),$

where d(u, V) = infvVd(u, v), d(U, v) = infuUd(u, v), and d(u, v) = u - v.

Let C be a nonempty closed convex subset of H. Let φ : C be a real-valued function, T : CCB(H) a multivalued mapping and Φ : H × C × C an equilibrium-like function, that is, Φ(w, u, v) + Φ(w, v, u) = 0 for all (w, u, v) H × C × C which satisfies the following conditions with respect to the multivalued map T : CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,

$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_{n},{u}_{n},v\right)\le \Phi \left(\omega ,u,v\right);$

(H 2) For each fixed (w, v) H × C, u Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) H × C, v Φ(w, u, v) is a convex function.

In 2009, Ceng et al.  introduced the following generalized equilibrium problem (GEP) as follows:

$\left(\text{GEP}\right)\left\{\begin{array}{c}\text{Find}\phantom{\rule{2.77695pt}{0ex}}u\in C\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}w\in T\left(u\right)\phantom{\rule{2.77695pt}{0ex}}\text{such}\phantom{\rule{2.77695pt}{0ex}}\text{that}\hfill \\ \Phi \left(w,u,v\right)+\phi \left(v\right)-\phi \left(u\right)\ge 0,\phantom{\rule{2.77695pt}{0ex}}\forall v\in C.\hfill \end{array}\right\$
(1.6)

The set of such solutions u C of (GEP) is denote by (GEP) s (Φ, φ).

In the case of φ ≡ 0 and Φ(w, u, v) ≡ G(u, v), then (GEP) s (Φ, φ) is denoted by EP(G). By using Nadler's theorem they introduced the following algorithm:

Let x1 C and w1 T(x1), there exists sequences {w n } H and {x n }, {u n } C such that

$\left\{\begin{array}{c}{w}_{n}\in T\left({x}_{n}\right),∥{w}_{n}-{w}_{n+1}∥\le \left(1+\frac{1}{n}\right)ℋ\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ \Phi \left({w}_{n},{u}_{n},v\right)+\phi \left(v\right)-\phi \left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},v-{u}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\forall u\in C,\hfill \\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)S{u}_{n},\phantom{\rule{1em}{0ex}}n=1,2,....\hfill \end{array}\right\$
(1.7)

They proved a strong convergence theorem of the sequence {x n } generated by (1.7) as follows:

Theorem 1.1. (See  ) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H and let φ : C be a lower semicontinuous and convex functional. Let T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ, Φ : H × C × C be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that $F\left(S\right)\cap {\left(GEP\right)}_{s}\left(\Phi ,\phi \right)\ne \varnothing$. Let f be a contraction of C into itself and let {x n }, {w n }, and {u n } be sequences generated by (1.7), where {α n } [0,1] and {r n } (0, ∞) satisfy

If there exists a constant λ > 0 such that

$\Phi \left({w}_{1},{T}_{{r}_{1}}\left({x}_{1}\right),{T}_{{r}_{2}}\left({x}_{2}\right)\right)+\Phi \left({w}_{2},{T}_{{r}_{2}}\left({x}_{2}\right),{T}_{{r}_{1}}\left({x}_{1}\right)\right)\le -\lambda {∥{T}_{{r}_{1}}\left({x}_{1}\right)-{T}_{{r}_{2}}\left({x}_{2}\right)∥}^{2}$
(1.8)

for all (r1, r2) Ξ × Ξ,(x1, x2) C × C and w i T(x i ), i = 1, 2, where Ξ = {r n : n ≥ 1}, then for $\stackrel{^}{x}={P}_{F\left(S\right)\cap {\left(GEP\right)}_{s}\left(\Phi ,\phi \right)}f\left(\stackrel{^}{x}\right)$, there exists $\stackrel{^}{w}\in T\left(\stackrel{^}{x}\right)$ such that $\left(\stackrel{^}{x},\stackrel{^}{w}\right)$ is a solution of (GEP) and

${x}_{n}\to \stackrel{^}{x},\phantom{\rule{2.77695pt}{0ex}}{w}_{n}\to \stackrel{^}{w}\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{u}_{n}\to \stackrel{^}{x}\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}n\to \infty .$

In 2011, Kangtunyakarn  proved the following theorem for strict pseudocontractive mapping in Hilbert space by using hybrid method as follows:

Theorem 1.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let F and G be bifunctions from C × C into satisfying (A1)-(A4), respectively. Let A : CH be a α-inverse strongly monotone mapping and let B : CH be a β-inverse strongly monotone mapping. Let T : CC be a κ-strict pseudo-contraction mapping with $F=F\left(T\right)\cap EP\left(F,A\right)\cap EP\left(G,B\right)\ne \varnothing$. Let {x n } be a sequence generated by x1 C = C1 and

$\left\{\begin{array}{c}F\left({u}_{n},u\right)+\left(A{x}_{n},u-{u}_{n}\right)+\frac{1}{{r}_{n}}⟨u-{u}_{n},{u}_{n}-{x}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall u\in C,\hfill \\ G\left({v}_{n},v\right)+\left(B{x}_{n},v-{v}_{n}\right)+\frac{1}{{s}_{n}}⟨v-{v}_{n},{v}_{n}-{x}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall v\in C,\hfill \\ {z}_{n}={\delta }_{n}{u}_{n}+\left(1-{\delta }_{n}\right){v}_{n}\hfill \\ {y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right)T{z}_{n}\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\forall n\ge 1,\hfill \end{array}\right\$
(1.9)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ is sequence in [0,1], r n [a, b] (0, 2α) and s n [c, d] (0, 2β) satisfy the following condition:

$\begin{array}{c}\left(i\right)\underset{n\to \infty }{\text{lim}}{\delta }_{n}=\delta \in \left(0,1\right)\\ \left(ii\right)\phantom{\rule{2.77695pt}{0ex}}0\le \kappa \le {\alpha }_{n}<1,\phantom{\rule{1em}{0ex}}\forall n\ge 1\end{array}$

Then x n converges strongly to ${P}_{F}{x}_{1}$.

From motivation of (1.7) and (1.9), we define the following algorithm as follows:

Algorithm 1.3. Let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i : i = 1,2,..., N} and let S n be the S-mappings generated by T1, T2, ..., T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},...,{\alpha }_{N}^{\left(n\right)}$ where ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right)\in I×I×I,I=\left[0,1\right],{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and $\kappa for all $j=1,2,...,N-1,\kappa for all j = 1,2,...,N. Let x1 C = C1 and ${w}_{1}^{1}\in T\left({x}_{1}\right),{w}_{1}^{2}\in D\left({x}_{1}\right)$, there exists sequence $\left\{{w}_{n}^{1}\right\},\left\{{w}_{n}^{2}\right\}\in H$ and {x n }, {u n }, {v n } C such that

$\left\{\begin{array}{c}{w}_{n}^{1}\in T\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}∥{w}_{n}^{1}-{w}_{n+1}^{1}∥\le \left(1+\frac{1}{n}\right)ℋ\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ {w}_{n}^{2}\in D\left({x}_{n}\right),\phantom{\rule{2.77695pt}{0ex}}∥{w}_{n}^{2}-{w}_{n+1}^{2}∥\le \left(1+\frac{1}{n}\right)ℋ\left(D\left({x}_{n}\right),D\left({x}_{n+1}\right)\right),\hfill \\ {\Phi }_{1}\left({w}_{n}^{1},{u}_{n},u\right)+{\phi }_{1}\left(u\right)-{\phi }_{1}\left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},u-{u}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall u\in C,\hfill \\ {\Phi }_{2}\left({w}_{n}^{2},{v}_{n},v\right)+{\phi }_{2}\left(v\right)-{\phi }_{2}\left({v}_{n}\right)+\frac{1}{{s}_{n}}⟨{v}_{n}-{x}_{n},v-{v}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall v\in C,\hfill \\ {z}_{n}={\delta }_{n}{P}_{C}\left(I-\lambda A\right){u}_{n}+\left(1-{\delta }_{n}\right){P}_{C}\left(I-\eta B\right){v}_{n},\hfill \\ {y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right){S}_{n}{z}_{n},\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\forall n\ge 1.\hfill \end{array}\right\$
(1.10)

where D, T : CCB(H) are $ℋ$-Lipschitz continuous with constant μ1, μ2, respectively, Φ1, Φ2 : H × C × C are equilibrium-like functions satisfying (H 1)-(H 3), A : CH is a α-inverse strongly monotone mapping and B : CH is a β-inverse strongly monotone mapping.

In this article, we prove under some control conditions on {δ n }, {α n }, {s n }, and {r n } that the sequence {x n } generated by (1.7) converges strongly to ${P}_{F}{x}_{1}$ where $F={\cap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{1},{\phi }_{1}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{2},{\phi }_{2}\right)\cap F\left({G}_{1}\right)\cap F\left({G}_{2}\right)$, G1, G2 : CC are defined by G1(x) = P C (x - λAx), G2(x) = P C (x - ηBx), x C and ${P}_{F}{x}_{1}$ is solution of the following system of variational inequality:

$\left\{\begin{array}{c}⟨A{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0.\hfill \end{array}\right\$

## 2 Preliminaries

In this section, we need the following lemmas and definition to prove our main result.

Let C be a nonempty closed convex subset of H. Then for any x H, there exists a unique nearest point in C, denoted by P C x, such that

$∥x-{P}_{C}x∥\le ∥x-y∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

The following lemma is a property of P C .

Lemma 2.1. (See .) Given x H and y C. Then P C x = y if and only if there holds the inequality

$⟨x-y,y-z⟩\ge 0\phantom{\rule{1em}{0ex}}\forall z\in C.$

Lemma 2.2. (See  ) Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n } be a sequence of nonexpansive mappings on C. Suppose ${\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$ is nonempty. Let {λ n } be a sequence of positive numbers with ${\sum }_{n=1}^{\infty }{\lambda }_{n}=1$. Then a mapping S on C defined by

$S\left(x\right)={\sum }_{n=1}^{\infty }{\lambda }_{n}{T}_{n}x$

for x C is well defined, nonexpansive and $F\left(S\right)={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$ hold.

The following lemma is well known.

Lemma 2.3. Let H be Hilbert space, C be a nonempty closed convex subset of H. Let T : CC be a κ-strictly pseudo-contractive, then the fixed point set F(T) of T is closed and convex so that the projection PF(T)is well defined.

In 2009, Kangtunyakarn and Suantai  introduced the S-mapping generated by a finite family of κ-strictly pseudo contractive mappings and real numbers as follows:

Definition 2.1. Let C be a nonempty convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of κ i -strict pseudo-contractions of C into itself. For each j = 1,2,..., N, let ${\alpha }_{j}=\left({a}_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$ where I [0,1] and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$. We define the mapping S : CC as follows:

$\begin{array}{ll}\hfill {U}_{0}& =I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{1}& ={\alpha }_{1}^{1}{T}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{2}& ={\alpha }_{1}^{2}{T}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{3}& ={\alpha }_{1}^{3}{T}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I\phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \hfill {U}_{N-1}& ={\alpha }_{1}^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I\phantom{\rule{2em}{0ex}}\\ \hfill S& ={U}_{N}={\alpha }_{1}^{N}{T}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I.\phantom{\rule{2em}{0ex}}\end{array}$
(2.1)

This mapping is called S-mapping generated by T1, ..., T N and α1, α2, ..., α N .

Lemma 2.4. (See  ) Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of κ-strict pseudo contraction mapping of C into C with ${\cap }_{i=1}^{N}F\left({T}_{i}\right)\ne 0̸$ and κ = max{κ i : i = 1, 2,..., N} and let ${\alpha }_{j}=\left({a}_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, j = 1,2,3,...,N, where $I=\left[0,1\right],\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1,\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{j},\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{3}^{j}\in \left(\kappa ,1\right)$ for all j = 1,2,...,N - 1 and ${\alpha }_{1}^{N}\in \left(\kappa ,1\right],{\alpha }_{3}^{N}\in \left[\kappa ,1\right)\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{2}^{j}\in \left[\kappa ,1\right)$ for all j = 1,2,..., N. Let S be the mapping generated by T1,....,T N and α1, α2,...,α N . Then $F\left(S\right)={\cap }_{i=1}^{N}F\left({T}_{i}\right)$ and S is a nonexpansive mapping.

Lemma 2.5. (See  ) Let C be a nonempty closed convex subset of a real Hilbert space H and S : CC be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition

$∥Sx-Sy∥\le \frac{1+\kappa }{1-\kappa }∥x-y∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

We prove the following lemma by using the concept of the S-mapping as follows:

Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T i , i = 1,2,...,N be κ i strictly pseudo-contraction mappings of C into itself and κ = max{κ i : i = 1,2,...,N} and let ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right),\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I=\left[0,1\right],\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$ such that ${\alpha }_{i}^{n,j}\to {\alpha }_{i}^{j}\in \left[0,1\right]$ as n → ∞ for i = 1, 3 and j = 1,2,3,..., N. For every n , let S and S n be the S-mapping generated by T1, T2,..., T N and α1, α2,...,α N and T1, T2,..., T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},\dots ,{\alpha }_{N}^{\left(n\right)}$, respectively. Then limn→∞S n x n - Sx n = 0 for every bounded sequence {x n } in C.

Proof. Let {x n } be bounded sequence in C, U k and Un,kbe generated by T1,T2,...,T N and α1,α2,...,α N and T1,T2,...,T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},\dots ,{\alpha }_{N}^{\left(n\right)}$, respectively. For each n , we have

$\begin{array}{ll}\hfill ∥{U}_{n,1}{x}_{n}-{U}_{1}{x}_{n}∥& =∥{\alpha }_{1}^{n,1}{T}_{1}{x}_{n}+\left(1-{\alpha }_{1}^{n,1}\right){x}_{n}-{\alpha }_{1}^{1}{T}_{1}{x}_{n}-\left(1-{\alpha }_{1}^{1}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥{\alpha }_{1}^{n,1}{T}_{1}{x}_{n}-{\alpha }_{1}^{n,1}{x}_{n}-{\alpha }_{1}^{1}{T}_{1}{x}_{n}+{\alpha }_{1}^{1}{x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥\left({\alpha }_{1}^{n,1}-{\alpha }_{1}^{1}\right){T}_{1}{x}_{n}-\left({\alpha }_{1}^{n,1}-{\alpha }_{1}^{1}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =\left|{a}_{1}^{n,1}-{\alpha }_{1}^{1}\right|∥{T}_{1}{x}_{n}-{x}_{n}∥\phantom{\rule{2em}{0ex}}\end{array}$
(2.2)

and for k {2, 3,..., N}, by using Lemma 2.5, we obtain

(2.3)

By (2.2) and (2.3), we have

This together with the assumption ${\alpha }_{i}^{n,j}\to {\alpha }_{i}^{j}$ as n → ∞ (i = 1, 3, j = 1,2,..., N), we can conclude that

$\underset{n\to \infty }{\text{lim}}∥{S}_{n}{x}_{n}-S{x}_{n}∥=0.$

Lemma 2.7. (See ) Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC be a nonexpansive mapping. Then I - S is demi-closed at zero.

Lemma 2.8. (See ) Let C be a closed convex subset of H. Let {x n } be a sequence in H and u H. Let q = P C u, if {x n } is such the ω(x n ) C and satisfy the condition

$∥{x}_{n}-u∥\le ∥u-q∥,\phantom{\rule{1em}{0ex}}\forall n\in ℕ.$

Then x n q, as n → ∞.

Definition 2.2. A multivalued map T : CCB(H) is say to be $ℋ$-Lipschitz continuous if there exists a constant μ > 0 such that

$ℋ\left(T\left(u\right)-T\left(v\right)\right)\le \mu ∥u-v∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall u,v\in C,$

where $ℋ\left(.,.\right)$ is the Hausdorff metric on CB(H).

Lemma 2.9. (Nadler's theorem, see ) Let (X, ) be a normed vector space and $ℋ\left(.,.\right)$ is the Hausdorff metric on CB(H). If U, V CB(X), then for any given ϵ > 0 and u U, there exists v V such that

$∥u-v∥\le \left(1+\epsilon \right)ℋ\left(U,V\right).$

Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ: CH be a real-valued function, T : CCB(H) be a multivalued map and Φ : H × C × C be an equilibrium-like function.

To solve the GEP, let us assume that the equilibrium-like function Φ : H × C × C satisfies the following conditions with respect to the multivalued map T: CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,

$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_{n},{u}_{n},v\right)\le \Phi \left(\omega ,u,v\right);$

(H 2) For each fixed (w, v) H × C, u Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) H × C, v Φ(w, u, v) is a convex function.

Theorem 2.10. (See ) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let φ : C be a lower semicontinuous and convex functional. Let T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ, and Φ : H × C × C be an equilibrium-like function satisfying (H 1)-(H 3). Let r > 0 be a constant. For each x C, take w x T(x) arbitrarily and define a mapping T r : CC as follows:

${T}_{r}\left(x\right)=\left\{u\in C:\Phi \left({w}_{x},u,v\right)+\phi \left(v\right)-\phi \left(u\right)+\frac{1}{r}⟨u-x,v-u⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall v\in C\right\}.$

Then, there hold the following:

(a) T r is single-valued;

(b) T r is firmly nonexpansive (that is, for any u, v C, T r u - T r v2 ≤ 〈T r u-T r v, u-v〉) if

$\Phi \left({w}_{1},{T}_{r}\left({x}_{1}\right),{T}_{r}\left({x}_{2}\right)\right)+\Phi \left({w}_{2},{T}_{r}\left({x}_{2}\right),{T}_{r}\left({x}_{1}\right)\right)\le 0,$

for all (x1, x2) C × C and all w i T(xi), i = 1,2;

(c) F(T r ) = (GEP) s (Φ, φ)

(d) (GEP) s (Φ, φ) is closed and convex.

Lemma 2.11. (See ) Let C be a nonempty closed convex subset of a Hilbert space H and let G : CC be defined by

$G\left(x\right)={P}_{C}\left(x-\lambda Ax\right),\phantom{\rule{1em}{0ex}}\forall x\in C,$

with λ > 0. Then x* VI (C, A) if and only if x* F(G).

## 3 Main results

In this section, we prove a strong convergence theorem of the sequence {x n } generated by (1.10) to ${P}_{F}{x}_{1}$.

Theorem 3.1. Let C be a nonempty bounded, closed, and convex subset of Hilbert space H and let φ1, φ2 : be a lower semicontinuous and convex function. Let D, T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ1, μ2, respectively, Φ12: H × C × C be equilibrium-like functions satisfying (H 1) - (H3). Let A: CH be a α-inverse strongly monotone mapping and B : CH be a β-inverse strongly monotone mapping, let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i :i = 1,2,..., N} with $F={\cap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{1},{\phi }_{1}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{2},{\phi }_{2}\right)\cap F\left({G}_{1}\right)\cap F\left({G}_{2}\right)$, where G1, G2 : CC are defined by G1(x) P C (x-λAx), G2(x) = P C (x-ηBx), x C. Let S n be the S-mappings generated by T1,T2,...,T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},...,{\alpha }_{N}^{\left(n\right)}$ where ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right)\in I×I×I,I=\left[0,1\right],{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and $\kappa for all $j=1,2,...,N-1,\kappa for all j = 1,2,...,N and let {x n }, {u n }, {v n }, $\left\{{w}_{n}^{1}\right\}$, and $\left\{{w}_{n}^{2}\right\}$ be sequences generated by (1.10), where {α n } is a sequence in [0,1], r n , λ [a, b] (0, ) and s n , η [c, d] (0, 2β), for every n and suppose the following conditions hold:

(i) $\underset{n\to \infty }{\text{lim}}{\delta }_{n}=\delta \in \left(0,1\right)$,

(ii) 0 ≤ κα n < 1, n ≥ 1,

(iii) $\sum _{n=1}^{\infty }\left|{\alpha }_{1}^{n+1,j}-{\alpha }_{1}^{n,j}\right|<\infty ,\sum _{n=1}^{\infty }\left|{\alpha }_{3}^{n+1,j}-{\alpha }_{3}^{n,j}\right|<\infty$, for all j {1,2,3,...,N}.

(iv) There exists λ1, λ2 such that

$\left\{\begin{array}{c}{\Phi }_{1}\left({w}_{1}^{1},{T}_{{r}_{1}}\left({x}_{1}\right),{T}_{{r}_{2}}\left({x}_{2}\right)\right)+{\Phi }_{1}\left({w}_{2}^{1},{T}_{{r}_{2}}\left({x}_{2}\right),{T}_{{r}_{1}}\left({x}_{1}\right)\right)\le -{\lambda }_{1}{∥{T}_{{r}_{1}}\left({x}_{1}\right)-{T}_{{r}_{2}}\left({x}_{2}\right)∥}^{2}and\hfill \\ {\Phi }_{2}\left({w}_{1}^{2},{T}_{{s}_{1}}\left({x}_{1}\right),{T}_{{s}_{2}}\left({x}_{2}\right)\right)+{\Phi }_{2}\left({w}_{2}^{2},{T}_{{s}_{2}}\left({x}_{2}\right),{T}_{{s}_{1}}\left({x}_{1}\right)\right)\le -{\lambda }_{2}{∥{T}_{{s}_{1}}\left({x}_{1}\right)-{T}_{{s}_{2}}\left({x}_{2}\right)∥}^{2}\hfill \end{array}\right\$
(3.1)

for all $\left({r}_{1},{r}_{2}\right)\in \Theta ×\Theta ,\phantom{\rule{2.77695pt}{0ex}}\left({s}_{1},{s}_{2}\right)\in \Xi ×\Xi ,\phantom{\rule{2.77695pt}{0ex}}{w}_{i}^{1}\in T\left({x}_{i}\right)$ and ${w}_{i}^{2}\in D\left({x}_{i}\right)$, for i = 1,2 where Θ = {r n : n ≥ 1} and Ξ = {s n : n ≥ 1}. Then {x n } converges strongly to ${P}_{F}{x}_{1}$ which is a solution of (3.2):

$\left\{\begin{array}{c}⟨A{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0.\hfill \end{array}\right\$
(3.2)

Proof. From (3.1) for every r Θ, we have

${\Phi }_{1}\left({w}_{1}^{1},{T}_{r}\left({x}_{1}\right),{T}_{r}\left({x}_{2}\right)\right)+{\Phi }_{1}\left({w}_{2}^{1},{T}_{r}\left({x}_{2}\right),{T}_{r}\left({x}_{1}\right)\right)\le -{\lambda }_{1}{∥{T}_{r}\left({x}_{1}\right)-{T}_{r}\left({x}_{2}\right)∥}^{2}\le 0,$
(3.3)

for all (x1, x2) C × C and ${w}_{i}^{1}\in T\left({x}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i=1,2$.

Similarly, for every s Ξ, we have

${\Phi }_{2}\left({w}_{1}^{2},{T}_{s}\left({x}_{1}\right),{T}_{s}\left({x}_{2}\right)\right)+{\Phi }_{2}\left({w}_{2}^{2},{T}_{s}\left({x}_{2}\right),{T}_{s}\left({x}_{1}\right)\right)\le -{\lambda }_{2}{∥{T}_{s}\left({x}_{1}\right)-{T}_{s}\left({x}_{2}\right)∥}^{2}\le 0.$
(3.4)

for all (x1, x2) C × C and ${w}_{i}^{2}\in D\left({x}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i=1,2$. From (3.3) and (3.4), we have Theorem 2.10 hold.

It is easy to see that I - λ A and I - η B are nonexpansive mapping. Indeed, since A is a α-inverse strongly monotone mapping with λ (0, 2α), we have

$\begin{array}{ll}\hfill {∥\left(I-\lambda A\right)x-\left(I-\lambda A\right)y∥}^{2}& ={∥x-y-\lambda \left(Ax-Ay\right)∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥x-y∥}^{2}-2\lambda ⟨x-y,Ax-Ay⟩+{\lambda }^{2}{∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥x-y∥}^{2}-2\alpha \lambda {∥Ax-Ay∥}^{2}+{\lambda }^{2}{∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥x-y∥}^{2}+\lambda \left(\lambda -2\alpha \right){∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥x-y∥}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

Thus (I - λA) is nonexpansive, so is I - ηB. Since

${\Phi }_{1}\left({w}_{n}^{1},{u}_{n},u\right)+{\phi }_{1}\left(u\right)-{\phi }_{1}\left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},u-{u}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall u\in C,$

and Theorem 2.10, we have ${u}_{n}={T}_{{r}_{n}}{x}_{n}$. Since

${\Phi }_{2}\left({w}_{n}^{2},{v}_{n},v\right)+{\phi }_{2}\left(v\right)-{\phi }_{2}\left({v}_{n}\right)+\frac{1}{{s}_{n}}⟨{v}_{n}-{x}_{n},v-{v}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall v\in C,$

and Theorem 2.10, we have ${v}_{n}={T}_{{s}_{n}}{x}_{n}$. Let $z\in F$, again by Theorem 2.10, we have $z={T}_{{r}_{n}}z={T}_{{s}_{n}}z={P}_{C}\left(I-\lambda A\right)z={P}_{C}\left(I-\eta B\right)z$. From nonexpansiveness of $\left\{{T}_{{r}_{n}}\right\},\phantom{\rule{2.77695pt}{0ex}}\left\{{T}_{{s}_{n}}\right\},\phantom{\rule{2.77695pt}{0ex}}\left\{I-\lambda A\right\}$, and {I - ηB}, we have

$\begin{array}{ll}\hfill ∥{z}_{n}-z∥& =∥{\delta }_{n}\left({P}_{C}\left(I-\lambda A\right){u}_{n}-z\right)+\left(1-{\delta }_{n}\right)\left({P}_{C}\left(I-\eta B\right){v}_{n}-z\right)∥\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}∥{P}_{C}\left(I-\lambda A\right){u}_{n}-z∥+\left(1-{\delta }_{n}\right)∥{P}_{C}\left(I-\eta B\right){v}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}∥{T}_{{r}_{n}}{x}_{n}-z∥+\left(1-{\delta }_{n}\right)∥{T}_{{s}_{n}}{x}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le ∥{x}_{n}-z∥.\phantom{\rule{2em}{0ex}}\end{array}$
(3.5)

By (3.5), we have

$\begin{array}{ll}\hfill ∥{y}_{n}-z∥& =∥{\alpha }_{n}\left({z}_{n}-z\right)+\left(1-{\alpha }_{n}\right)\left({S}_{n}{z}_{n}-z\right)∥\phantom{\rule{2em}{0ex}}\\ \le {\alpha }_{n}∥{z}_{n}-z∥+\left(1-{\alpha }_{n}\right)∥{S}_{n}{z}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le ∥{z}_{n}-z∥\le ∥{x}_{n}-z∥.\phantom{\rule{2em}{0ex}}\end{array}$
(3.6)

Next, we show that C n is closed and convex for every n . It is obvious that C n is closed. In fact, we know that, for z C n ,

$∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\phantom{\rule{2.77695pt}{0ex}}\text{is}\phantom{\rule{2.77695pt}{0ex}}\text{equivalent}\phantom{\rule{2.77695pt}{0ex}}\text{to}\phantom{\rule{2.77695pt}{0ex}}{∥{y}_{n}-{x}_{n}∥}^{2}+2⟨{y}_{n}-{x}_{n},{x}_{n}-z⟩\le 0.$

So, we have that z1, z2 C n and t (0,1), it follows that

$\begin{array}{ll}\hfill {∥{y}_{n}-{x}_{n}∥}^{2}& +2⟨{y}_{n}-{x}_{n},{x}_{n}-\left(t{z}_{1}+\left(1-t\right){z}_{2}\right)⟩\phantom{\rule{2em}{0ex}}\\ =t\left(2⟨{y}_{n}-{x}_{n},{x}_{n}-{z}_{1}⟩+{∥{y}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(1-t\right)\left(2⟨{y}_{n}-{x}_{n},{x}_{n}-{z}_{2}⟩+{∥{y}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \le 0,\phantom{\rule{2em}{0ex}}\end{array}$

then, we have C n is convex. By Theorem 2.10 and Lemma 2.3, we conclude that $F$ is closed and convex. This implies that ${P}_{F}$ is well defined. Next, we show that $F\subset {C}_{n}$ for every n . Putting $q\in F$, by (3.6), it is easy to see that q C n , then we have $F\subset {C}_{n}$ for all n . Since ${x}_{n}={P}_{{C}_{n}}{x}_{1}$, for every w C n , we have

$∥{x}_{n}-{x}_{1}∥\le ∥w-{x}_{1}∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall n\in ℕ.$
(3.7)

In particular, we have

$∥{x}_{n}-{x}_{1}∥\le ∥{P}_{F}{x}_{1}-{x}_{1}∥.$
(3.8)

Since C is bounded, we have {x n } is bounded, so are {u n }, {v n }, {z n }, and {y n }. Since ${x}_{n+1}={P}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$ and ${x}_{n}={P}_{{C}_{n}}{x}_{1}$, we have

$\begin{array}{ll}\hfill 0& \le ⟨{x}_{1}-{x}_{n},{x}_{n}-{x}_{n+1}⟩\phantom{\rule{2em}{0ex}}\\ =⟨{x}_{1}-{x}_{n},{x}_{n}-{x}_{1}+{x}_{1}-{x}_{n+1}⟩\phantom{\rule{2em}{0ex}}\\ \le -{∥{x}_{n}-{x}_{1}∥}^{2}+∥{x}_{n}-{x}_{1}∥∥{x}_{1}-{x}_{n+1}∥,\phantom{\rule{2em}{0ex}}\end{array}$

it implies that

$∥{x}_{n}-{x}_{1}∥\le ∥{x}_{n+1}-{x}_{1}∥.$

Hence, we have limn→∞x n - x1 exists. Since

$\begin{array}{ll}\hfill {∥{x}_{n}-{x}_{n+1}∥}^{2}& ={∥{x}_{n}-{x}_{1}+{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{1}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{1}-{x}_{n}+{x}_{n}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}-2{∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{n}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥{x}_{1}-{x}_{n+1}∥}^{2}-{∥{x}_{n}-{x}_{1}∥}^{2},\phantom{\rule{2em}{0ex}}\end{array}$
(3.9)

it implies that

$\underset{n\to \infty }{\text{lim}}∥{x}_{n}-{x}_{n+1}∥=0.$
(3.10)

Since ${x}_{n+1}={P}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}$, we have

$∥{y}_{n}-{x}_{n+1}∥\le ∥{x}_{n}-{x}_{n+1}∥,$

by (3.10), we have

$\underset{n\to \infty }{\text{lim}}∥{y}_{n}-{x}_{n+1}∥=0.$
(3.11)

Since

$∥{y}_{n}-{x}_{n}∥\le ∥{y}_{n}-{x}_{n+1}∥+∥{x}_{n+1}-{x}_{n}∥,$

by (3.10) and (3.11), we have

$\underset{n\to \infty }{\text{lim}}∥{y}_{n}-{x}_{n}∥=0.$
(3.12)

Next, we show that

$\underset{n\to \infty }{\text{lim}}∥{z}_{n}-{S}_{n}{z}_{n}∥=0.$
(3.13)

By definition of y n , we have

${y}_{n}-{z}_{n}=\left(1-{\alpha }_{n}\right)\left({S}_{n}{z}_{n}-{z}_{n}\right).$
(3.14)

Claim that

$\underset{n\to \infty }{\text{lim}}∥{z}_{n}-{x}_{n}∥=0.$
(3.15)

Putting M n = P C (I - λA)u n and N n = P C (I - ηB)v n , we have

$∥{z}_{n}-{x}_{n}∥\le {\delta }_{n}∥{M}_{n}-{x}_{n}∥+\left(1-{\delta }_{n}\right)∥{N}_{n}-{x}_{n}∥.$
(3.16)

Let $z\in F$. Since ${T}_{{r}_{n}}$ is firmly nonexpansive mapping and ${T}_{{r}_{n}}{x}_{n}={u}_{n}$, we have

$\begin{array}{ll}\hfill {∥z-{u}_{n}∥}^{2}& ={∥{T}_{{r}_{n}}z-{T}_{{r}_{n}}{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le ⟨{T}_{{r}_{n}}z-{T}_{{r}_{n}}{x}_{n},z-{x}_{n}⟩\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2}\left({∥{u}_{n}-z∥}^{2}+{∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Hence

${∥{u}_{n}-z∥}^{2}\le {∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}.$
(3.17)

Since ${T}_{{r}_{n}}$ is firmly nonexpansive mapping and ${T}_{{s}_{n}}{x}_{n}={v}_{n}$, by using the same method as (3.17), we have

${∥{v}_{n}-z∥}^{2}\le {∥{x}_{n}-z∥}^{2}-{∥{v}_{n}-{x}_{n}∥}^{2}.$
(3.18)

By nonexpansiveness of S n and (3.17), (3.18), we have

$\begin{array}{ll}\hfill {∥{y}_{n}-z∥}^{2}& \le {∥{z}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}{∥{u}_{n}-z∥}^{2}+\left(1-{\delta }_{n}\right){∥{v}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}\left({∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}\right)+\left(1-{\delta }_{n}\right)\left({∥{x}_{n}-z∥}^{2}-{∥{v}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-z∥}^{2}-{\delta }_{n}{∥{u}_{n}-{x}_{n}∥}^{2}-\left(1-{\delta }_{n}\right){∥{v}_{n}-{x}_{n}∥}^{2},\phantom{\rule{2em}{0ex}}\end{array}$

it implies that

$\begin{array}{ll}\hfill {\delta }_{n}{∥{u}_{n}-{x}_{n}∥}^{2}& \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}-\left(1-{\delta }_{n}\right){∥{v}_{n}-{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(∥{x}_{n}-z∥+∥{y}_{n}-z∥\right)∥{x}_{n}-{y}_{n}∥,\phantom{\rule{2em}{0ex}}\end{array}$

by (3.12) and condition (i), we have

$\underset{n\to \infty }{\text{lim}}∥{u}_{n}-{x}_{n}∥=0.$
(3.19)

By using the same method as (3.19), we have

$\underset{n\to \infty }{\text{lim}}∥{v}_{n}-{x}_{n}∥=0.$
(3.20)

Since

(3.21)

Claim that

$\underset{n\to \infty }{\text{lim}}∥A{u}_{n}-Az∥=\underset{n\to \infty }{\text{lim}}∥B{v}_{n}-Bz∥=0.$

By nonexpansiveness of P C , we have

$\begin{array}{c}{‖{y}_{n}-z‖}^{2}\le {‖{z}_{n}-z‖}^{2}\\ \le {\delta }_{n}{‖{P}_{C}\left(I-\lambda A\right){u}_{n}-{P}_{C}\left(I-\lambda A\right)z‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right){‖{P}_{C}\left(I-\eta B\right){v}_{n}-{P}_{C}\left(I-\eta B\right)z‖}^{2}\\ \le {\delta }_{n}{‖\left(I-\lambda A\right){u}_{n}-\left(I-\lambda A\right)z‖}^{2}+\left(1-{\delta }_{n}\right){‖\left(I-\eta B\right){v}_{n}-\left(I-\eta B\right)z‖}^{2}\\ \le {\delta }_{n}{‖\left({u}_{n}-\lambda A{u}_{n}-\left(z-\lambda Az\right)‖}^{2}+\left(1-{\delta }_{n}\right){‖\left({v}_{n}-\eta B{v}_{n}-\left(z-\eta Bz\right)‖}^{2}\\ ={\delta }_{n}{‖\left({u}_{n}-z\right)-\lambda \left(A{u}_{n}-Az\right)‖}^{2}+\left(1-{\delta }_{n}\right){‖\left({v}_{n}-z\right)-\eta \left(B{v}_{n}-Bz\right)‖}^{2}\\ ={\delta }_{n}\left({‖{u}_{n}-z‖}^{2}+{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}-2\lambda 〈{u}_{n}-z,A{u}_{n}-Az〉\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{v}_{n}-z‖}^{2}+{\eta }^{2}{‖B{v}_{n}-Bz‖}^{2}-2\eta 〈{v}_{n}-z,B{v}_{n}-Bz〉\right)\\ \le {\delta }_{n}\left({‖{u}_{n}-z‖}^{2}+{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}-2\lambda \alpha {‖A{u}_{n}-Az‖}^{2}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{v}_{n}-z‖}^{2}+{\eta }^{2}{‖B{v}_{n}-Bz‖}^{2}-2\eta \beta {‖B{v}_{n}-Bz‖}^{2}\right)\\ \le {\delta }_{n}\left({‖{x}_{n}-z‖}^{2}+\lambda \left(\lambda -2\alpha \right){‖A{u}_{n}-Az‖}^{2}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{x}_{n}-z‖}^{2}+\eta \left(\eta -2\beta \right){‖B{v}_{n}-Bz‖}^{2}\right)\\ ={‖{x}_{n}-z‖}^{2}-{\delta }_{n}\lambda \left(2\alpha -\lambda \right){‖A{u}_{n}-Az‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}-\left(1-{\delta }_{n}\right)\eta \left(2\beta -\eta \right){‖B{v}_{n}-Bz‖}^{2},\end{array}$

it follows that

$\begin{array}{ll}\hfill {\delta }_{n}\lambda \left(2\alpha -\lambda \right){∥A{u}_{n}-Az∥}^{2}& \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left(1-{\delta }_{n}\right)\eta \left(2\beta -\eta \right){∥B{v}_{n}-Bz∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(∥{x}_{n}-z∥+∥{y}_{n}-z∥\right)∥{y}_{n}-{x}_{n}∥,\phantom{\rule{2em}{0ex}}\end{array}$
(3.22)

by conditions (i), (ii), λ (0, 2α) and (3.12), it implies that

$\underset{n\to \infty }{\text{lim}}∥A{u}_{n}-Az∥=0.$
(3.23)

By using the same method as (3.23), we have

$\underset{n\to \infty }{\text{lim}}∥B{v}_{n}-Bz∥=0.$
(3.24)

By nonexpansiveness of ${T}_{{r}_{n}}$, we have

$\begin{array}{c}{‖{M}_{n}-z‖}^{2}={‖{P}_{C}\left({u}_{n}-\lambda A{u}_{n}\right)-{P}_{C}\left(z-\lambda Az\right)‖}^{2}\\ \le 〈\left({u}_{n}-\lambda A{u}_{n}\right)-\left(z-\lambda Az\right),{M}_{n}-z〉\\ =\frac{1}{2}\left({‖\left({u}_{n}-\lambda A{u}_{n}\right)-\left(z-\lambda Az\right)‖}^{2}+{‖{M}_{n}-z‖}^{2}-‖\left({u}_{n}-\lambda A{u}_{n}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}{-\left(z-\lambda Az\right)-\left({M}_{n}-z\right)‖}^{2}\right)\\ \le \frac{1}{2}\left({‖{u}_{n}-z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖\left({u}_{n}-{M}_{n}\right)-\lambda \left(A{u}_{n}-Az\right)‖}^{2}\right)\\ =\frac{1}{2}\left({‖{T}_{{r}_{n}}{x}_{n}-{T}_{{r}_{n}}z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖{u}_{n}-{M}_{n}‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+2\lambda 〈{u}_{n}-{M}_{n},A{u}_{n}-Az〉-{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}\right)\\ \le \frac{1}{2}\left({‖{x}_{n}-z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖{u}_{n}-{M}_{n}‖}^{2}+2\lambda 〈{u}_{n}-{M}_{n},A{u}_{n}-Az〉\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}-{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}\right).\end{array}$

Hence, we have

$\begin{array}{ll}\hfill {∥{M}_{n}-z∥}^{2}& \le {∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{M}_{n}∥}^{2}+2\lambda ⟨{u}_{n}-{M}_{n}\end{array}$