# On generalized asymptotically quasi-*ϕ*-nonexpansive mappings and a Ky Fan inequality

- Jianmin Song
^{1}Email author and - Minjiang Chen
^{1}

**2013**:237

https://doi.org/10.1186/1687-1812-2013-237

© Song and Chen; licensee Springer. 2013

**Received: **24 May 2013

**Accepted: **12 August 2013

**Published: **23 September 2013

## Abstract

Generalized asymptotically quasi-*ϕ*-nonexpansive mappings and a Ky Fan inequality are investigated. A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-*ϕ*-nonexpansive mappings and a Ky Fan inequality is established in a Banach space.

**MSC:**47H05, 47J25, 90C33.

### Keywords

asymptotically quasi-*ϕ*-nonexpansive mapping generalized asymptotically quasi-

*ϕ*-nonexpansive mapping generalized projection equilibrium problem fixed point

## 1 Introduction

Iterative algorithms have been studied by many authors. The applications of iterative algorithms are found in a wide range of areas, including economics, image recovery and signal processing. Many well-known problems can be studied by using algorithms which are iterative in their nature; see [1–14] and the references therein. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set, in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point.

Mann iteration, introduced by Mann [15], is an efficient tool to study fixed point problems of asymptotical nonexpansive mappings. However, Mann iteration is only weak convergence in infinite-dimensional spaces; see [10] and the references therein. The importance of strong convergence is underlined in [16], where a convex function *f* is minimized via the proximal-point algorithm: it is shown that the rate of convergence of the value sequence $\{f({x}_{n})\}$ is better when $\{{x}_{n}\}$ converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite-dimensional space. To obtain strong convergence of Mann iteration, projection methods, which were first introduced by Haugazeau [17], have been considered for modifying Mann iteration to obtain strong convergence. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.

The organization of this paper is as follows. In Section 2, we provide some necessary concepts and lemmas. In Section 3, fixed point problems of generalized asymptotically quasi-*ϕ*-nonexpansive mappings and solutions of a Ky Fan inequality are investigated. A strong convergence theorem is established in a Banach space.

## 2 Preliminaries

*J*from

*E*to ${2}^{{E}^{\ast}}$ is defined by

*E*. Then the Banach space

*E*is said to be smooth iff

exists for each $x,y\in {U}_{E}$. It is also said to be uniformly smooth iff 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 *E* is said to be strictly convex iff $\parallel \frac{x+y}{2}\parallel <1$ 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 iff ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$ for any two sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in *E* such that $\parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1$ and ${lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1$. Recall that *E* enjoys the Kadec-Klee property if for any sequence $\{{x}_{n}\}\subset E$, and $x\in E$ with ${x}_{n}\rightharpoonup x$, 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, readers can refer to [18] and the references therein. It is well known that if *E* is a uniformly convex Banach space, then *E* enjoys the Kadec-Klee property.

*E*is a smooth Banach space. Consider the functional defined by

*H*, the equality is reduced to $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$. 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 [19] 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. Recall that 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 $\varphi (x,y)$, that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

*J*; see, for example, [18]. In Hilbert spaces, ${\mathrm{\Pi}}_{C}={P}_{C}$. It is obvious from the definition of a function

*ϕ*that

**Remark 2.1** If *E* is a reflexive, strictly convex and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$; for more details, see [18] and the reference therein.

*C*be a nonempty subset of

*E*and $T:C\to C$ be a mapping. In this paper, we use $F(T)$ 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*,

*T* is said to be closed if for any sequence $\{{x}_{n}\}\subset C$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0}$ and ${lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}$, then $T{x}_{0}={y}_{0}$. In this paper, we use → and ⇀ to denote strong convergence and weak convergence, respectively. Recall that a point *p* in *C* is said to be an asymptotic fixed point of *T* iff *C* contains a sequence $\{{x}_{n}\}$ which converges weakly to *p* so 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 $\tilde{F}(T)$.

*T*is said to be relatively nonexpansive iff

*T*is said to be relatively asymptotically nonexpansive iff

where $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ is a sequence such that ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$.

*T*is said to be quasi-

*ϕ*-nonexpansive iff

*T*is said to be asymptotically quasi-

*ϕ*-nonexpansive iff there exists a sequence $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that

**Remark 2.2** The class of asymptotically quasi-*ϕ*-nonexpansive mappings was considered in Zhou *et al.* [20] and Qin *et al.* [21]; see also [22] and [23].

**Remark 2.3** The class of quasi-*ϕ*-nonexpansive mappings and the class of asymptotically quasi-*ϕ*-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings [24]. Quasi-*ϕ*-nonexpansive mappings and asymptotically quasi-*ϕ*-nonexpansive mappings do not require the restriction $F(T)=\tilde{F}(T)$.

**Remark 2.4** The class of quasi-*ϕ*-nonexpansive mappings and the class of asymptotically quasi-*ϕ*-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

Recall that *T* is said to be generalized asymptotically quasi-*ϕ*-nonexpansive if $F(T)\ne \mathrm{\varnothing}$, and there exists a sequence $\{{\mu}_{n}\}\subset [1,\mathrm{\infty})$ with ${\mu}_{n}\to 1$ as $n\to \mathrm{\infty}$ and a sequence $\{{\nu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\nu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that $\varphi (p,Tx)\le {\mu}_{n}\varphi (p,x)+{\nu}_{n}$ for all $x\in C$, $p\in F(T)$ and $n\ge 1$.

**Remark 2.5** The class of generalized asymptotically quasi-*ϕ*-nonexpansive mappings was considered in Qin *et al.* [25]; see also [26].

*f*be a bifunction from $C\times C$ to ℝ, where ℝ denotes the set of real numbers, and let $A:C\to {E}^{\ast}$ be a mapping. Consider the following Ky Fan inequality which is known as a generalized equilibrium problem. Find $p\in C$ such that

*α*-inverse-strongly monotone if there exists $\alpha >0$ such that

*α*-inverse-strongly monotone and the bifunction $f:C\times C\to \mathbb{R}$ satisfies the following conditions:

- (A1)
$F(x,x)=0$, $\mathrm{\forall}x\in C$;

- (A2)
*F*is monotone,*i.e.*, $F(x,y)+F(y,x)\le 0$, $\mathrm{\forall}x,y\in C$; - (A3)$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y,z\in C;$
- (A4)
for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semicontinuous.

Recently, many authors investigated the solutions of problems (2.3), (2.4) and (2.5) based on iterative methods; see [27–37]. In this paper, we investigate generalized asymptotically quasi-*ϕ*-nonexpansive mappings and problem (2.3). A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-*ϕ*-nonexpansive mappings and problem (2.3) is established in a Banach space.

In order to state our main results, we need the following lemmas, which play an import role in the paper.

**Lemma 2.6** [28]

*Let*

*E*

*be a smooth*,

*strictly convex and reflexive Banach space and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$A:C\to {E}^{\ast}$

*be an*

*α*-

*inverse*-

*strongly monotone mapping and*

*f*

*be a bifunction satisfying conditions*(A1)-(A4).

*Let*$r>0$

*be any given number and*$x\in E$

*be any given point*.

*Then there exists*$p\in C$

*such that*

**Lemma 2.7** [28]

*Let*

*E*

*be a smooth*,

*strictly convex and reflexive Banach space and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$A:C\to {E}^{\ast}$

*be an*

*α*-

*inverse*-

*strongly monotone mapping and*

*f*

*be a bifunction satisfying conditions*(A1)-(A4).

*Let*$r>0$

*be any given number and*$x\in E$

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

*as follows*:

*for any*$x\in C$,

*Then the following conclusions hold*:

- (1)
${K}_{r}$

*is single*-*valued*; - (2)${K}_{r}$
*is a firmly nonexpansive*-*type mapping*,*i*.*e*.,*for all*$x,y\in E$,$\u3008{K}_{r}x-{K}_{r}y,J{K}_{r}x-J{K}_{r}y\u3009\le \u3008{S}_{r}x-{S}_{r}y,Jx-Jy\u3009;$ - (3)
$F({K}_{r})=S(f,A)$;

- (4)
${K}_{r}$

*is quasi*-*ϕ*-*nonexpansive*; - (5)$\varphi (q,{K}_{r}x)+\varphi ({K}_{r}x,x)\le \varphi (q,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}q\in F({K}_{r});$
- (6)
$S(f,A)$

*is closed and convex*.

**Lemma 2.8** [19]

*Let*

*C*

*be a nonempty closed convex subset of a smooth Banach space*

*E*

*and*$x\in E$.

*Then*${x}_{0}={\mathrm{\Pi}}_{C}x$

*if and only if*

**Lemma 2.9** [19]

*Let*

*E*

*be a reflexive*,

*strictly convex and smooth Banach space*,

*C*

*be a nonempty closed convex subset of*

*E*

*and*$x\in E$.

*Then*

**Lemma 2.10** [25]

*Let* *E* *be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-*Klee property and* *C* *be a nonempty closed convex subset of* *E*. *Let* $T:C\to C$ *be a closed generalized asymptotically quasi*-*ϕ*-*nonexpansive mapping*. *Then* $F(T)$ *is closed and convex*.

**Lemma 2.11** [38]

*Let*

*E*

*be a smooth and uniformly convex Banach space and let*$r>0$.

*Then there exists a strictly increasing*,

*continuous and convex function*$g:[0,2r]\to R$

*such that*$g(0)=0$

*and*

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

## 3 Main results

**Theorem 3.1**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$T:C\to C$

*be a generalized asymptotically quasi*-

*ϕ*-

*nonexpansive mapping*.

*Let*

*f*

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

*to*ℝ

*satisfying*(A1)-(A4)

*and*$A:C\to {E}^{\ast}$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

*Assume that*

*T*

*is closed and asymptotically regular on*

*C*,

*and*$F(T)\cap S(f,A)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* ${W}_{n}=sup\{\varphi (p,{x}_{n}):p\in F(T)\cap S(f,A)\}$, $\{{\alpha}_{n}\}$ *is a real number sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ *and* $\{{r}_{n}\}$ *is a real number sequence such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)\cap S(f,A)}{x}_{1}$, *where* ${\mathrm{\Pi}}_{F(T)\cap S(f,A)}$ *is the generalized projection from* *E* *onto* $F(T)\cap S(f,A)$.

*Proof*First, we prove ${C}_{n}$ is closed and convex so that the projection is well defined. We see that ${C}_{1}=C$ is closed and convex. Assume that ${C}_{m}$ is closed and convex for some positive integer

*m*. For $k\in {C}_{m}$, we find that

It is easy to see that ${C}_{m+1}$ is closed and convex. This proves that ${C}_{n}$ is closed and convex so that ${\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}$ is well defined. Set ${u}_{n}={K}_{{r}_{n}}{y}_{n}$. It follows from Lemma 2.7 that ${K}_{{r}_{n}}$ is quasi-*ϕ*-nonexpansive.

*m*. Then, for $\mathrm{\forall}e\in F(T)\cap S(f,A)\subset {C}_{m}$, we have

*E*is a uniform space, we find that

*E*is reflexive. We may assume, without loss of generality, that ${x}_{n}\rightharpoonup \stackrel{\u02c6}{x}$. Next, we prove that $\stackrel{\u02c6}{x}\in F(T)\cap S(f,A)$. Since ${C}_{n}$ is closed and convex, we find that $\stackrel{\u02c6}{x}\in {C}_{n}$. This implies from ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}$ that $\varphi ({x}_{n},{x}_{1})\le \varphi (\stackrel{\u02c6}{x},{x}_{1})$. On the other hand, we see from the weakly lower semicontinuity of the norm $\parallel \cdot \parallel $ that

*E*enjoys the Kadec-Klee property, we find that ${x}_{n}\to \stackrel{\u02c6}{x}$ as $n\to \mathrm{\infty}$. In the light of ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}$ and ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$, we find that $\varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1})$. This shows that $\{\varphi ({x}_{n},{x}_{1})\}$ is nondecreasing. We obtain that ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})$ exists. It follows that

*E*and ${E}^{\ast}$ are uniform, we find that both

*E*and ${E}^{\ast}$ are reflexive. We may assume, without loss of generality, that $J{u}_{n}\rightharpoonup {u}^{\ast}\in {E}^{\ast}$. In view of the reflexivity of

*E*, we see that $J(E)={E}^{\ast}$. This shows that there exists an element $u\in E$ such that $Ju={u}^{\ast}$. It follows that

*E*is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. Therefore, ${E}^{\ast}$ enjoys the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty}}J{u}_{n}=J\stackrel{\u02c6}{x}$. Since ${J}^{-1}:{E}^{\ast}\to E$ is demicontinuous and

*E*enjoys the Kadec-Klee property, we obtain that ${u}_{n}\to \stackrel{\u02c6}{x}$ as $n\to \mathrm{\infty}$. Note that

*E*is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. In the light of Lemma 2.11, we find that

*E*has the Kadec-Klee property, we obtain that ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{n}{x}_{n}-\stackrel{\u02c6}{x}\parallel =0$. Notice that

*T*that

*T*, we find $\stackrel{\u02c6}{x}=T\stackrel{\u02c6}{x}$. This proves $\stackrel{\u02c6}{x}\in F(T)$. Next, we show that $\stackrel{\u02c6}{x}\in S(f,A)$. It follows from Lemma 2.9 and (3.1) that

*E*enjoys the Kadec-Klee property, we obtain that ${y}_{n}\to \stackrel{\u02c6}{x}$ as $n\to \mathrm{\infty}$. Note that

*J*is uniformly norm-to-norm continuous on any bounded sets, we have ${lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n}-J{y}_{n}\parallel =0$. In view of the restriction ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$, we see that

Letting $t\downarrow 0$, we obtain from (A3) that $F(\stackrel{\u02c6}{x},q)\ge 0$, $\mathrm{\forall}q\in C$. This implies that $\stackrel{\u02c6}{x}\in S(f,A)$. This completes the proof $\stackrel{\u02c6}{x}\in F(T)\cap S(f,A)$.

Finally, what we need to prove is $\stackrel{\u02c6}{x}={\mathrm{\Pi}}_{F(T)\cap S(f,A)}{x}_{1}$.

From Lemma 2.8, we immediately find that $\stackrel{\u02c6}{x}={\mathrm{\Pi}}_{F(T)\cap S(f,A)}{x}_{1}$. This completes the whole proof. □

**Remark 3.2** Since the class of generalized asymptotically quasi-*ϕ*-nonexpansive mappings is a generalization of the class of asymptotically quasi-*ϕ*-nonexpansive mappings, Theorem 3.1 includes Kim’s [36] results as a special case.

**Remark 3.3** Notice that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and every uniformly convex Banach space enjoys the Kadec-Klee property. We find that Theorem 3.1 is still valid in the framework of every uniformly smooth and uniformly convex space.

Next, we consider the solution of problem (2.4).

If the mapping *T* is closed quasi-*ϕ*-nonexpansive, which is more general than relatively nonexpansive mappings, we have the following.

**Corollary 3.4**

*Let*

*E*

*be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec*-

*Klee property and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$T:C\to C$

*be a quasi*-

*ϕ*-

*nonexpansive mapping and*

*f*

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

*to*ℝ

*satisfying*(A1)-(A4).

*Assume that*

*T*

*is closed and*$F(T)\cap S(f)$

*is nonempty*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* $\{{\alpha}_{n}\}$ *is a real number sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ *and* $\{{r}_{n}\}$ *is a real number sequence such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)\cap S(f)}{x}_{1}$, *where* ${\mathrm{\Pi}}_{F(T)\cap S(f)}$ *is the generalized projection from* *E* *onto* $F(T)\cap S(f)$.

In the framework of Hilbert spaces, we find from Theorem 3.1 the following.

**Theorem 3.5**

*Let*

*E*

*be a Hilbert space and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$T:C\to C$

*be a generalized asymptotically quasi*-

*nonexpansive mapping*.

*Let*

*f*

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

*to*ℝ

*satisfying*(A1)-(A4),

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

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

*Assume that*

*T*

*is closed and asymptotically regular on*

*C*,

*and*$F(T)\cap S(f,A)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* ${W}_{n}=sup\{{\parallel p-{x}_{n}\parallel}^{2}:p\in F(T)\cap S(f,A)\}$, $\{{\alpha}_{n}\}$ *is a real number sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ *and* $\{{r}_{n}\}$ *is a real number sequence such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)\cap S(f,A)}{x}_{1}$, *where* ${P}_{F(T)\cap S(f,A)}$ *is the metric projection from* *E* *onto* $F(T)\cap S(f,A)$.

*Proof* In the framework of Hilbert spaces, we see that $\varphi (x,y)={\parallel x-y\parallel}^{2}$ and the mapping *J* is reduced to the identity mapping. The desired conclusion can be immediately drawn from Theorem 3.1. □

For problem (2.4), we have the following result.

**Corollary 3.6**

*Let*

*E*

*be Hilbert space and*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*$T:C\to C$

*be a generalized asymptotically quasi*-

*nonexpansive mapping*.

*Let*

*f*

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

*to*ℝ

*satisfying*(A1)-(A4).

*Assume that*

*T*

*is closed and asymptotically regular on*

*C*,

*and*$F(T)\cap S(f)$

*is nonempty and bounded*.

*Let*$\{{x}_{n}\}$

*be a sequence generated in the following manner*:

*where* ${W}_{n}=sup\{{\parallel p-{x}_{n}\parallel}^{2}:p\in F(T)\cap S(f)\}$, $\{{\alpha}_{n}\}$ *is a real number sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ *and* $\{{r}_{n}\}$ *is a real number sequence such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)\cap S(f)}{x}_{1}$, *where* ${P}_{F(T)\cap S(f)}$ *is the metric projection from* *E* *onto* $F(T)\cap S(f)$.

## Declarations

### Acknowledgements

The authors are grateful to the editor and the anonymous reviewers’ suggestions which improved the contents of the article.

## Authors’ Affiliations

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