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# On generalized quasi-ϕ-nonexpansive mappings and their projection algorithms

Fixed Point Theory and Applications20132013:204

https://doi.org/10.1186/1687-1812-2013-204

• Received: 29 April 2013
• Accepted: 11 July 2013
• Published:

## Abstract

A fixed point problem of a generalized asymptotically quasi-ϕ-nonexpansive mapping and an equilibrium problem are investigated. A strong convergence theorem for solutions of the fixed point problem and the equilibrium problem is established in a Banach space.

## Keywords

• asymptotically quasi-ϕ-nonexpansive mapping
• equilibrium problem
• fixed point
• generalized asymptotically quasi-ϕ-nonexpansive mapping
• generalized projection

## 1 Introduction and preliminaries

Let E be a real Banach space, and let ${E}^{\ast }$ be the dual space of E. We denote by J the normalized duality mapping from E to ${2}^{{E}^{\ast }}$ defined by
$Jx=\left\{{f}^{\ast }\in {E}^{\ast }:〈x,{f}^{\ast }〉={\parallel x\parallel }^{2}={\parallel {f}^{\ast }\parallel }^{2}\right\},$
where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. A Banach space E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. It is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ in E such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1$. Let ${U}_{E}=\left\{x\in E:\parallel x\parallel =1\right\}$ be the unit sphere of E. Then the Banach space E is said to be smooth provided
$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$

exists for each $x,y\in {U}_{E}$. It is also said to be uniformly smooth if the above limit is attained uniformly for $x,y\in {U}_{E}$. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if ${E}^{\ast }$ is uniformly convex.

Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence $\left\{{x}_{n}\right\}\subset E$, and $x\in E$ with ${x}_{n}⇀x$, and $\parallel {x}_{n}\parallel \to \parallel x\parallel$, then $\parallel {x}_{n}-x\parallel \to 0$ as $n\to \mathrm{\infty }$. For more details on the Kadec-Klee property, the readers can refer to  and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Let C be a nonempty subset of E. Let f be a bifunction from $C×C$ to , where denotes the set of real numbers. In this paper, we investigate the following equilibrium problem. Find $p\in C$ such that
$f\left(p,y\right)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(1.1)
We use $EP\left(f\right)$ to denote the solution set of the equilibrium problem (1.1). That is,
$EP\left(f\right)=\left\{p\in C:f\left(p,y\right)\ge 0,\mathrm{\forall }y\in C\right\}.$
Given a mapping $Q:C\to {E}^{\ast }$, let
$f\left(x,y\right)=〈Qx,y-x〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$
Then $p\in EP\left(f\right)$ iff p is a solution of the following variational inequality. Find p such that
$〈Qp,y-p〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(1.2)
In order to study the solution problem of the equilibrium problem (1.1), we assume that f satisfies the following conditions:
1. (A1)

$f\left(x,x\right)=0$, $\mathrm{\forall }x\in C$;

2. (A2)

f is monotone, i.e., $f\left(x,y\right)+f\left(y,x\right)\le 0$, $\mathrm{\forall }x,y\in C$;

3. (A3)
$\underset{t↓0}{lim sup}f\left(tz+\left(1-t\right)x,y\right)\le f\left(x,y\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y,z\in C;$

4. (A4)

for each $x\in C$, $y↦f\left(x,y\right)$ is convex and weakly lower semi-continuous.

As we all know, if C is a nonempty closed convex subset of a Hilbert space H and ${P}_{C}:H\to C$ is the metric projection of H onto C, then ${P}_{C}$ is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber  recently introduced a generalized projection operator ${\mathrm{\Pi }}_{C}$ in a Banach space E, which is an analogue of the metric projection ${P}_{C}$ in Hilbert spaces.

Next, we assume that E is a smooth Banach space. Consider the functional defined by
$\varphi \left(x,y\right)={\parallel x\parallel }^{2}-2〈x,Jy〉+{\parallel y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E.$
Observe that in a Hilbert space H, the equality is reduced to $\varphi \left(x,y\right)={\parallel x-y\parallel }^{2}$, $x,y\in H$. The generalized projection ${\mathrm{\Pi }}_{C}:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi \left(x,y\right)$, that is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem
$\varphi \left(\overline{x},x\right)=\underset{y\in C}{min}\varphi \left(y,x\right).$
Existence and uniqueness of the operator ${\mathrm{\Pi }}_{C}$ follows from the properties of the functional $\varphi \left(x,y\right)$ and strict monotonicity of the mapping J; see, for example,  and . In Hilbert spaces, ${\mathrm{\Pi }}_{C}={P}_{C}$. It is obvious from the definition of function ϕ that
${\left(\parallel x\parallel -\parallel y\parallel \right)}^{2}\le \varphi \left(x,y\right)\le {\left(\parallel y\parallel +\parallel x\parallel \right)}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in E$
(1.3)
and
$\varphi \left(x,y\right)=\varphi \left(x,z\right)+\varphi \left(z,y\right)+2〈x-z,Jz-Jy〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y,z\in E.$
(1.4)

Remark 1.1 If E is a reflexive, strictly convex and smooth Banach space, then $\varphi \left(x,y\right)=0$ if and only if $x=y$; for more details, see  and .

Let $T:C\to C$ be a mapping. In this paper, we use $F\left(T\right)$ to denote the fixed point set of T. T is said to be asymptotically regular on C if, for any bounded subset K of C, ${lim}_{n\to \mathrm{\infty }}{sup}_{x\in K}\parallel {T}^{n+1}x-{T}^{n}x\parallel =0$. T is said to be closed if, for any sequence $\left\{{x}_{n}\right\}\subset C$ such that ${lim}_{n\to \mathrm{\infty }}{x}_{n}={x}_{0}$ and ${lim}_{n\to \mathrm{\infty }}T{x}_{n}={y}_{0}$, $T{x}_{0}={y}_{0}$. In this paper, we use → and to denote the strong convergence and weak convergence, respectively.

A point p in C is said to be an asymptotic fixed point of T  iff C contains a sequence $\left\{{x}_{n}\right\}$ which converges weakly to p such that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The set of asymptotic fixed points of T will be denoted by $\stackrel{˜}{F}\left(T\right)$. T is said to be relatively nonexpansive [4, 5] iff $\stackrel{˜}{F}\left(T\right)=F\left(T\right)\ne \mathrm{\varnothing }$ and $\varphi \left(p,Tx\right)\le \varphi \left(p,x\right)$ for all $x\in C$ and $p\in F\left(T\right)$. T is said to be relatively asymptotically nonexpansive [6, 7] iff $\stackrel{˜}{F}\left(T\right)=F\left(T\right)\ne \mathrm{\varnothing }$ and there exists a sequence $\left\{{\mu }_{n}\right\}\subset \left[1,\mathrm{\infty }\right)$ with ${\mu }_{n}\to 1$ as $n\to \mathrm{\infty }$ such that $\varphi \left(p,Tx\right)\le {\mu }_{n}\varphi \left(p,x\right)$ for all $x\in C$, $p\in F\left(T\right)$ and $n\ge 1$. T is said to be quasi-ϕ-nonexpansive [8, 9] iff $F\left(T\right)\ne \mathrm{\varnothing }$ and $\varphi \left(p,Tx\right)\le \varphi \left(p,x\right)$ for all $x\in C$ and $p\in F\left(T\right)$. T is said to be asymptotically quasi-ϕ-nonexpansive  iff $F\left(T\right)\ne \mathrm{\varnothing }$ and there exists a sequence $\left\{{\mu }_{n}\right\}\subset \left[1,\mathrm{\infty }\right)$ with ${\mu }_{n}\to 1$ as $n\to \mathrm{\infty }$ such that $\varphi \left(p,Tx\right)\le {\mu }_{n}\varphi \left(p,x\right)$ for all $x\in C$, $p\in F\left(T\right)$ and $n\ge 1$.

Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings which requires the restriction $F\left(T\right)=\stackrel{˜}{F}\left(T\right)$.

Remark 1.3 The classes of asymptotically quasi-ϕ-nonexpansive mappings and quasi-ϕ-nonexpansive mappings are the generalizations of asymptotically quasi-nonexpansive mappings and quasi-nonexpansive mappings in Hilbert spaces.

Recently, Qin et al.  introduced a class of generalized asymptotically quasi-ϕ-nonexpansive mappings. Recall that a mapping T is said to be generalized asymptotically quasi-ϕ-nonexpansive iff $F\left(T\right)\ne \mathrm{\varnothing }$ and there exist a sequence $\left\{{\mu }_{n}\right\}\subset \left[1,\mathrm{\infty }\right)$ with ${\mu }_{n}\to 1$ as $n\to \mathrm{\infty }$ and a sequence $\left\{{\nu }_{n}\right\}\subset \left[0,\mathrm{\infty }\right)$ with ${\nu }_{n}\to 0$ as $n\to \mathrm{\infty }$ such that $\varphi \left(p,Tx\right)\le {\mu }_{n}\varphi \left(p,x\right)+{\nu }_{n}$ for all $x\in C$, $p\in F\left(T\right)$ and $n\ge 1$.

Remark 1.4 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings which was studied in .

Recently, fixed point and equilibrium problems (1.1) have been intensively investigated based on iterative methods; see . The projection method which grants strong convergence of the iterative sequences is one of efficient methods for the problems. In this paper, we investigate the equilibrium problem (1.1) and a fixed point problem of the generalized quasi-ϕ-nonexpansive mapping based on a projection method. A strong convergence theorem for solutions of the equilibrium and the fixed point problem is established in a Banach space.

In order to state our main results, we need the following lemmas.

Lemma 1.5 

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and $x\in E$. Then
$\varphi \left(y,{\mathrm{\Pi }}_{C}x\right)+\varphi \left({\mathrm{\Pi }}_{C}x,x\right)\le \varphi \left(y,x\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

Lemma 1.6 

Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and $x\in E$. Then ${x}_{0}={\mathrm{\Pi }}_{C}x$ if and only if
$〈{x}_{0}-y,Jx-J{x}_{0}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

Lemma 1.7 

Let E be a reflexive, strictly convex, and smooth Banach space such that both E and ${E}^{\ast }$ have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let $T:C\to C$ be a closed asymptotically quasi-ϕ-nonexpansive mapping. Then $F\left(T\right)$ is a closed convex subset of C.

Lemma 1.8 [29, 30]

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4). Let $r>0$ and $x\in E$. Then there exists $z\in C$ such that $f\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0$, $\mathrm{\forall }y\in C$. Define a mapping ${S}_{r}:E\to C$ by ${S}_{r}x=\left\{z\in C:f\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C\right\}$. Then the following conclusions hold:
1. (1)
${S}_{r}$ is a single-valued and firmly nonexpansive-type mapping, i.e., for all $x,y\in E$,
$〈{S}_{r}x-{S}_{r}y,J{S}_{r}x-J{S}_{r}y〉\le 〈{S}_{r}x-{S}_{r}y,Jx-Jy〉;$

2. (2)

$F\left({S}_{r}\right)=EP\left(f\right)$ is closed and convex;

3. (3)

${S}_{r}$ is quasi-ϕ-nonexpansive;

4. (4)

$\varphi \left(q,{S}_{r}x\right)+\varphi \left({S}_{r}x,x\right)\le \varphi \left(q,x\right)$, $\mathrm{\forall }q\in F\left({S}_{r}\right)$.

Lemma 1.9 

Let E be a smooth and uniformly convex Banach space, and let $r>0$. Then there exists a strictly increasing, continuous and convex function $g:\left[0,2r\right]\to R$ such that $g\left(0\right)=0$ and
${\parallel tx+\left(1-t\right)y\parallel }^{2}\le t{\parallel x\parallel }^{2}+\left(1-t\right){\parallel y\parallel }^{2}-t\left(1-t\right)g\left(\parallel x-y\parallel \right)$

for all $x,y\in {B}_{r}=\left\{x\in E:\parallel x\parallel \le r\right\}$ and $t\in \left[0,1\right]$.

## 2 Main results

Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4), and let $T:C\to C$ be a closed generalized asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that $\mathcal{F}=F\left(T\right)\cap EP\left(f\right)$ is nonempty and bounded. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:
$\left\{\begin{array}{c}{x}_{0}\in E\phantom{\rule{1em}{0ex}}\mathit{\text{chosen arbitrarily}},\hfill \\ {C}_{1}=C,\hfill \\ {x}_{1}={\mathrm{\Pi }}_{{C}_{1}}{x}_{0},\hfill \\ {y}_{n}={J}^{-1}\left({\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)J{T}^{n}{x}_{n}\right),\hfill \\ {u}_{n}\in C\phantom{\rule{1em}{0ex}}\mathit{\text{such that}}\phantom{\rule{1em}{0ex}}f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left(z,{u}_{n}\right)\le \varphi \left(z,{x}_{n}\right)+\left({\mu }_{n}-1\right){M}_{n}+{\nu }_{n}\right\},\hfill \\ {x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1},\hfill \end{array}$

where ${M}_{n}=sup\left\{\varphi \left(z,{x}_{n}\right):z\in \mathcal{F}\right\}$, $\left\{{\alpha }_{n}\right\}$ is a real sequence in $\left[0,1\right]$ such that ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}\right)>0$, and $\left\{{r}_{n}\right\}$ is a real sequence in $\left[a,\mathrm{\infty }\right)$, where a is some positive real number. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_{1}$.

Proof In view of Lemma 1.7 and Lemma 1.8, we find that is closed and convex, so that ${\mathrm{\Pi }}_{\mathcal{F}}x$ is well defined for any $x\in C$. Next, we show that ${C}_{n}$ is closed and convex. It is obvious that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{m}$ is closed and convex for some $m\in \mathbb{N}$. We now show that ${C}_{m+1}$ is also closed and convex. For ${z}_{1},{z}_{2}\in {C}_{m+1}$, we see that ${z}_{1},{z}_{2}\in {C}_{m}$. It follows that $z=t{z}_{1}+\left(1-t\right){z}_{2}\in {C}_{m}$, where $t\in \left(0,1\right)$. Notice that
$\varphi \left({z}_{1},{u}_{m}\right)\le \varphi \left({z}_{1},{x}_{m}\right)+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m}$
and
$\varphi \left({z}_{1},{u}_{h}\right)\le \varphi \left({z}_{1},{x}_{m}\right)+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m}.$
The above inequalities are equivalent to
$2〈{z}_{1},J{x}_{m}-J{u}_{m}〉\le {\parallel {x}_{m}\parallel }^{2}-{\parallel {u}_{m}\parallel }^{2}+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m}$
(2.1)
and
$2〈{z}_{2},J{x}_{m}-J{u}_{m}〉\le {\parallel {x}_{m}\parallel }^{2}-{\parallel {u}_{m}\parallel }^{2}+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m}.$
(2.2)
Multiplying t and $\left(1-t\right)$ on both sides of (2.1) and (2.2), respectively, yields that
$2〈z,J{x}_{m}-J{u}_{m}〉\le {\parallel {x}_{m}\parallel }^{2}-{\parallel {y}_{m}\parallel }^{2}+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m}.$
That is,
$\varphi \left(z,{u}_{m}\right)\le \varphi \left(z,{x}_{m}\right)+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m},$
(2.3)

where $z\in {C}_{m}$. This gives that ${C}_{m+1}$ is closed and convex. Then ${C}_{n}$ is closed and convex. This shows that ${\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}$ is well defined.

Next, we show that $\mathcal{F}\subset {C}_{n}$. $\mathcal{F}\subset {C}_{1}=C$ is obvious. Suppose that $\mathcal{F}\subset {C}_{m}$ for some $m\in \mathbb{N}$. Fix $w\in \mathcal{F}\subset {C}_{m}$. It follows that
$\begin{array}{rcl}\varphi \left(w,{u}_{m}\right)& =& \varphi \left(w,{S}_{{r}_{m}}{y}_{m}\right)\\ \le & \varphi \left(w,{y}_{m}\right)\\ =& \varphi \left(w,{J}^{-1}\left({\alpha }_{m}J{x}_{m}+\left(1-{\alpha }_{m}\right)J{T}^{m}{x}_{m}\right)\right)\\ =& {\parallel w\parallel }^{2}-2〈w,{\alpha }_{m}J{x}_{m}+\left(1-{\alpha }_{m}\right)J{T}^{m}{x}_{m}〉+{\parallel {\alpha }_{m}J{x}_{m}+\left(1-{\alpha }_{m}\right)J{T}^{m}{x}_{m}\parallel }^{2}\\ \le & {\parallel w\parallel }^{2}-2{\alpha }_{m}〈w,J{x}_{m}〉-2\left(1-{\alpha }_{m}\right)〈w,J{T}^{m}{x}_{m}〉+{\alpha }_{m}{\parallel {x}_{m}\parallel }^{2}+\left(1-{\alpha }_{m}\right){\parallel {T}^{m}{x}_{m}\parallel }^{2}\\ =& {\alpha }_{m}\varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right)\varphi \left(w,{T}^{m}{x}_{m}\right)\\ \le & {\alpha }_{m}\varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right){\mu }_{m}\varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right){\nu }_{m}\\ =& \varphi \left(w,{x}_{m}\right)-\left(1-{\alpha }_{m}\right)\varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right){\mu }_{m}\varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right){\nu }_{m}\\ \le & \varphi \left(w,{x}_{m}\right)+\left(1-{\alpha }_{m}\right)\left({\mu }_{m}-1\right)\varphi \left(w,{x}_{m}\right)+{\nu }_{m}\\ \le & \varphi \left(w,{x}_{m}\right)+\left({\mu }_{m}-1\right){M}_{m}+{\nu }_{m},\end{array}$
(2.4)
which shows that $w\in {C}_{m+1}$. This implies that $\mathcal{F}\subset {C}_{n}$ for each $n\ge 1$. In view of ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$, from Lemma 1.6 we find that $〈{x}_{n}-z,J{x}_{1}-J{x}_{n}〉\ge 0$ for any $z\in {C}_{n}$. It follows from $\mathcal{F}\subset {C}_{n}$ that
$〈{x}_{n}-w,J{x}_{1}-J{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }w\in \mathcal{F}.$
(2.5)
It follows from Lemma 1.5 that
$\begin{array}{rl}\varphi \left({x}_{n},{x}_{1}\right)& =\varphi \left({\mathrm{\Pi }}_{{C}_{n}}{x}_{1},{x}_{1}\right)\\ \le \varphi \left({\mathrm{\Pi }}_{\mathcal{F}}{x}_{1},{x}_{1}\right)-\varphi \left({\mathrm{\Pi }}_{\mathcal{F}}{x}_{1},{x}_{n}\right)\\ \le \varphi \left({\mathrm{\Pi }}_{\mathcal{F}}{x}_{1},{x}_{1}\right).\end{array}$
This implies that the sequence $\left\{\varphi \left({x}_{n},{x}_{1}\right)\right\}$ is bounded. It follows from (1.3) that the sequence $\left\{{x}_{n}\right\}$ is also bounded. Since the space is reflexive, we may assume that ${x}_{n}⇀\overline{x}$. Since ${C}_{n}$ is closed and convex, we find that $\overline{x}\in {C}_{n}$. On the other hand, we see from the weak lower semicontinuity of the norm that
$\begin{array}{rl}\varphi \left(\overline{x},{x}_{1}\right)& ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},J{x}_{1}〉+{\parallel {x}_{1}\parallel }^{2}\\ \le \underset{n\to \mathrm{\infty }}{lim inf}\left({\parallel {x}_{n}\parallel }^{2}-2〈{x}_{n},J{x}_{1}〉+{\parallel {x}_{1}\parallel }^{2}\right)\\ =\underset{n\to \mathrm{\infty }}{lim inf}\varphi \left({x}_{n},{x}_{1}\right)\\ \le \underset{n\to \mathrm{\infty }}{lim sup}\varphi \left({x}_{n},{x}_{1}\right)\\ \le \varphi \left(\overline{x},{x}_{1}\right),\end{array}$

which implies that ${lim}_{n\to \mathrm{\infty }}\varphi \left({x}_{n},{x}_{1}\right)=\varphi \left(\overline{x},{x}_{1}\right)$. Hence, we have ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}\parallel =\parallel \overline{x}\parallel$. In view of the Kadec-Klee property of E, we find that ${x}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$.

Now, we are in a position to prove that $\overline{x}\in F\left(T\right)$. Since ${x}_{n}={\mathrm{\Pi }}_{{C}_{n}}{x}_{1}$ and ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$, we find that $\varphi \left({x}_{n},{x}_{1}\right)\le \varphi \left({x}_{n+1},{x}_{1}\right)$. This shows that $\left\{\varphi \left({x}_{n},{x}_{1}\right)\right\}$ is nondecreasing. It follows from the boundedness that ${lim}_{n\to \mathrm{\infty }}\varphi \left({x}_{n},{x}_{1}\right)$ exists. In view of the construction of ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$, we arrive at
$\begin{array}{rl}\varphi \left({x}_{n+1},{x}_{n}\right)& =\varphi \left({x}_{n+1},{\mathrm{\Pi }}_{{C}_{n}}{x}_{1}\right)\\ \le \varphi \left({x}_{n+1},{x}_{1}\right)-\varphi \left({\mathrm{\Pi }}_{{C}_{n}}{x}_{1},{x}_{1}\right)\\ =\varphi \left({x}_{n+1},{x}_{1}\right)-\varphi \left({x}_{n},{x}_{1}\right).\end{array}$
This implies that
$\underset{n\to \mathrm{\infty }}{lim}\varphi \left({x}_{n+1},{x}_{n}\right)=0.$
(2.6)
In light of ${x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}$, we find that
$\varphi \left({x}_{n+1},{u}_{n}\right)\le \varphi \left({x}_{n+1},{x}_{n}\right)+\left({\mu }_{n}-1\right){M}_{n}+{\nu }_{n}.$
Thanks to (2.6), we find that
$\underset{n\to \mathrm{\infty }}{lim}\varphi \left({x}_{n+1},{u}_{n}\right)=0.$
(2.7)
In view of (1.3), we see that ${lim}_{n\to \mathrm{\infty }}\left(\parallel {x}_{n+1}\parallel -\parallel {u}_{n}\parallel \right)=0$. It follows that ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}\parallel =\parallel \overline{x}\parallel$. This is equivalent to
$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_{n}\parallel =\parallel J\overline{x}\parallel .$
(2.8)
This implies that $\left\{J{u}_{n}\right\}$ is bounded. Note that both E and ${E}^{\ast }$ are reflexive. We may assume that $J{u}_{n}⇀{u}^{\ast }\in {E}^{\ast }$. In view of the reflexivity of E, we see that $J\left(E\right)={E}^{\ast }$. This shows that there exists an element $u\in E$ such that $Ju={u}^{\ast }$. It follows that
$\begin{array}{rl}\varphi \left({x}_{n+1},{u}_{n}\right)& ={\parallel {x}_{n+1}\parallel }^{2}-2〈{x}_{n+1},J{u}_{n}〉+{\parallel {u}_{n}\parallel }^{2}\\ ={\parallel {x}_{n+1}\parallel }^{2}-2〈{x}_{n+1},J{u}_{n}〉+{\parallel J{u}_{n}\parallel }^{2}.\end{array}$
Taking ${lim inf}_{n\to \mathrm{\infty }}$ on both sides of the equality above yields that
$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^{2}-2〈\overline{x},{u}^{\ast }〉+{\parallel {u}^{\ast }\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Ju〉+{\parallel Ju\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Ju〉+{\parallel u\parallel }^{2}\\ =\varphi \left(\overline{x},u\right).\end{array}$
That is, $\overline{x}=u$, which in turn implies that ${u}^{\ast }=J\overline{x}$. It follows that $J{u}_{n}⇀J\overline{x}\in {E}^{\ast }$. Since ${E}^{\ast }$ enjoys the Kadec-Klee property, we obtain from (2.8) that
$\underset{n\to \mathrm{\infty }}{lim}J{u}_{n}=J\overline{x}.$
Since E enjoys the Kadec-Klee property, we obtain that ${u}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$. Note that $\parallel {x}_{n}-{u}_{n}\parallel \le \parallel {x}_{n}-\overline{x}\parallel +\parallel \overline{x}-{u}_{n}\parallel$. It follows that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{u}_{n}\parallel =0.$
(2.9)
This gives that
$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n}-J{u}_{n}\parallel =0.$
(2.10)
Notice that
$\begin{array}{rl}\varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)& ={\parallel {x}_{n}\parallel }^{2}-{\parallel {u}_{n}\parallel }^{2}-2〈w,J{x}_{n}-J{u}_{n}〉\\ \le \parallel {x}_{n}-{u}_{n}\parallel \left(\parallel {x}_{n}\parallel +\parallel {u}_{n}\parallel \right)+2\parallel w\parallel \parallel J{x}_{n}-J{u}_{n}\parallel .\end{array}$
It follows from (2.9) and (2.10) that
$\underset{n\to \mathrm{\infty }}{lim}\left(\varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)\right)=0.$
(2.11)
Since E is uniformly smooth, we know that ${E}^{\ast }$ is uniformly convex. In view of Lemma 1.9, we see that
$\begin{array}{r}\varphi \left(w,{u}_{n}\right)\\ \phantom{\rule{1em}{0ex}}=\varphi \left(w,{S}_{{r}_{n}}{y}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\le \varphi \left(w,{y}_{n}\right)\\ \phantom{\rule{1em}{0ex}}=\varphi \left(w,{J}^{-1}\left[{\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)J{T}^{n}{x}_{n}\right]\right)\\ \phantom{\rule{1em}{0ex}}={\parallel w\parallel }^{2}-2〈w,{\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)J{T}^{n}{x}_{n}〉+{\parallel {\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)J{T}^{n}{x}_{n}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel w\parallel }^{2}-2{\alpha }_{n}〈w,J{x}_{n}〉-2\left(1-{\alpha }_{n}\right)〈w,J{T}^{n}{x}_{n}〉+{\alpha }_{n}{\parallel {x}_{n}\parallel }^{2}+\left(1-{\alpha }_{n}\right){\parallel {T}^{n}{x}_{n}\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J{x}_{n}-J\left({T}^{n}{x}_{n}\right)\parallel \right)\\ \phantom{\rule{1em}{0ex}}={\alpha }_{n}\varphi \left(w,{x}_{n}\right)+\left(1-{\alpha }_{n}\right)\varphi \left(w,{T}^{n}{x}_{n}\right)-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J{x}_{n}-J\left({T}^{n}{x}_{n}\right)\parallel \right)\\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\varphi \left(w,{x}_{n}\right)+\left(1-{\alpha }_{n}\right){\mu }_{n}\varphi \left(w,{x}_{n}\right)+{\nu }_{n}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J{x}_{n}-J\left({T}^{n}{x}_{n}\right)\parallel \right)\\ \phantom{\rule{1em}{0ex}}\le \varphi \left(w,{x}_{n}\right)+\left(1-{\alpha }_{n}\right)\left({\mu }_{n}-1\right)\varphi \left(w,{x}_{n}\right)+{\nu }_{n}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J{x}_{n}-J\left({T}^{n}{x}_{n}\right)\parallel \right).\end{array}$
This implies that
${\alpha }_{n}\left(1-{\alpha }_{n}\right)g\left(\parallel J{x}_{n}-J\left({T}^{n}{x}_{n}\right)\parallel \right)\le \varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)+\left(1-{\alpha }_{n}\right)\left({\mu }_{n}-1\right)\varphi \left(w,{x}_{n}\right)+{\nu }_{n}.$
In view of the restrictions on the sequence $\left\{{\alpha }_{n}\right\}$, we find from (2.11) that
$\underset{n\to \mathrm{\infty }}{lim}\parallel J\left({T}^{n}{x}_{n}\right)-J{x}_{n}\parallel =0.$
(2.12)
Notice that
$\parallel J\left({T}^{n}{x}_{n}\right)-J\overline{x}\parallel \le \parallel J\left({T}^{n}{x}_{n}\right)-J{x}_{n}\parallel +\parallel J{x}_{n}-J\overline{x}\parallel .$
It follows from (2.12) that
$\underset{n\to \mathrm{\infty }}{lim}\parallel J\left({T}^{n}{x}_{n}\right)-J\overline{x}\parallel =0.$
(2.13)
The demicontinuity of ${J}^{-1}:{E}^{\ast }\to E$ implies that ${T}^{n}{x}_{n}⇀\overline{x}$. Note that
$|\parallel {T}^{n}{x}_{n}\parallel -\parallel \overline{x}\parallel |=|\parallel J\left({T}^{n}{x}_{n}\right)\parallel -\parallel J\overline{x}\parallel |\le \parallel J\left({T}^{n}{x}_{n}\right)-J\overline{x}\parallel .$
This implies from (2.13) that ${lim}_{n\to \mathrm{\infty }}\parallel {T}^{n}{x}_{n}\parallel =\parallel \overline{x}\parallel$. Since E has the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty }}\parallel {T}^{n}{x}_{n}-\overline{x}\parallel =0$. Since
$\parallel {T}^{n+1}{x}_{n}-\overline{x}\parallel \le \parallel {T}^{n+1}{x}_{n}-{T}^{n}{x}_{n}\parallel +\parallel {T}^{n}{x}_{n}-\overline{x}\parallel ,$

we find from the asymptotic regularity of T that ${lim}_{n\to \mathrm{\infty }}\parallel {T}^{n+1}{x}_{n}-\overline{x}\parallel =0$, that is, $T{T}^{n}{x}_{n}-\overline{x}\to 0$ as $n\to \mathrm{\infty }$. It follows from the closedness of T that $T\overline{x}=\overline{x}$.

Next, we show that $\overline{x}\in EP\left(f\right)$. In view of Lemma 1.8, we find from (2.4) that
$\begin{array}{rl}\varphi \left({u}_{n},{y}_{n}\right)& \le \varphi \left(w,{y}_{n}\right)-\varphi \left(w,{u}_{n}\right)\\ \le \varphi \left(w,{x}_{n}\right)+\left({\mu }_{n}-1\right){M}_{n}+{\nu }_{n}-\varphi \left(w,{u}_{n}\right)\\ =\varphi \left(w,{x}_{n}\right)-\varphi \left(w,{u}_{n}\right)+\left({k}_{n}-1\right){M}_{n}.\end{array}$
(2.14)
It follows from (2.11) that ${lim}_{n\to \mathrm{\infty }}\varphi \left({u}_{n},{y}_{n}\right)=0$. This implies that ${lim}_{n\to \mathrm{\infty }}\left(\parallel {u}_{n}\parallel -\parallel {y}_{n}\parallel \right)=0$. In view of (2.9), we see that ${u}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$. This implies that $\parallel {y}_{n}\parallel -\parallel \overline{x}\parallel \to 0$ as $n\to \mathrm{\infty }$. It follows that ${lim}_{n\to \mathrm{\infty }}\parallel J{y}_{n}\parallel =\parallel J\overline{x}\parallel$. Since ${E}^{\ast }$ is reflexive, we may assume that $J{y}_{n}⇀{r}^{\ast }\in {E}^{\ast }$. In view of $J\left(E\right)={E}^{\ast }$, we see that there exists $r\in E$ such that $Jr={r}^{\ast }$. It follows that
$\begin{array}{rl}\varphi \left({u}_{n},{y}_{n}\right)& ={\parallel {u}_{n}\parallel }^{2}-2〈{u}_{n},J{y}_{n}〉+{\parallel {y}_{n}\parallel }^{2}\\ ={\parallel {u}_{n}\parallel }^{2}-2〈{u}_{n},J{y}_{n}〉+{\parallel J{y}_{n}\parallel }^{2}.\end{array}$
Taking ${lim inf}_{n\to \mathrm{\infty }}$ on both sides of the equality above yields that
$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^{2}-2〈\overline{x},{r}^{\ast }〉+{\parallel {r}^{\ast }\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Jr〉+{\parallel Jr\parallel }^{2}\\ ={\parallel \overline{x}\parallel }^{2}-2〈\overline{x},Jr〉+{\parallel r\parallel }^{2}\\ =\varphi \left(\overline{x},r\right).\end{array}$
That is, $\overline{x}=r$, which in turn implies that ${r}^{\ast }=J\overline{x}$. It follows that $J{y}_{n}⇀J\overline{x}\in {E}^{\ast }$. Since ${E}^{\ast }$ enjoys the Kadec-Klee property, we obtain that $J{y}_{n}-J\overline{x}\to 0$ as $n\to \mathrm{\infty }$. Note that ${J}^{-1}:{E}^{\ast }\to E$ is demicontinuous. It follows that ${y}_{n}⇀\overline{x}$. Since E enjoys the Kadec-Klee property, we obtain that ${y}_{n}\to \overline{x}$ as $n\to \mathrm{\infty }$. Note that
$\parallel {u}_{n}-{y}_{n}\parallel \le \parallel {u}_{n}-\overline{x}\parallel +\parallel \overline{x}-{y}_{n}\parallel .$
This implies that
$\underset{n\to \mathrm{\infty }}{lim}\parallel {u}_{n}-{y}_{n}\parallel =0.$
(2.15)
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_{n}-J{y}_{n}\parallel =0.$
From the assumption ${r}_{n}\ge a$, we see that
$\underset{n\to \mathrm{\infty }}{lim}\frac{\parallel J{u}_{n}-J{y}_{n}\parallel }{{r}_{n}}=0.$
(2.16)
In view of ${u}_{n}={S}_{{r}_{n}}{y}_{n}$, we see that
$f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
It follows from (A2) that
$\parallel y-{u}_{n}\parallel \frac{\parallel J{u}_{n}-J{y}_{n}\parallel }{{r}_{n}}\ge \frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge -f\left({u}_{n},y\right)\ge f\left(y,{u}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
By taking the limit as $n\to \mathrm{\infty }$ in the above inequality, from (A4) and (2.16) we obtain that
$f\left(y,\overline{x}\right)\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
For $0 and $y\in C$, define ${y}_{t}=ty+\left(1-t\right)\overline{x}$. It follows that ${y}_{t}\in C$, which yields that $f\left({y}_{t},\overline{x}\right)\le 0$. It follows from (A1) and (A4) that
$0=f\left({y}_{t},{y}_{t}\right)\le tf\left({y}_{t},y\right)+\left(1-t\right)f\left({y}_{t},\overline{x}\right)\le tf\left({y}_{t},y\right).$
That is,
$f\left({y}_{t},y\right)\ge 0.$

Letting $t↓0$, we obtain from (A3) that $f\left(\overline{x},y\right)\ge 0$, $\mathrm{\forall }y\in C$. This implies that $\overline{x}\in EP\left(f\right)$. This shows that $\overline{x}\in \mathcal{F}=EP\left(f\right)\cap F\left(T\right)$.

Finally, we prove that $\overline{x}={\mathrm{\Pi }}_{\mathcal{F}}{x}_{1}$. Letting $n\to \mathrm{\infty }$ in (2.5), we see that
$〈\overline{x}-w,J{x}_{1}-J\overline{x}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }w\in \mathcal{F}.$

In view of Lemma 1.6, we find that $\overline{x}={\mathrm{\Pi }}_{\mathcal{F}}{x}_{1}$. This completes the proof. □

If T is asymptotically quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.

Corollary 2.2 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4), and let $T:C\to C$ be a closed asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that $\mathcal{F}=F\left(T\right)\cap EP\left(f\right)$ is nonempty and bounded. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:
$\left\{\begin{array}{c}{x}_{0}\in E\phantom{\rule{1em}{0ex}}\mathit{\text{chosen arbitrarily}},\hfill \\ {C}_{1}=C,\hfill \\ {x}_{1}={\mathrm{\Pi }}_{{C}_{1}}{x}_{0},\hfill \\ {y}_{n}={J}^{-1}\left({\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)J{T}^{n}{x}_{n}\right),\hfill \\ {u}_{n}\in C\phantom{\rule{1em}{0ex}}\mathit{\text{such that}}\phantom{\rule{1em}{0ex}}f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left(z,{u}_{n}\right)\le \varphi \left(z,{x}_{n}\right)+\left({\mu }_{n}-1\right){M}_{n}\right\},\hfill \\ {x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1},\hfill \end{array}$

where ${M}_{n}=sup\left\{\varphi \left(z,{x}_{n}\right):z\in \mathcal{F}\right\}$, $\left\{{\alpha }_{n}\right\}$ is a real sequence in $\left[0,1\right]$ such that ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}\right)>0$, and $\left\{{r}_{n}\right\}$ is a real sequence in $\left[a,\mathrm{\infty }\right)$, where a is some positive real number. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_{1}$.

If T is quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.

Corollary 2.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4), and let $T:C\to C$ be a closed quasi-ϕ-nonexpansive mapping. Assume that $\mathcal{F}=F\left(T\right)\cap EP\left(f\right)$ is nonempty. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:
$\left\{\begin{array}{c}{x}_{0}\in E\phantom{\rule{1em}{0ex}}\mathit{\text{chosen arbitrarily}},\hfill \\ {C}_{1}=C,\hfill \\ {x}_{1}={\mathrm{\Pi }}_{{C}_{1}}{x}_{0},\hfill \\ {y}_{n}={J}^{-1}\left({\alpha }_{n}J{x}_{n}+\left(1-{\alpha }_{n}\right)JT{x}_{n}\right),\hfill \\ {u}_{n}\in C\phantom{\rule{1em}{0ex}}\mathit{\text{such that}}\phantom{\rule{1em}{0ex}}f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{y}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left(z,{u}_{n}\right)\le \varphi \left(z,{x}_{n}\right)\right\},\hfill \\ {x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1},\hfill \end{array}$

where $\left\{{\alpha }_{n}\right\}$ is a real sequence in $\left[0,1\right]$ such that ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}\right)>0$, and $\left\{{r}_{n}\right\}$ is a real sequence in $\left[a,\mathrm{\infty }\right)$, where a is some positive real number. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_{1}$.

If T is the identity, then Theorem 2.1 is reduced to the following.

Corollary 2.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from $C×C$ to satisfying (A1)-(A4). Assume that $EP\left(f\right)$ is nonempty. Let $\left\{{x}_{n}\right\}$ be a sequence generated in the following manner:
$\left\{\begin{array}{c}{x}_{0}\in E\phantom{\rule{1em}{0ex}}\mathit{\text{chosen arbitrarily}},\hfill \\ {C}_{1}=C,\hfill \\ {x}_{1}={\mathrm{\Pi }}_{{C}_{1}}{x}_{0},\hfill \\ {u}_{n}\in C\phantom{\rule{1em}{0ex}}\mathit{\text{such that}}\phantom{\rule{1em}{0ex}}f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},J{u}_{n}-J{x}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\varphi \left(z,{u}_{n}\right)\le \varphi \left(z,{x}_{n}\right)+\left({\mu }_{n}-1\right){M}_{n}+{\nu }_{n}\right\},\hfill \\ {x}_{n+1}={\mathrm{\Pi }}_{{C}_{n+1}}{x}_{1},\hfill \end{array}$

where ${M}_{n}=sup\left\{\varphi \left(z,{x}_{n}\right):z\in \mathcal{F}\right\}$, $\left\{{\alpha }_{n}\right\}$ is a real sequence in $\left[0,1\right]$ such that ${lim inf}_{n\to \mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}\right)>0$, and $\left\{{r}_{n}\right\}$ is a real sequence in $\left[a,\mathrm{\infty }\right)$, where a is some positive real number. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${\mathrm{\Pi }}_{EP\left(f\right)}{x}_{1}$.

## Declarations

### Acknowledgements

The study was supported by the Natural Science Foundation of Zhejiang Province (Y6110270).

## Authors’ Affiliations

(1)
School of Mathematics Physics and Information Science, Zhejiang Ocean University, Zhoushan, 316004, China

## References 