# An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces

- Pattanapong Tianchai
^{1}Email author

**2012**:214

https://doi.org/10.1186/1687-1812-2012-214

© Tianchai; licensee Springer 2012

**Received: **2 May 2012

**Accepted: **7 November 2012

**Published: **26 November 2012

## Abstract

We introduce a new iterative scheme by modifying Mann’s iteration method to find a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu (Thai J. Math. 9(2):399-409, 2011) and many others.

**MSC:**46C05, 47H09, 47H10, 49J30, 49J40.

## Keywords

## 1 Introduction

Throughout this paper, we always assume that *C* is a nonempty closed convex subset of a real Hilbert space *H* with inner product and norm denoted by $\u3008\cdot ,\cdot \u3009$ and $\parallel \cdot \parallel $, respectively. For a sequence $\{{x}_{n}\}$ in *H*, we denote the strong convergence and the weak convergence of $\{{x}_{n}\}$ to $x\in H$ by ${x}_{n}\to x$ and ${x}_{n}\rightharpoonup x$, respectively.

*metric projection*of

*H*onto

*C*; that is, for each $x\in H$, there exists the unique point ${P}_{C}x\in C$ such that $\parallel x-{P}_{C}x\parallel ={min}_{y\in C}\parallel x-y\parallel $. A mapping $T:C\to C$ is called

*nonexpansive*if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in C$, and

*uniformly*

*L-Lipschitzian*if there exists a constant $L>0$ such that for each $n\in \mathbb{N}$, $\parallel {T}^{n}x-{T}^{n}y\parallel \le L\parallel x-y\parallel $ for all $x,y\in C$, and a mapping $f:C\to C$ is called a

*contraction*if there exists a constant $\alpha \in (0,1)$ such that $\parallel f(x)-f(y)\parallel \le \alpha \parallel x-y\parallel $ for all $x,y\in C$. A point $x\in C$ is a

*fixed point*of

*T*provided that $Tx=x$. We denote by $F(T)$ the set of fixed points of

*T*; that is, $F(T)=\{x\in C:Tx=x\}$. If

*C*is a nonempty bounded closed convex subset of

*H*and

*T*is a nonexpansive mapping of

*C*into itself, then $F(T)$ is nonempty (see [1]). Recall that a mapping $T:C\to C$ is said to be

- (i)
*monotone*if$\u3008Tx-Ty,x-y\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,$(1.1) - (ii)
*k-Lipschitz continuous*if there exists a constant $k>0$ such that$\parallel Tx-Ty\parallel \le k\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,$(1.2)

*A*is nonexpansive,

- (iii)
*α-strongly monotone*if there exists a constant $\alpha >0$ such that$\u3008Tx-Ty,x-y\u3009\ge \alpha {\parallel x-y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,$(1.3) - (iv)
*α-inverse-strongly monotone*(*or**α-cocoercive*) if there exists a constant $\alpha >0$ such that$\u3008Tx-Ty,x-y\u3009\ge \alpha {\parallel Tx-Ty\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,$(1.4)

*T*is called

*firmly nonexpansive*; it is obvious that any

*α*-inverse-strongly monotone mapping

*T*is monotone and $(1/\alpha )$-Lipschitz continuous,

- (v)
*κ-strictly pseudocontractive*[2] if there exists a constant $\kappa \in [0,1)$ such that${\parallel Tx-Ty\parallel}^{2}\le {\parallel x-y\parallel}^{2}+\kappa {\parallel (I-T)x-(I-T)y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$(1.5)

*κ*-SPC to denote

*κ*-strictly pseudocontractive. It is obvious that

*T*is nonexpansive if and only if

*T*is 0-SPC,

- (vi)
*asymptotically**κ-SPC*[3] if there exists a constant $\kappa \in [0,1)$ and a sequence $\{{\gamma}_{n}\}$ of nonnegative real numbers with ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$ such that${\parallel {T}^{n}x-{T}^{n}y\parallel}^{2}\le (1+{\gamma}_{n}){\parallel x-y\parallel}^{2}+\kappa {\parallel (I-{T}^{n})x-(I-{T}^{n})y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,$(1.6)

*T*is asymptotically nonexpansive with ${k}_{n}=\sqrt{1+{\gamma}_{n}}$ for all $n\in \mathbb{N}$; that is,

*T*is

*asymptotically nonexpansive*[4] if there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that

*κ*-SPC mappings and the class of asymptotically

*κ*-SPC mappings are independent (see [5]). And the class of asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate sense with ${\u03f5}_{n}={\gamma}_{n}K$ for all $n\in \mathbb{N}$ and for some $K>0$; that is,

*T*is an

*asymptotically nonexpansive mapping in the intermediate sense*if there exists a sequence $\{{\u03f5}_{n}\}$ of nonnegative real numbers with ${lim}_{n\to \mathrm{\infty}}{\u03f5}_{n}=0$ such that

- (vii)
*asymptotically**κ-SPC mapping in the intermediate sense*[6] if there exists a constant $\kappa \in [0,1)$ and a sequence $\{{\gamma}_{n}\}$ of nonnegative real numbers with ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$ such that$\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}({\parallel {T}^{n}x-{T}^{n}y\parallel}^{2}-(1+{\gamma}_{n}){\parallel x-y\parallel}^{2}-\kappa {\parallel (I-{T}^{n})x-(I-{T}^{n})y\parallel}^{2})\le 0.$

for all $n\in \mathbb{N}$. It is obvious that if ${\tau}_{n}=0$ for all $n\in \mathbb{N}$, then the class of asymptotically *κ*-SPC mappings in the intermediate sense is reduced to the class of asymptotically *κ*-SPC mappings; and if ${\tau}_{n}=\kappa =0$ for all $n\in \mathbb{N}$, then the class of asymptotically *κ*-SPC mappings in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings; and if ${\gamma}_{n}={\tau}_{n}=\kappa =0$ for all $n\in \mathbb{N}$, then the class of asymptotically *κ*-SPC mappings in the intermediate sense is reduced to the class of nonexpansive mappings; and the class of asymptotically nonexpansive mappings in the intermediate sense with $\{{\u03f5}_{n}\}$ of nonnegative real numbers such that ${lim}_{n\to \mathrm{\infty}}{\u03f5}_{n}=0$ is reduced to the class of asymptotically *κ*-SPC mappings in the intermediate sense with ${\tau}_{n}={\u03f5}_{n}K$ for all $n\in \mathbb{N}$ and for some $K>0$. Some methods have been proposed to solve the fixed point problem of an asymptotically *κ*-SPC mapping in the intermediate sense (1.9); related work can also be found in [6–13] and the references therein.

**Example 1.1** (Sahu *et al.* [6])

- (1)
*T*is an asymptotically nonexpansive mapping in the intermediate sense. Therefore,*T*is an asymptotically*κ*-SPC mapping in the intermediate sense. - (2)
*T*is not continuous. Therefore,*T*is not an asymptotically*κ*-SPC and asymptotically nonexpansive mapping.

**Example 1.2** (Hu and Cai [7])

- (1)
*T*is an asymptotically nonexpansive mapping in the intermediate sense. Therefore,*T*is an asymptotically*κ*-SPC mapping in the intermediate sense. - (2)
*T*is continuous but not uniformly*L*-Lipschitzian. Therefore,*T*is not an asymptotically*κ*-SPC mapping.

**Example 1.3**Let $X=\mathbb{R}$ and $C=[0,1]$. For each $x\in C$, we define $T:C\to C$ by

- (1)
*T*is an asymptotically nonexpansive mapping in the intermediate sense. Therefore,*T*is an asymptotically*κ*-SPC mapping in the intermediate sense. - (2)
*T*is not continuous. Therefore,*T*is not an asymptotically*κ*-SPC and asymptotically nonexpansive mapping.

*T*is a contraction. Picard’s method generates a sequence $\{{x}_{n}\}$ successively as ${x}_{n+1}=T{x}_{n}$ for all $n\in \mathbb{N}$ with ${x}_{1}=x$ chosen arbitrarily, and this sequence converges in norm to the unique fixed point of

*T*. However, if

*T*is not a contraction (for instance, if

*T*is nonexpansive), then Picard’s successive iteration fails, in general, to converge. Instead, Mann’s iteration method for a nonexpansive mapping

*T*(see [14]) prevails, generates a sequence $\{{x}_{n}\}$ recursively by

where ${x}_{1}=x\in C$ chosen arbitrarily and the sequence $\{{\alpha}_{n}\}$ lies in the interval $[0,1]$.

Mann’s algorithm for nonexpansive mappings has been extensively investigated (see [2, 15, 16] and the references therein). One of the well-known results is proven by Reich [16] for a nonexpansive mapping *T* on *C*, which asserts the weak convergence of the sequence $\{{x}_{n}\}$ generated by (1.10) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})=\mathrm{\infty}$. Recently, Marino and Xu [17] developed and extended Reich’s result to a SPC mapping in a Hilbert space setting. More precisely, they proved the weak convergence of Mann’s iteration process (1.10) for a *κ*-SPC mapping *T* on *C*, and subsequently, this result was improved and carried over the class of asymptotically *κ*-SPC mappings by Kim and Xu [18].

It is known that Mann’s iteration (1.10) is in general not strongly convergent (see [19]). The way to guarantee strong convergence has been proposed by Nakajo and Takahashi [20]. They modified Mann’s iteration method (1.10), which is to find a fixed point of a nonexpansive mapping by the hybrid method, called the shrinking projection method (or the CQ method), as the following theorem.

**Theorem NT**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*

*T*

*be a nonexpansive mapping of*

*C*

*into itself such that*$F(T)\ne \mathrm{\varnothing}$.

*Suppose that*${x}_{1}=x\in C$

*chosen arbitrarily and*$\{{x}_{n}\}$

*is the sequence defined by*

*where* $0\le {\alpha}_{n}\le \alpha <1$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)}({x}_{1})$.

Subsequently, Marino and Xu [21] introduced an iterative scheme for finding a fixed point of a *κ*-SPC mapping as the following theorem.

**Theorem MX**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*$T:C\to C$

*be a*

*κ*-

*SPC mapping for some*$0\le \kappa <1$.

*Assume that*$F(T)\ne \mathrm{\varnothing}$.

*Suppose that*${x}_{1}=x\in C$

*chosen arbitrarily and*$\{{x}_{n}\}$

*is the sequence defined by*

*where* $0\le {\alpha}_{n}<1$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)}({x}_{1})$.

Quite recently, Kim and Xu [18] have improved and carried Theorem MX over a wider class of asymptotically *κ*-SPC mappings as the following theorem.

**Theorem KX**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*$T:C\to C$

*be an asymptotically*

*κ*-

*SPC mapping for some*$0\le \kappa <1$

*with a bounded sequence*$\{{\gamma}_{n}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$.

*Assume that*$F(T)$

*is a nonempty bounded subset of*

*C*.

*Suppose that*${x}_{1}=x\in C$

*chosen arbitrarily and*$\{{x}_{n}\}$

*is the sequence defined by*

*where* ${\theta}_{n}={\mathrm{\Delta}}_{n}^{2}(1-{\alpha}_{n}){\gamma}_{n}\to 0$ *as* $n\to \mathrm{\infty}$, ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in F(T)\}<\mathrm{\infty}$ *and* $0\le {\alpha}_{n}<1$ *such that* ${lim\hspace{0.17em}sup}_{n\u27f6\mathrm{\infty}}{\alpha}_{n}<1-\kappa $. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)}({x}_{1})$.

*domain*of the function $\phi :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is the set

*mixed equilibrium problem*is to find $x\in C$ such that

*x*is a solution of problem (1.11), then $x\in dom\phi $. If $\phi =0$, then problem (1.11) is reduced to finding $x\in C$ such that

We denote by $EP(\mathrm{\Phi})$ the set of solutions of the *equilibrium problem*. The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity, and optimization and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis, and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and have proved to be productive and innovative. Problem (1.12) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [22, 23] and the references therein. Some methods have been proposed to solve equilibrium problem (1.12); related work can also be found in [7, 12, 24–34].

For solving the mixed equilibrium problem, let us assume that the bifunction $\mathrm{\Phi}:C\times C\to \mathbb{R}$, the function $\phi :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$, and the set *C* satisfy the following conditions:

(A1) $\mathrm{\Phi}(x,x)=0$ for all $x\in C$;

(A2) Φ is monotone; that is, $\mathrm{\Phi}(x,y)+\mathrm{\Phi}(y,x)\le 0$ for all $x,y\in C$;

(A4) for each $x\in C$, $y\mapsto \mathrm{\Phi}(x,y)$ is convex and lower semicontinuous;

(A5) for each $y\in C$, $x\mapsto \mathrm{\Phi}(x,y)$ is weakly upper semicontinuous;

(B2) *C* is a bounded set.

Variational inequality theory provides us with a simple, natural, general, and unified framework for studying a wide class of unrelated problems arising in elasticity, structural analysis, economics, optimization, oceanography, and regional, physical, and engineering sciences, *etc.* (see [35–41] and the references therein). In recent years, variational inequalities have been extended and generalized in different directions, using novel and innovative techniques, both for their own sake and for their applications. A useful and important generalization of variational inequalities is a variational inclusion.

*quasivariational inclusion problem*, which is to find a point $x\in H$ such that

where *θ* is the zero vector in *H*. The set of solutions of problem (1.13) is denoted by $\mathit{VI}(H,B,M)$.

A set-valued mapping $T:H\to {2}^{H}$ is called *monotone* if for all $x,y\in H$, $f\in Tx$ and $g\in Ty$ imply $\u3008x-y,f-g\u3009\ge 0$. A monotone mapping $T:H\to {2}^{H}$ is maximal if the graph of $G(T)$ of *T* is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping *T* is maximal if and only if for $(x,f)\in H\times H$, $\u3008x-y,f-g\u3009\ge 0$ for all $(y,g)\in G(T)$ implies $f\in Tx$.

**Definition 1.4** (see [42])

Let $M:H\to {2}^{H}$ be a multi-valued maximal monotone mapping. Then the single-valued mapping ${J}_{M,\lambda}:H\to H$ defined by ${J}_{M,\lambda}(u)={(I+\lambda M)}^{-1}(u)$, for all $u\in H$, is called *the resolvent operator associated with* *M*, where *λ* is any positive number and *I* is the identity mapping.

*et al.*[43] introduced the following algorithm for a finite family of asymptotically ${\kappa}_{i}$-SPC mappings ${T}_{i}$ on

*C*. Let ${x}_{0}\in C$ and ${\{{\alpha}_{n}\}}_{n=0}^{\mathrm{\infty}}$ be a sequence in $[0,1]$. Let $N\ge 1$ be an integer. The sequence $\{{x}_{n}\}$ generated in the following way:

is called the explicit iterative scheme of a finite family of asymptotically ${\kappa}_{i}$-SPC mappings ${T}_{i}$ on *C*, where $i=1,2,\dots ,N$. Since, for each $n\in \mathbb{N}$, it can be written as $n=(h-1)N+i$, where $i=i(n)\in \{1,2,\dots ,N\}$, $h=h(n)\ge 1$ is a positive integer and $h(n)\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$.

To be more precise, they introduced an iterative scheme for finding a common fixed point of a finite family of asymptotically ${\kappa}_{i}$-SPC mappings as the following theorem.

**Theorem QCKS**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*$N\ge 1$

*be an integer*.

*For each*$i=1,2,\dots ,N$,

*let*$\{{T}_{i}:C\to C\}$

*be a finite family of asymptotically*${\kappa}_{i}$-

*SPC mappings defined as in*(1.15),

*when*${\kappa}_{i}\in [0,1)$

*with the sequence*$\{{\gamma}_{n,i}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}=0$.

*Let*$\kappa =max\{{\kappa}_{i}:1\le i\le N\}$

*and*${\gamma}_{n}=max\{\sqrt{1+{\gamma}_{n,i}}:1\le i\le N\}$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{N}F({T}_{i})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{0}=x\in C$

*chosen arbitrarily*,

*suppose that*$\{{x}_{n}\}$

*is generated iteratively by*

*where* ${\theta}_{n-1}=({\gamma}_{h(n)}^{2}-1)(1-{\alpha}_{n-1})\cdot {\mathrm{\Delta}}_{n-1}^{2}\to 0$ *as* $n\to \mathrm{\infty}$ *such that* ${\mathrm{\Delta}}_{n-1}=sup\{\parallel {x}_{n-1}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$. *If the control sequence* ${\{{\alpha}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is chosen such that* $0\le {\alpha}_{n}\le \alpha <1$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{\mathrm{\Omega}}({x}_{0})$.

*C*is said to be an

*asymptotically*

*κ-SPC semigroup*[44] if the following conditions are satisfied:

- (1)
${T}_{0}=I$, where

*I*denotes the identity operator on*C*; - (2)
${T}_{n+m}x={T}_{n}{T}_{m}x$, $\mathrm{\forall}n,m\ge 0$, $\mathrm{\forall}x\in C$;

- (3)there exists a constant $\kappa \in [0,1)$ and a sequence $\{{\gamma}_{n}\}$ of nonnegative real numbers with ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$ such that$\begin{array}{rcl}{\parallel {T}_{n}x-{T}_{n}y\parallel}^{2}& \le & (1+{\gamma}_{n}){\parallel x-y\parallel}^{2}\\ +\kappa {\parallel (I-{T}_{n})x-(I-{T}_{n})y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C,\end{array}$(1.16)

for all $n\ge 0$. Note that, for a single asymptotically *κ*-SPC mapping $T:C\to C$, (1.16) immediately reduces to (1.6) by taking ${T}_{n}={T}^{n}$ for all $n\ge 0$ such that ${T}^{0}=I$.

On the other hand, Tianchai [24] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of common fixed points for a discrete asymptotically *κ*-SPC semigroup which is a subclass of the class of infinite families for the asymptotically *κ*-SPC mapping as the following theorem.

**Theorem T**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*, Φ

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying the conditions*(A1)-(A5),

*and let*$\phi :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a proper lower semicontinuous and convex function with the assumption that either*(B1)

*or*(B2)

*holds*.

*Let*$\mathcal{S}=\{{T}_{n}:n\ge 0\}$

*be an asymptotically*

*κ*-

*SPC semigroup on*

*C*

*for some*$\kappa \in [0,1)$

*with the sequence*$\{{\gamma}_{n}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0$.

*Assume that*$\mathrm{\Omega}:=F(\mathcal{S})\cap MEP(\mathrm{\Phi},\phi )={\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})\phantom{\rule{0.2em}{0ex}}\cap MEP(\mathrm{\Phi},\phi )$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{0}=x\in C$

*chosen arbitrarily*,

*suppose that*$\{{x}_{n}\}$, $\{{y}_{n}\}$

*and*$\{{u}_{n}\}$

*are generated iteratively by*

*where* ${\theta}_{n}={\gamma}_{n}\cdot sup\{{\parallel {x}_{n}-z\parallel}^{2}:z\in \mathrm{\Omega}\}<\mathrm{\infty}$, $\{{\alpha}_{n}\}\subset [a,b]$ *such that* $\kappa <a<b<1$, $\{{r}_{n}\}\subset [r,\mathrm{\infty})$ *for some* $r>0$ *and* ${\sum}_{n=0}^{\mathrm{\infty}}|{r}_{n+1}-{r}_{n}|<\mathrm{\infty}$. *Then the sequences* $\{{x}_{n}\}$, $\{{y}_{n}\}$, *and* $\{{u}_{n}\}$ *converge strongly to* $w={P}_{\mathrm{\Omega}}({x}_{0})$.

Recently, Saha *et al.* [6] modified Mann’s iteration method (1.10) for finding a fixed point of the asymptotically *κ*-SPC mapping in the intermediate sense which is not necessarily uniformly Lipschitzian (see, *e.g.*, [6, 7]) as the following theorem.

**Theorem SXY**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*$T:C\to C$

*be a uniformly continuous and asymptotically*

*κ*-

*SPC mapping in the intermediate sense defined as in*(1.9),

*when*$\kappa \in [0,1)$,

*with the sequences*$\{{\gamma}_{n}\},\{{\tau}_{n}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}={lim}_{n\to \mathrm{\infty}}{\tau}_{n}=0$.

*Assume that*$F(T)$

*is a nonempty bounded subset of*

*C*.

*Suppose that*${x}_{1}=x\in C$

*chosen arbitrarily and*$\{{x}_{n}\}$

*is the sequence defined by*

*where* ${\theta}_{n}={\gamma}_{n}\cdot {\mathrm{\Delta}}_{n}+{\tau}_{n}\to 0$ *as* $n\to \mathrm{\infty}$, ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in F(T)\}<\mathrm{\infty}$ *and* $\{{\alpha}_{n}\}\subset (0,1]$ *such that* $0<\alpha \le {\alpha}_{n}\le 1-\kappa $. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)}({x}_{1})$.

for all $n\in \mathbb{N}\cup \{0\}$ such that $n=(h-1)N+i$, where $i=i(n)\in \{0,1,\dots ,N-1\}$, $h=h(n)\ge 1$ is a positive integer and $h(n)\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$.

Subsequently, Hu and Cai [7] modified Ishikawa’s iteration method (see [45]) for finding a common element of the set of common fixed points for a finite family of asymptotically ${\kappa}_{i}$-SPC mappings in the intermediate sense and the set of solutions of the equilibrium problems (see also Duan and Zhao [8]) as the following theorem.

**Theorem HC**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*, $\mathrm{\Phi}:C\times C\to \mathbb{R}$

*be a bifunction satisfying the conditions*(A1)-(A4),

*and let*$A:C\to H$

*be a*

*ξ*-

*cocoercive mapping*.

*Let*$N\ge 1$

*be an integer*.

*For each*$i=0,1,\dots ,N-1$,

*let*$\{{T}_{i}:C\to C\}$

*be a finite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.17)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*$\{{\gamma}_{n,i}\},\{{\tau}_{n,i}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Let*$\kappa =max\{{\kappa}_{i}:0\le i\le N-1\}$, ${\gamma}_{n}=max\{{\gamma}_{n,i}:0\le i\le N-1\}$

*and*${\tau}_{n}=max\{{\tau}_{n,i}:0\le i\le N-1\}$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=0}^{N-1}F({T}_{i})\cap EP(\mathrm{\Phi})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{0}=x\in C$

*chosen arbitrarily*,

*suppose that*$\{{x}_{n}\}$

*and*$\{{u}_{n}\}$

*are generated iteratively by*

*where* ${\theta}_{n}={\gamma}_{h(n)}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{h(n)}\to 0$ *as* $n\to \mathrm{\infty}$ *such that* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$. *If the control sequences* $\{{\alpha}_{n}\},\{{\beta}_{n}\}\subset (0,1]$, *and* $\{{r}_{n}\}\subset [a,b]$ *are chosen such that* $0<\alpha \le {\alpha}_{n}\le 1$, $0<\beta \le {\beta}_{n}\phantom{\rule{0.25em}{0ex}}\le 1-\kappa $, *and* $0<a\le {r}_{n}\le b<2\xi $, *then the sequences* $\{{x}_{n}\}$ *and* $\{{u}_{n}\}$ *converge strongly to* ${P}_{\mathrm{\Omega}}({x}_{0})$.

Quite recently, Ezeora and Shehu [9] introduced an iterative scheme for finding a common fixed point of an infinite family of asymptotically ${\kappa}_{i}$-SPC mappings in the intermediate sense as the following theorem.

**Theorem ES**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:C\to C\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{1}=x\in C$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $), ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *and* ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset (0,1]$ ($i=1,2,\dots $) *such that* $0<{\alpha}_{i}\le {\alpha}_{n,i}\le 1-{\kappa}_{i}$. *Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* ${P}_{\mathrm{\Omega}}({x}_{1})$.

Inspired and motivated by the works mentioned above, in this paper, we introduce a new iterative scheme (3.1) below by modifying Mann’s iteration method (1.10) to find a common element for the set of common fixed points of an infinite family of asymptotically ${\kappa}_{i}$-SPC mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu [9] and many others.

## 2 Preliminaries

We collect the following lemmas which be used in the proof of the main results in the next section.

**Lemma 2.1** (see [1])

*Let*

*C*

*be a nonempty closed convex subset of a Hilbert space*

*H*.

*Then the following inequality holds*:

**Lemma 2.2** (see [46])

*Let*

*H*

*be a Hilbert space*.

*For all*$x,y,z\in H$

*and*$\alpha ,\beta ,\gamma \in [0,1]$

*such that*$\alpha +\beta +\gamma =1$,

*one has*

**Lemma 2.3** (see [25])

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*, $\mathrm{\Phi}:C\times C\to \mathbb{R}$

*satisfying the conditions*(A1)-(A5),

*and let*$\phi :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a proper lower semicontinuous and convex function*.

*Assume that either*(B1)

*or*(B2)

*holds*.

*For*$r>0$,

*define a mapping*${S}_{r}:C\to C$

*as follows*:

*for all*$x\in C$.

*Then the following statements hold*:

- (1)
*for each*$x\in C$, ${S}_{r}(x)\ne \mathrm{\varnothing}$; - (2)
${S}_{r}$

*is single*-*valued*; - (3)${S}_{r}$
*is firmly nonexpansive*;*that is*,*for any*$x,y\in C$,${\parallel {S}_{r}x-{S}_{r}y\parallel}^{2}\le \u3008{S}_{r}x-{S}_{r}y,x-y\u3009;$ - (4)
$F({S}_{r})=MEP(\mathrm{\Phi},\phi )$;

- (5)
$MEP(\mathrm{\Phi},\phi )$

*is closed and convex*.

**Lemma 2.4** (see [42])

*Let*$M:H\to {2}^{H}$

*be a maximal monotone mapping and let*$B:H\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

*Then*,

*the following statements hold*:

- (1)
*B**is a*$(1/\alpha )$-*Lipschitz continuous and monotone mapping*. - (2)$u\in H$
*is a solution of quasivariational inclusion*(1.13)*if and only if*$u={J}_{M,\lambda}(u-\lambda Bu)$,*for all*$\lambda >0$,*that is*,$\mathit{VI}(H,B,M)=F({J}_{M,\lambda}(I-\lambda B)),\phantom{\rule{1em}{0ex}}\mathrm{\forall}\lambda >0.$ - (3)
*If*$\lambda \in (0,2\alpha ]$,*then*$\mathit{VI}(H,B,M)$*is a closed convex subset in**H*,*and the mapping*$I-\lambda B$*is nonexpansive*,*where**I**is the identity mapping on**H*. - (4)
*The resolvent operator*${J}_{M,\lambda}$*associated with**M**is single*-*valued and nonexpansive for all*$\lambda >0$. - (5)
*The resolvent operator*${J}_{M,\lambda}$*is*1-*inverse*-*strongly monotone*;*that is*,${\parallel {J}_{M,\lambda}(x)-{J}_{M,\lambda}(y)\parallel}^{2}\le \u3008x-y,{J}_{M,\lambda}(x)-{J}_{M,\lambda}(y)\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in H.$

**Lemma 2.5** (see [47])

*Let* $M:H\to {2}^{H}$ *be a maximal monotone mapping and let* $B:H\to H$ *be a Lipschitz continuous mapping*. *Then the mapping* $S=M+B:H\to {2}^{H}$ *is a maximal monotone mapping*.

**Lemma 2.6** (see [6])

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*$T:C\to C$

*be a uniformly continuous and asymptotically*

*κ*-

*strictly pseudocontractive mapping in the intermediate sense defined as in*(1.9)

*when*$\kappa \in [0,1)$

*with the sequences*$\{{\gamma}_{n}\},\{{\tau}_{n}\}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}={lim}_{n\to \mathrm{\infty}}{\tau}_{n}=0$.

*Then the following statements hold*:

- (1)
$\parallel {T}^{n}x-{T}^{n}y\parallel \le \frac{1}{1-\kappa}(\kappa \parallel x-y\parallel +\sqrt{(1+(1-\kappa ){\gamma}_{n}){\parallel x-y\parallel}^{2}+(1-\kappa ){\tau}_{n}})$,

*for all*$x,y\in C$*and*$n\in \mathbb{N}$. - (2)
*If*$\{{x}_{n}\}$*is a sequence in**C**such that*${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}_{n+1}\parallel =0$*and*${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}^{n}{x}_{n}\parallel =0$.*Then*${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$. - (3)
$I-T$

*is demiclosed at zero in the sense that if*$\{{x}_{n}\}$*is a sequence in**C**such that*${x}_{n}\rightharpoonup x\in C$*as*$n\to \mathrm{\infty}$,*and*${lim\hspace{0.17em}sup}_{m\to \mathrm{\infty}}{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}^{m}{x}_{n}\parallel =0$,*then*$(I-T)x=0$. - (4)
$F(T)$

*is closed and convex*.

## 3 Main results

**Theorem 3.1**

*Let*

*H*

*be a real Hilbert space*, Φ

*be a bifunction from*$H\times H$

*into*ℝ

*satisfying the conditions*(A1)-(A5),

*and let*$\phi :H\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a proper lower semicontinuous and convex function with the assumption that either*(B1)

*or*(B2)

*holds*.

*Let*$M:H\to {2}^{H}$

*be a maximal monotone mapping and let*$B:H\to H$

*be a*

*ξ*-

*cocoercive mapping*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:H\to H\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap \mathit{VI}(H,B,M)\cap MEP(\mathrm{\Phi},\phi )$

*is a nonempty bounded subset of*

*H*.

*For*${x}_{1}=x\in H$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

(C1) ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{1,i},{b}_{1,i}]$ *and* ${\{{\beta}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{2,i},{b}_{2,i}]$ ($i=1,2,\dots $) *such that* $0<{a}_{j,i}<{b}_{j,i}<1$ *for each* $j=1,2$, *and* ${b}_{1,i}+{b}_{2,i}<1-{\kappa}_{i}$ ($i=1,2,\dots $);

(C2) $\lambda \in (0,2\xi ]$ *and* ${\{{r}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset [r,\mathrm{\infty})$ *for some* $r>0$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

*Proof*Pick $p\in \mathrm{\Omega}$ and fix $i=1,2,\dots $ . From (3.1), by the definition of ${S}_{{r}_{n}}$ in Lemma 2.3, we have

where ${\theta}_{n,i}:={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ and ${\mathrm{\Delta}}_{n}:=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}$.

for all $n\in \mathbb{N}$. Since ${C}_{1}\cap {Q}_{1}$ is convex, by putting $n=1$ in (3.7), (3.8), and (3.9), we have that ${C}_{2}\cap {Q}_{2}$ is convex. Suppose that ${x}_{k}$ is given and ${C}_{k}\cap {Q}_{k}$ is convex for some $k\ge 2$. It follows by putting $n=k$ in (3.7), (3.8), and (3.9) that ${C}_{k+1}\cap {Q}_{k+1}$ is convex. Therefore, by mathematical induction, we have that ${C}_{n}\cap {Q}_{n}$ is convex for all $n\ge 2$, that is, ${C}_{n}\cap {Q}_{n}$ is convex for all $n\in \mathbb{N}$. Hence, we obtain that ${C}_{n}\cap {Q}_{n}$ is closed and convex for all $n\in \mathbb{N}$.

*i*, and so $p\in {C}_{2}$, and note that $p\in H={Q}_{2}$, and so $p\in {C}_{2}\cap {Q}_{2}$. Hence, we have $\mathrm{\Omega}\subset {C}_{2}\cap {Q}_{2}$. Since ${C}_{2}\cap {Q}_{2}$ is a nonempty closed convex subset of

*H*, there exists a unique element ${x}_{2}\in {C}_{2}\cap {Q}_{2}$ such that ${x}_{2}={P}_{{C}_{2}\cap {Q}_{2}}({x}_{1})$. Suppose that ${x}_{k}\in {C}_{k}\cap {Q}_{k}$ is given such that ${x}_{k}={P}_{{C}_{k}\cap {Q}_{k}}({x}_{1})$, and $p\in \mathrm{\Omega}\subset {C}_{k}\cap {Q}_{k}$ for some $k\ge 2$. Therefore, by (3.1) and (3.6), we have $p\in {C}_{k+1,i}$ for all

*i*, and so $p\in {C}_{k+1}$. Since ${x}_{k}={P}_{{C}_{k}\cap {Q}_{k}}({x}_{1})$, therefore, by Lemma 2.1, we have

for all $z\in {C}_{k}\cap {Q}_{k}$. Thus, by (3.1), we have $p\in {Q}_{k+1}$, and so $p\in {C}_{k+1}\cap {Q}_{k+1}$. Hence, we have $\mathrm{\Omega}\subset {C}_{k+1}\cap {Q}_{k+1}$. Since ${C}_{k+1}\cap {Q}_{k+1}$ is a nonempty closed convex subset of *H*, there exists a unique element ${x}_{k+1}\in {C}_{k+1}\cap {Q}_{k+1}$ such that ${x}_{k+1}={P}_{{C}_{k+1}\cap {Q}_{k+1}}({x}_{1})$. Therefore, by mathematical induction, we obtain that $\mathrm{\Omega}\subset {C}_{n}\cap {Q}_{n}$ for all $n\ge 2$, and so $\mathrm{\Omega}\subset {C}_{n}\cap {Q}_{n}$ for all $n\in \mathbb{N}$, and we can define ${x}_{n+1}={P}_{{C}_{n+1}\cap {Q}_{n+1}}({x}_{1})$ for all $n\in \mathbb{N}$. Hence, we obtain that the iteration (3.1) is well defined, and by Lemmas 2.3(5), 2.4(3), and 2.6(4), we also obtain that ${P}_{\mathrm{\Omega}}({x}_{1})$ is well defined.

for all $z\in {C}_{n}\cap {Q}_{n}$. It follows by $p\in \mathrm{\Omega}\subset {C}_{n}\cap {Q}_{n}$ that $\parallel {x}_{n}-{x}_{1}\parallel \le \parallel p-{x}_{1}\parallel $ for all $n\in \mathbb{N}$. This implies that $\{{x}_{n}\}$ is bounded, and so are $\{{u}_{n}\}$, $\{{t}_{n}\}$, and $\{{y}_{n,i}\}$ for each $i=1,2,\dots $ .

Hence, by (3.11), we obtain that $\parallel {x}_{n+m}-{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, which implies that $\{{x}_{n}\}$ is a Cauchy sequence in *H*, and then there exists a point $w\in H$ such that ${x}_{n}\to w$ as $n\to \mathrm{\infty}$.

*I*is the identity mapping on

*H*. Therefore, by (3.27) and (3.28), we obtain

From (3.22) and ${x}_{n}\to w$, we have ${u}_{n}\rightharpoonup w$ as $n\to \mathrm{\infty}$. Therefore, from (3.29), by Lemma 2.6(3), we obtain that $w\in F({T}_{i})$ for all $i=1,2,\dots $ ; that is, $w\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$.

*t*with $0<t<1$ and $y\in H$, let ${y}_{t}=ty+(1-t)w$. Since $y,w\in H$, thus, ${y}_{t}\in H$. So, from (3.31), we have

*φ*, we have

Therefore, by the condition (A3) and the weakly lower semicontinuity of *φ*, we have $\mathrm{\Phi}(w,y)+\phi (y)-\phi (w)\ge 0$ as $t\to 0$ for all $y\in H$, and hence we obtain that $w\in MEP(\mathrm{\Phi},\phi )$.

*B*is $(1/\xi )$-Lipschitz continuous. Therefore, by Lemma 2.5, we have that $M+B$ is maximal monotone. Let $(y,g)\in G(M+B)$, that is,

*M*is maximal monotone, we have

*B*that

From (3.33), we have $\parallel B{t}_{n}-B{u}_{n}\parallel \to 0$, and since ${x}_{n}\to w$, by (3.22) and (3.33), we have ${t}_{n}\to w$ as $n\to \mathrm{\infty}$. Therefore, by (3.33) and (3.36), we obtain that $\u3008y-w,g\u3009\ge 0$ as $n\to \mathrm{\infty}$. It follows from the maximal monotonicity of $M+B$ that $0\in (M+B)w$; that is, $w\in \mathit{VI}(H,B,M)$, and so $w\in \mathrm{\Omega}$.

Therefore, by Lemma 2.1, we obtain that $w={P}_{\mathrm{\Omega}}({x}_{1})$. This completes the proof. □

**Remark 3.2**The iteration (3.1) is different from the iterative scheme of Ezeora and Shehu [9] as follows:

- 1.The sequence $\{{x}_{n}\}$ is a projection sequence of ${x}_{1}$ onto ${C}_{n}\cap {Q}_{n}$ for all $n\in \mathbb{N}$ such that${C}_{1}\cap {Q}_{1}\supset {C}_{2}\cap {Q}_{2}\supset \cdots \supset {C}_{n}\cap {Q}_{n}\supset \cdots \supset \mathrm{\Omega}.$
- 2.
The proof to the strong convergence of the sequence $\{{x}_{n}\}$ is simple by a Cauchy sequence.

- 3.
An approximate solution to a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems by iteration is obtained.

We define the condition (B3) as the condition (B1) such that $\phi =0$. If $\phi =0$, then Theorem 3.1 is reduced immediately to the following result.

**Corollary 3.3**

*Let*

*H*

*be a real Hilbert space and let*Φ

*be a bifunction from*$H\times H$

*into*ℝ

*satisfying the conditions*(A1)-(A5)

*with the assumption that either*(B2)

*or*(B3)

*holds*.

*Let*$M:H\to {2}^{H}$

*be a maximal monotone mapping and let*$B:H\to H$

*be a*

*ξ*-

*cocoercive mapping*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:H\to H\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap \mathit{VI}(H,B,M)\cap EP(\mathrm{\Phi})$

*is a nonempty bounded subset of*

*H*.

*For*${x}_{1}=x\in H$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

(C1) ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{1,i},{b}_{1,i}]$ *and* ${\{{\beta}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{2,i},{b}_{2,i}]$ ($i=1,2,\dots $) *such that* $0<{a}_{j,i}<{b}_{j,i}<1$ *for each* $j=1,2$, *and* ${b}_{1,i}+{b}_{2,i}<1-{\kappa}_{i}$ ($i=1,2,\dots $);

(C2) $\lambda \in (0,2\xi ]$ *and* ${\{{r}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset [r,\mathrm{\infty})$ *for some* $r>0$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

If $\mathrm{\Phi}=0$, then Corollary 3.3 is reduced immediately to the following result.

**Corollary 3.4**

*Let*

*H*

*be a real Hilbert space*, $M:H\to {2}^{H}$

*be a maximal monotone mapping*,

*and let*$B:H\to H$

*be a*

*ξ*-

*cocoercive mapping*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:H\to H\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap \mathit{VI}(H,B,M)$

*is a nonempty bounded subset of*

*H*.

*For*${x}_{1}=x\in H$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

(C1) ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{1,i},{b}_{1,i}]$ *and* ${\{{\beta}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{2,i},{b}_{2,i}]$ ($i=1,2,\dots $) *such that* $0<{a}_{j,i}<{b}_{j,i}<1$ *for each* $j=1,2$, *and* ${b}_{1,i}+{b}_{2,i}<1-{\kappa}_{i}$ ($i=1,2,\dots $);

(C2) $\lambda \in (0,2\xi ]$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

If $B=0$ and $M=0$, then Theorem 3.1 is reduced immediately to the following result.

**Corollary 3.5**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*, Φ

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying the conditions*(A1)-(A5),

*and let*$\phi :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a proper lower semicontinuous and convex function with the assumption that either*(B1)

*or*(B2)

*holds*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:C\to C\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap MEP(\mathrm{\Phi},\phi )$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{1}=x\in C$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

(C1) ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [a,b]$ ($i=1,2,\dots $) *such that* $0<a<b<1-{\kappa}_{i}$;

(C2) ${\{{r}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset [r,\mathrm{\infty})$ *for some* $r>0$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

If $\phi =0$, then Corollary 3.5 is reduced immediately to the following result.

**Corollary 3.6**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*Φ

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying the conditions*(A1)-(A5)

*with the assumption that either*(B2)

*or*(B3)

*holds*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:C\to C\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap \phantom{\rule{0.2em}{0ex}}EP(\mathrm{\Phi})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{1}=x\in C$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

(C1) ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [a,b]$ ($i=1,2,\dots $) *such that* $0<a<b<1-{\kappa}_{i}$;

(C2) ${\{{r}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset [r,\mathrm{\infty})$ *for some* $r>0$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

If $\mathrm{\Phi}=0$, then Corollary 3.6 is reduced immediately to the following result.

**Corollary 3.7**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:C\to C\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{1}=x\in C$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $), ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *and* ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [a,b]$ ($i=1,2,\dots $) *such that* $0<a<b<1-{\kappa}_{i}$. *Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

*asymptotically nonexpansive*if there exists a sequence $\{{\gamma}_{n,i}\}\subset [0,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}=0$ such that

for all $n\in \mathbb{N}$. If ${\kappa}_{i}=0$ and ${\tau}_{n,i}=0$ for all $i=1,2,\dots $ and $n\in \mathbb{N}$, then Corollary 3.7 is reduced immediately to the following result.

**Corollary 3.8**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:C\to C\}$

*be an infinite family of asymptotically nonexpansive mappings defined as in*(3.37)

*with the sequence*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$

*is a nonempty bounded subset of*

*C*.

*For*${x}_{1}=x\in C$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}$ ($i=1,2,\dots $), ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$, *and* ${\{{\alpha}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [a,b]$ ($i=1,2,\dots $) *such that* $0<a<b<1$. *Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

## 4 Applications

where *C* is a nonempty closed convex subset of a real Hilbert space *H* and $\zeta :C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is proper convex and lower semicontinuous. We denote by $Argmin(\zeta )$ the set of solutions of problem (4.1). We define the condition (B4) as the condition (B3) such that $\mathrm{\Phi}:C\times C\to \mathbb{R}$ is a bifunction defined by $\mathrm{\Phi}(x,y)=\zeta (y)-\zeta (x)$ for all $x,y\in C$. Observe that $EP(\mathrm{\Phi})=Argmin(\zeta )$. We obtain that Corollary 3.3 is reduced immediately to the following result.

**Theorem 4.1**

*Let*

*H*

*be a real Hilbert space and let*$\zeta :H\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a proper lower semicontinuous and convex function with the assumption that either*(B2)

*or*(B4)

*holds*.

*Let*$M:H\to {2}^{H}$

*be a maximal monotone mapping and let*$B:H\to H$

*be a*

*ξ*-

*cocoercive mapping*.

*For each*$i=1,2,\dots $ ,

*let*$\{{T}_{i}:H\to H\}$

*be an infinite family of uniformly continuous and asymptotically*${\kappa}_{i}$-

*SPC mappings in the intermediate sense defined as in*(1.18)

*when*${\kappa}_{i}\in [0,1)$

*with the sequences*${\{{\gamma}_{n,i}\}}_{n=1}^{\mathrm{\infty}},{\{{\tau}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [0,\mathrm{\infty})$

*such that*${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}={lim}_{n\to \mathrm{\infty}}{\tau}_{n,i}=0$.

*Assume that*$\mathrm{\Omega}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap \mathit{VI}(H,B,M)\cap Argmin(\zeta )$

*is a nonempty bounded subset of H*.

*For*${x}_{1}=x\in H$

*chosen arbitrarily*,

*suppose that*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is generated iteratively by*

*where* ${\theta}_{n,i}={\gamma}_{n,i}\cdot {\mathrm{\Delta}}_{n}^{2}+{\tau}_{n,i}$ ($i=1,2,\dots $) *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in \mathrm{\Omega}\}<\mathrm{\infty}$ *satisfying the following conditions*:

*and* ${\{{\beta}_{n,i}\}}_{n=1}^{\mathrm{\infty}}\subset [{a}_{2,i},{b}_{2,i}]$ ($i=1,2,\dots $) *such that* $0<{a}_{j,i}<{b}_{j,i}<1$ *for each* $j=1,2$, *and* ${b}_{1,i}+{b}_{2,i}<1-{\kappa}_{i}$ ($i=1,2,\dots $);

(C2) $\lambda \in (0,2\xi ]$ *and* ${\{{r}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset [r,\mathrm{\infty})$ *for some* $r>0$.

*Then the sequence* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* $w={P}_{\mathrm{\Omega}}({x}_{1})$.

## Declarations

### Acknowledgements

The author would like to thank the Faculty of Science, Maejo University for its financial support.

## Authors’ Affiliations

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