# An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces

## Abstract

We introduce an implicit method for finding an element of the set of common fixed points of a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the common fixed point of a representation of nonexpansive mappings.

MSC:90C33, 47H10.

## 1 Introduction

Let C be a nonempty closed and convex subset of a Banach space E and ${E}^{âˆ—}$ be the dual space of E. Let $ã€ˆâ‹\dots ,â‹\dots ã€‰$ denote the pairing between E and ${E}^{âˆ—}$. The normalized duality mapping $J:Eâ†’{E}^{âˆ—}$ is defined by

$J\left(x\right)=\left\{fâˆˆ{E}^{âˆ—}:ã€ˆx,fã€‰={âˆ¥xâˆ¥}^{2}={âˆ¥fâˆ¥}^{2}\right\}$

for all $xâˆˆE$. In the sequel, we use j to denote the single-valued normalized duality mapping. Let $U=\left\{xâˆˆE:âˆ¥xâˆ¥=1\right\}$. E is said to be smooth or to have a GÃ¢teaux differentiable norm if the limit

$\underset{tâ†’0}{lim}\frac{âˆ¥x+tyâˆ¥âˆ’âˆ¥xâˆ¥}{t}$

exists for each $x,yâˆˆU$. E is said to have a uniformly GÃ¢teaux differentiable norm if for each $yâˆˆU$, the limit is attained uniformly for all $xâˆˆU$. E is said to be uniformly smooth or is said to have a uniformly FÃ©chet differentiable norm if the limit is attained uniformly for $x,yâˆˆU$. It is known that if the norm of E is uniformly GÃ¢teaux differentiable, then the duality mapping J is single-valued and uniformly norm to weakâˆ— continuous on each bounded subset of E. A Banach space E is smooth if the duality mapping J of E is single-valued. We know that if E is smooth, then J is norm to weak-star continuous; for more details, see [1].

Let C be a nonempty closed and convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if $âˆ¥Txâˆ’Tyâˆ¥â‰¤âˆ¥xâˆ’yâˆ¥$ for all $x,yâˆˆC$, and a mapping f is an Î±-contraction on E if $âˆ¥f\left(x\right)âˆ’f\left(y\right)âˆ¥â‰¤\mathrm{Î±}âˆ¥xâˆ’yâˆ¥$, $x,yâˆˆE$ such that $0â‰¤\mathrm{Î±}<1$.

In this paper, motivated by Lashkarizadeh Bami and Soori [2] and Hussain and Takahashi [3], we introduce the following general implicit algorithm for finding a common element of the set of fixed points of a representation $\mathcal{S}=\left\{{T}_{t}:tâˆˆS\right\}$ of a semigroup S as nonexpansive mappings from C into itself, with respect to a left regular sequence of means defined on an appropriate subspace of bounded real-valued functions of the semigroup. On the other hand, our goal is to prove that there exists a sunny nonexpansive retraction P of C onto $Fix\left(\mathcal{S}\right)$ and $xâˆˆC$ such that the following sequence $\left\{{z}_{n}\right\}$ converges strongly to Px:

${z}_{n}={\mathrm{Ïµ}}_{n}f\left({z}_{n}\right)+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right){T}_{{\mathrm{Î¼}}_{n}}{z}_{n}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N}\right).$

## 2 Preliminaries

Let S be a semigroup. We denote by $B\left(S\right)$ the Banach space of all bounded real-valued functions defined on S with supremum norm. For each $sâˆˆS$ and $fâˆˆB\left(S\right)$, we define ${l}_{s}$ and ${r}_{s}$ in $B\left(S\right)$ by

$\left({l}_{s}f\right)\left(t\right)=f\left(st\right),\phantom{\rule{2em}{0ex}}\left({r}_{s}f\right)\left(t\right)=f\left(ts\right)\phantom{\rule{1em}{0ex}}\left(tâˆˆS\right).$

Let X be a subspace of $B\left(S\right)$ containing 1, and let ${X}^{âˆ—}$ be its topological dual. An element Î¼ of ${X}^{âˆ—}$ is said to be a mean on X if $âˆ¥\mathrm{Î¼}âˆ¥=\mathrm{Î¼}\left(1\right)=1$. We often write ${\mathrm{Î¼}}_{t}\left(f\left(t\right)\right)$ instead of $\mathrm{Î¼}\left(f\right)$ for $\mathrm{Î¼}âˆˆ{X}^{âˆ—}$ and $fâˆˆX$. Let X be left invariant (resp. right invariant), i.e., ${l}_{s}\left(X\right)âŠ‚X$ (resp. ${r}_{s}\left(X\right)âŠ‚X$) for each $sâˆˆS$. A mean Î¼ on X is said to be left invariant (resp. right invariant) if $\mathrm{Î¼}\left({l}_{s}f\right)=\mathrm{Î¼}\left(f\right)$ (resp. $\mathrm{Î¼}\left({r}_{s}f\right)=\mathrm{Î¼}\left(f\right)$) for each $sâˆˆS$ and $fâˆˆX$. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, $B\left(S\right)$ is amenable when S is a commutative semigroup (see p.29 of [1]). A net $\left\{{\mathrm{Î¼}}_{\mathrm{Î±}}\right\}$ of means on X is said to be left regular if

$\underset{\mathrm{Î±}}{lim}âˆ¥{l}_{s}^{âˆ—}{\mathrm{Î¼}}_{\mathrm{Î±}}âˆ’{\mathrm{Î¼}}_{\mathrm{Î±}}âˆ¥=0$

for each $sâˆˆS$, where ${l}_{s}^{âˆ—}$ is the adjoint operator of ${l}_{s}$.

Let f be a function of the semigroup S into a reflexive Banach space E such that the weak closure of $\left\{f\left(t\right):tâˆˆS\right\}$ is weakly compact, and let X be a subspace of $B\left(S\right)$ containing all the functions $tâ†’ã€ˆf\left(t\right),{x}^{âˆ—}ã€‰$ with ${x}^{âˆ—}âˆˆ{E}^{âˆ—}$. We know from [4] that for any $\mathrm{Î¼}âˆˆ{X}^{âˆ—}$, there exists a unique element ${f}_{\mathrm{Î¼}}$ in E such that $ã€ˆ{f}_{\mathrm{Î¼}},{x}^{âˆ—}ã€‰={\mathrm{Î¼}}_{t}ã€ˆf\left(t\right),{x}^{âˆ—}ã€‰$ for all ${x}^{âˆ—}âˆˆ{E}^{âˆ—}$. We denote such ${f}_{\mathrm{Î¼}}$ by $âˆ«f\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\mathrm{Î¼}\left(t\right)$. Moreover, if Î¼ is a mean on X, then from [5], $âˆ«f\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}\mathrm{Î¼}\left(t\right)âˆˆ\stackrel{Â¯}{co}\left\{f\left(t\right):tâˆˆS\right\}$.

Let C be a nonempty closed and convex subset of E. Then a family $\mathcal{S}=\left\{{T}_{s}:sâˆˆS\right\}$ of mappings from C into itself is said to be a representation of S as a nonexpansive mapping on C into itself if $\mathcal{S}$ satisfies the following:

1. (1)

${T}_{st}x={T}_{s}{T}_{t}x$ for all $s,tâˆˆS$ and $xâˆˆC$;

2. (2)

for every $sâˆˆS$, the mapping ${T}_{s}:Câ†’C$ is nonexpansive.

We denote by $Fix\left(\mathcal{S}\right)$ the set of common fixed points of $\mathcal{S}$, that is, $Fix\left(\mathcal{S}\right)={â‹‚}_{sâˆˆS}\left\{xâˆˆC:{T}_{s}x=x\right\}$.

Theorem 2.1 [6]

Let S be a semigroup, let C be a closed, convex subset of a reflexive Banach space E, $\mathcal{S}=\left\{{T}_{s}:sâˆˆS\right\}$ be a representation of S as a nonexpansive mapping from C into itself such that weak closure of $\left\{{T}_{t}x:tâˆˆS\right\}$ is weakly compact for each $xâˆˆC$, and let X be a subspace of $B\left(S\right)$ such that $1âˆˆX$ and the mapping $tâ†’ã€ˆT\left(t\right)x,{x}^{âˆ—}ã€‰$ be an element of X for each $xâˆˆC$ and ${x}^{âˆ—}âˆˆE$, and Î¼ be a mean on X. If we write ${T}_{\mathrm{Î¼}}x$ instead of $âˆ«{T}_{t}x\phantom{\rule{0.2em}{0ex}}\mathrm{d}\mathrm{Î¼}\left(t\right)$, then the following hold.

1. (i)

${T}_{\mathrm{Î¼}}$ is a nonexpansive mapping from C into C.

2. (ii)

${T}_{\mathrm{Î¼}}x=x$ for each $xâˆˆFix\left(\mathcal{S}\right)$.

3. (iii)

${T}_{\mathrm{Î¼}}xâˆˆ\stackrel{Â¯}{co}\left\{{T}_{t}x:tâˆˆS\right\}$ for each $xâˆˆC$.

4. (iv)

If X is ${r}_{s}$-invariant for each $sâˆˆS$ and Î¼ is right invariant, then ${T}_{\mathrm{Î¼}}{T}_{t}={T}_{\mathrm{Î¼}}$ for each $tâˆˆS$.

Remark From Theorem 4.1.6 in [1], every uniformly convex Banach space is strictly convex and reflexive.

Let D be a subset of B, where B is a subset of a Banach space E, and let P be a retraction of B onto D, that is, $Px=x$ for each $xâˆˆD$. Then P is said to be sunny if for each $xâˆˆB$ and $tâ‰¥0$ with $Px+t\left(xâˆ’Px\right)âˆˆB$, $P\left(Px+t\left(xâˆ’Px\right)\right)=Px$. A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D. We know that if E is smooth and P is a retraction of B onto D, then P is sunny and nonexpansive if and only if for each $xâˆˆB$ and $zâˆˆD$, $ã€ˆxâˆ’Px,J\left(zâˆ’Px\right)ã€‰â‰¤0$. For more details, see [1].

Lemma 2.2 [7]

Let S be a semigroup, and let C be a compact convex subset of a real strictly convex and smooth Banach space E. Suppose that $\mathcal{S}=\left\{{T}_{s}:sâˆˆS\right\}$ is a representation of S as a nonexpansive mapping from C into itself. Let X be a left invariant subspace of $B\left(S\right)$ such that $1âˆˆX$, and the function $tâ†¦ã€ˆ{T}_{t}x,{x}^{âˆ—}ã€‰$ is an element of X for each $xâˆˆC$ and ${x}^{âˆ—}âˆˆ{E}^{âˆ—}$. If Î¼ is a left invariant mean on X, then $Fix\left({T}_{\mathrm{Î¼}}\right)={T}_{\mathrm{Î¼}}C=Fix\left(\mathcal{S}\right)$ and there exists a unique sunny nonexpansive retraction from C onto $Fix\left(\mathcal{S}\right)$.

Throughout the rest of this paper, the open ball of radius r centered at 0 is denoted by ${B}_{r}$. Let C be a nonempty closed convex subset of a Banach space E. For $\mathrm{Ïµ}>0$ and a mapping $T:Câ†’C$, we let ${F}_{\mathrm{Ïµ}}\left(T\right)$ be the set of Ïµ-approximate fixed points of T, i.e., ${F}_{\mathrm{Ïµ}}\left(T\right)=\left\{xâˆˆC:âˆ¥xâˆ’Txâˆ¥â‰¤\mathrm{Ïµ}\right\}$.

## 3 Main result

In this section, we deal with a strong convergence approximation scheme for finding a common element of the set of common fixed points of a representation of nonexpansive mappings.

Theorem 3.1 Let S be a semigroup. Let C be a nonempty compact convex subset of a real strictly convex and reflexive and smooth Banach space E. Suppose that $\mathcal{S}=\left\{{T}_{s}:sâˆˆS\right\}$ is a representation of S as a nonexpansive mapping from C into itself such that . Let X be a left invariant subspace of $B\left(S\right)$ such that $1âˆˆX$, and the function $tâ†¦ã€ˆ{T}_{t}x,{x}^{âˆ—}ã€‰$ is an element of X for each $xâˆˆC$ and ${x}^{âˆ—}âˆˆ{E}^{âˆ—}$. Let $\left\{{\mathrm{Î¼}}_{n}\right\}$ be a left regular sequence of means on X. Suppose that f is an Î±-contraction on C. Let ${\mathrm{Ïµ}}_{n}$ be a sequence in $\left(0,1\right)$ such that ${lim}_{n}{\mathrm{Ïµ}}_{n}=0$. Then there exists a unique sunny nonexpansive retraction P of C onto $Fix\left(\mathcal{S}\right)$ and $xâˆˆC$ such that the following sequence $\left\{{z}_{n}\right\}$ generated by

${z}_{n}={\mathrm{Ïµ}}_{n}f\left({z}_{n}\right)+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right){T}_{{\mathrm{Î¼}}_{n}}{z}_{n}\phantom{\rule{1em}{0ex}}\left(nâˆˆ\mathbb{N}\right)$
(1)

strongly converges to Px.

Proof By Proposition 1.7.3 and Theorem 1.9.21 in [8], any compact subset C of a reflexive Banach space E is weakly compact, and from Proposition 1.9.18 in [8], any closed convex subset of a weakly compact subset C of a Banach space E is itself weakly compact, and by Proposition 1.9.13 in [8], any convex subset C of a normed space E is weakly closed if and only if C is closed. Therefore, weak closure of $\left\{{T}_{t}x:tâˆˆS\right\}$ is weakly compact for each $xâˆˆC$.

We divide the proof into five steps.

Step 1. The existence of ${z}_{n}$ which satisfies (1).

This follows immediately from the fact that for every $nâˆˆ\mathbb{N}$, the mapping ${N}_{n}$ given by

${N}_{n}x:={\mathrm{Ïµ}}_{n}f\left(x\right)+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right){T}_{{\mathrm{Î¼}}_{n}}x\phantom{\rule{1em}{0ex}}\left(xâˆˆC\right)$

is a contraction. To see this, put ${\mathrm{Î²}}_{n}=\left(1+{\mathrm{Ïµ}}_{n}\left(\mathrm{Î±}âˆ’1\right)\right)$, then $0â‰¤{\mathrm{Î²}}_{n}<1$ ($nâˆˆ\mathbb{N}$). Then we have

$\begin{array}{rl}âˆ¥{N}_{n}xâˆ’{N}_{n}yâˆ¥& â‰¤{\mathrm{Ïµ}}_{n}âˆ¥f\left(x\right)âˆ’f\left(y\right)âˆ¥+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right)âˆ¥{T}_{{\mathrm{Î¼}}_{n}}xâˆ’{T}_{{\mathrm{Î¼}}_{n}}yâˆ¥\\ â‰¤{\mathrm{Ïµ}}_{n}\mathrm{Î±}âˆ¥xâˆ’yâˆ¥+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right)âˆ¥xâˆ’yâˆ¥\\ =\left(1+{\mathrm{Ïµ}}_{n}\left(\mathrm{Î±}âˆ’1\right)\right)âˆ¥xâˆ’yâˆ¥={\mathrm{Î²}}_{n}âˆ¥xâˆ’yâˆ¥.\end{array}$

Therefore, by the Banach contraction principle [1], there exists a unique point ${z}_{n}âˆˆC$ such that ${N}_{n}{z}_{n}={z}_{n}$.

Step 2. ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{z}_{n}âˆ’{T}_{t}{z}_{n}âˆ¥=0$ for all $tâˆˆS$.

Consider $tâˆˆS$ and let $\mathrm{Ïµ}>0$. By Lemma 1 in [9], there exists $\mathrm{Î´}>0$ such that $\stackrel{Â¯}{co}\phantom{\rule{0.2em}{0ex}}{F}_{\mathrm{Î´}}\left({T}_{t}\right)+2{B}_{\mathrm{Î´}}âŠ†{F}_{\mathrm{Ïµ}}\left({T}_{t}\right)$. By Corollary 2.8 in [10], there also exists a natural number N such that

$âˆ¥\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}yâˆ’{T}_{t}\left(\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}y\right)âˆ¥â‰¤\mathrm{Î´}$
(2)

for all $sâˆˆS$ and $yâˆˆC$. Let $pâˆˆFix\left(\mathcal{S}\right)$ and ${M}_{0}$ be a positive number such that ${sup}_{yâˆˆC}âˆ¥yâˆ¥â‰¤{M}_{0}$. Let $tâˆˆS$, since $\left\{{\mathrm{Î¼}}_{n}\right\}$ is strongly left regular, there exists ${N}_{0}âˆˆ\mathbb{N}$ such that $âˆ¥{\mathrm{Î¼}}_{n}âˆ’{l}_{{t}^{i}}^{âˆ—}{\mathrm{Î¼}}_{n}âˆ¥â‰¤\frac{\mathrm{Î´}}{\left(3{M}_{0}\right)}$ for $nâ‰¥{N}_{0}$ and $i=1,2,â€¦,N$. Then we have

$\begin{array}{r}\underset{yâˆˆC}{sup}âˆ¥{T}_{{\mathrm{Î¼}}_{n}}yâˆ’âˆ«\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}y\phantom{\rule{0.2em}{0ex}}\mathrm{d}{\mathrm{Î¼}}_{n}\left(s\right)âˆ¥\\ \phantom{\rule{1em}{0ex}}=\underset{yâˆˆC}{sup}\underset{âˆ¥{x}^{âˆ—}âˆ¥=1}{sup}|ã€ˆ{T}_{{\mathrm{Î¼}}_{n}}y,{x}^{âˆ—}ã€‰âˆ’ã€ˆâˆ«\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}y\phantom{\rule{0.2em}{0ex}}\mathrm{d}{\mathrm{Î¼}}_{n}\left(s\right),{x}^{âˆ—}ã€‰|\\ \phantom{\rule{1em}{0ex}}=\underset{yâˆˆC}{sup}\underset{âˆ¥{x}^{âˆ—}âˆ¥=1}{sup}|\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{\left({\mathrm{Î¼}}_{n}\right)}_{s}ã€ˆ{T}_{s}y,{x}^{âˆ—}ã€‰âˆ’\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{\left({\mathrm{Î¼}}_{n}\right)}_{s}ã€ˆ{T}_{{t}^{i}s}y,{x}^{âˆ—}ã€‰|\\ \phantom{\rule{1em}{0ex}}â‰¤\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}\underset{yâˆˆC}{sup}\underset{âˆ¥{x}^{âˆ—}âˆ¥=1}{sup}|{\left({\mathrm{Î¼}}_{n}\right)}_{s}ã€ˆ{T}_{s}y,{x}^{âˆ—}ã€‰âˆ’{\left({l}_{{t}^{i}}^{âˆ—}{\mathrm{Î¼}}_{n}\right)}_{s}ã€ˆ{T}_{s}y,{x}^{âˆ—}ã€‰|\\ \phantom{\rule{1em}{0ex}}â‰¤\underset{i=1,2,â€¦,N}{max}âˆ¥{\mathrm{Î¼}}_{n}âˆ’{l}_{{t}^{i}}^{âˆ—}{\mathrm{Î¼}}_{n}âˆ¥\left({M}_{0}+2âˆ¥pâˆ¥\right)\\ \phantom{\rule{1em}{0ex}}â‰¤\underset{i=1,2,â€¦,N}{max}âˆ¥{\mathrm{Î¼}}_{n}âˆ’{l}_{{t}^{i}}^{âˆ—}{\mathrm{Î¼}}_{n}âˆ¥\left(3{M}_{0}\right)\\ \phantom{\rule{1em}{0ex}}â‰¤\mathrm{Î´}\phantom{\rule{1em}{0ex}}\left(nâ‰¥{N}_{0}\right).\end{array}$
(3)

By Theorem 2.1 we have

$âˆ«\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}y\phantom{\rule{0.2em}{0ex}}\mathrm{d}{\mathrm{Î¼}}_{n}\left(s\right)âˆˆ\stackrel{Â¯}{co}\left\{\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}}\left({T}_{s}y\right):sâˆˆS\right\}.$
(4)

It follows from (2)-(4) that

$\begin{array}{rl}{T}_{{\mathrm{Î¼}}_{n}}y& âˆˆ\stackrel{Â¯}{co}\left\{\frac{1}{N+1}\underset{i=0}{\overset{N}{âˆ‘}}{T}_{{t}^{i}s}y:sâˆˆS\right\}+{B}_{\mathrm{Î´}}\\ âŠ‚\stackrel{Â¯}{co}\phantom{\rule{0.2em}{0ex}}{F}_{\mathrm{Î´}}\left({T}_{t}\right)+2{B}_{\mathrm{Î´}}âŠ‚{F}_{\mathrm{Ïµ}}\left({T}_{t}\right)\end{array}$

for all $yâˆˆC$ and $nâ‰¥{N}_{0}$. Therefore, ${limâ€‰sup}_{nâ†’\mathrm{âˆž}}{sup}_{yâˆˆC}âˆ¥{T}_{t}\left({T}_{{\mathrm{Î¼}}_{n}}y\right)âˆ’{T}_{{\mathrm{Î¼}}_{n}}yâˆ¥â‰¤\mathrm{Ïµ}$. Since $\mathrm{Ïµ}>0$ is arbitrary, we have

$\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}\underset{yâˆˆC}{sup}âˆ¥{T}_{t}\left({T}_{{\mathrm{Î¼}}_{n}}y\right)âˆ’{T}_{{\mathrm{Î¼}}_{n}}yâˆ¥=0.$
(5)

Let $tâˆˆS$ and $\mathrm{Ïµ}>0$, then there exists $\mathrm{Î´}>0$ which satisfies (2). Take ${L}_{0}=\left(1+\mathrm{Î±}\right)2{M}_{0}+âˆ¥f\left(p\right)âˆ’pâˆ¥$. Now, from the condition ${lim}_{n}{\mathrm{Ïµ}}_{n}=0$ and from (5), there exists a natural number ${N}_{1}$ such that ${T}_{{\mathrm{Î¼}}_{n}}yâˆˆ{F}_{\mathrm{Î´}}\left({T}_{t}\right)$ for all $yâˆˆC$ and ${\mathrm{Ïµ}}_{n}<\frac{\mathrm{Î´}}{2{L}_{0}}$ for all $nâ‰¥{N}_{1}$. Since $pâˆˆFix\left(\mathcal{S}\right)$, we have

$\begin{array}{rcl}{\mathrm{Ïµ}}_{n}âˆ¥f\left({z}_{n}\right)âˆ’{T}_{{\mathrm{Î¼}}_{n}}{z}_{n}âˆ¥& â‰¤& {\mathrm{Ïµ}}_{n}\left(âˆ¥f\left({z}_{n}\right)âˆ’f\left(p\right)âˆ¥+âˆ¥f\left(p\right)âˆ’pâˆ¥+âˆ¥{T}_{{\mathrm{Î¼}}_{n}}pâˆ’{T}_{{\mathrm{Î¼}}_{n}}{z}_{n}âˆ¥\right)\\ â‰¤& {\mathrm{Ïµ}}_{n}\left(\mathrm{Î±}âˆ¥{z}_{n}âˆ’pâˆ¥+âˆ¥f\left(p\right)âˆ’pâˆ¥+âˆ¥Aâˆ¥âˆ¥{z}_{n}âˆ’pâˆ¥\right)\\ â‰¤& {\mathrm{Ïµ}}_{n}\left(\mathrm{Î±}âˆ¥{z}_{n}âˆ’pâˆ¥+âˆ¥f\left(p\right)âˆ’pâˆ¥+âˆ¥{z}_{n}âˆ’pâˆ¥\right)\\ â‰¤& {\mathrm{Ïµ}}_{n}\left(\left(1+\mathrm{Î±}\right)âˆ¥{z}_{n}âˆ’pâˆ¥+âˆ¥f\left(p\right)âˆ’pâˆ¥\right)\\ â‰¤& {\mathrm{Ïµ}}_{n}\left(\left(1+\mathrm{Î±}\right)2{M}_{0}+âˆ¥f\left(p\right)âˆ’pâˆ¥\right)\\ =& {\mathrm{Ïµ}}_{n}{L}_{0}â‰¤\frac{\mathrm{Î´}}{2}\end{array}$

for all $nâ‰¥{N}_{1}$. Observe that

$\begin{array}{rl}{z}_{n}& ={\mathrm{Ïµ}}_{n}f\left({z}_{n}\right)+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right){T}_{{\mathrm{Î¼}}_{n}}{z}_{n}\\ ={T}_{{\mathrm{Î¼}}_{n}}{z}_{n}+{\mathrm{Ïµ}}_{n}\left(f\left({z}_{n}\right)âˆ’{T}_{{\mathrm{Î¼}}_{n}}{z}_{n}\right)\\ âˆˆ{F}_{\mathrm{Î´}}\left({T}_{t}\right)+{B}_{\frac{\mathrm{Î´}}{2}}\\ âŠ†{F}_{\mathrm{Î´}}\left({T}_{t}\right)+2{B}_{\mathrm{Î´}}\\ âŠ†{F}_{\mathrm{Ïµ}}\left({T}_{t}\right)\end{array}$

for all $nâ‰¥{N}_{1}$. This shows that

$âˆ¥{z}_{n}âˆ’{T}_{t}{z}_{n}âˆ¥â‰¤\mathrm{Ïµ}\phantom{\rule{1em}{0ex}}\left(nâ‰¥{N}_{1}\right).$

Since $\mathrm{Ïµ}>0$ is arbitrary, we get ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{z}_{n}âˆ’{T}_{t}{z}_{n}âˆ¥=0$.

Step 3. $\mathfrak{S}\left\{{z}_{n}\right\}âŠ‚Fix\left(\mathcal{S}\right)$, where $\mathfrak{S}\left\{{z}_{n}\right\}$ denotes the set of strongly limit points of $\left\{{z}_{n}\right\}$.

Let $zâˆˆ\mathfrak{S}\left\{{z}_{n}\right\}$, and let $\left\{{z}_{{n}_{j}}\right\}$ be a subsequence of $\left\{{z}_{n}\right\}$ such that ${z}_{{n}_{j}}â†’z$,

$\begin{array}{rl}âˆ¥{T}_{t}zâˆ’zâˆ¥& â‰¤âˆ¥{T}_{t}zâˆ’{T}_{t}{z}_{{n}_{j}}âˆ¥+âˆ¥{T}_{t}{z}_{{n}_{j}}âˆ’{z}_{{n}_{j}}âˆ¥+âˆ¥{z}_{{n}_{j}}âˆ’zâˆ¥\\ â‰¤2âˆ¥{z}_{{n}_{j}}âˆ’zâˆ¥+âˆ¥{T}_{t}{z}_{{n}_{j}}âˆ’{z}_{{n}_{j}}âˆ¥,\end{array}$

then by Step 2,

$âˆ¥{T}_{t}zâˆ’zâˆ¥â‰¤2\underset{j}{lim}âˆ¥{z}_{{n}_{j}}âˆ’zâˆ¥+\underset{j}{lim}âˆ¥{T}_{t}{z}_{{n}_{j}}âˆ’{z}_{{n}_{j}}âˆ¥=0,$

therefore $zâˆˆFix\left(\mathcal{S}\right)$.

Step 4. There exists a unique sunny nonexpansive retraction P of C onto $Fix\left(\mathcal{S}\right)$ and $xâˆˆC$ such that

$\mathrm{Î“}:=\underset{n}{limâ€‰sup}ã€ˆxâˆ’Px,J\left({z}_{n}âˆ’Px\right)ã€‰â‰¤0.$
(6)

By Lemma 2.2 there exists a unique sunny nonexpansive retraction P of C onto $Fix\left(\mathcal{S}\right)$. The Banach contraction mapping principle guarantees that fP has a unique fixed point $xâˆˆC$. We show that

$\mathrm{Î“}:=\underset{n}{limâ€‰sup}ã€ˆxâˆ’Px,J\left({z}_{n}âˆ’Px\right)ã€‰â‰¤0.$

Note that from the definition of Î“ and the fact that C is a compact subset of E, we can select a subsequence $\left\{{z}_{{n}_{j}}\right\}$ of $\left\{{z}_{n}\right\}$ with the following properties:

1. (i)

${lim}_{j}ã€ˆxâˆ’Px,J\left({z}_{{n}_{j}}âˆ’Px\right)ã€‰=\mathrm{Î“}$;

2. (ii)

$\left\{{z}_{{n}_{j}}\right\}$ converges strongly to a point z.

By Step 3, we have $zâˆˆFix\left(\mathcal{S}\right)$. Since E is smooth, we have

$\mathrm{Î“}=\underset{j}{lim}ã€ˆxâˆ’Px,J\left({z}_{{n}_{j}}âˆ’Px\right)ã€‰=ã€ˆxâˆ’Px,J\left(zâˆ’Px\right)ã€‰â‰¤0.$

Since $fPx=x$, we have $\left(fâˆ’I\right)Px=xâˆ’Px$. From Theorem 4.2.1(v) in [1], for $x,yâˆˆE$ and $fâˆˆJ\left(y\right)$, ${âˆ¥xâˆ¥}^{2}âˆ’{âˆ¥yâˆ¥}^{2}â‰¥2\left(xâˆ’y,f\right)$. Therefore, for each $nâˆˆ\mathbb{N}$, we have

$\begin{array}{c}{\mathrm{Ïµ}}_{n}\left(\mathrm{Î±}âˆ’1\right){âˆ¥{z}_{n}âˆ’Pxâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¥{\left[{\mathrm{Ïµ}}_{n}\mathrm{Î±}âˆ¥{z}_{n}âˆ’Pxâˆ¥+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right)âˆ¥{z}_{n}âˆ’Pxâˆ¥\right]}^{2}âˆ’{âˆ¥{z}_{n}âˆ’Pxâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¥{\left[{\mathrm{Ïµ}}_{n}âˆ¥f\left({z}_{n}\right)âˆ’f\left(Px\right)âˆ¥+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right)âˆ¥{T}_{{\mathrm{Î¼}}_{n}}{z}_{n}âˆ’Pxâˆ¥\right]}^{2}âˆ’{âˆ¥{z}_{n}âˆ’Pxâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¥2ã€ˆ{\mathrm{Ïµ}}_{n}\left(f\left({z}_{n}\right)âˆ’f\left(Px\right)\right)+\left(1âˆ’{\mathrm{Ïµ}}_{n}\right)\left({T}_{{\mathrm{Î¼}}_{n}}{z}_{n}âˆ’Px\right)âˆ’\left({z}_{n}âˆ’Px\right),J\left({z}_{n}âˆ’Px\right)ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=âˆ’2{\mathrm{Ïµ}}_{n}ã€ˆ\left(fâˆ’I\right)Px,J\left({z}_{n}âˆ’Px\right)ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=âˆ’2{\mathrm{Ïµ}}_{n}ã€ˆxâˆ’Px,J\left({z}_{n}âˆ’Px\right)ã€‰.\hfill \end{array}$

Hence

${âˆ¥{z}_{n}âˆ’Pxâˆ¥}^{2}â‰¤\frac{2}{1âˆ’\mathrm{Î±}}ã€ˆxâˆ’Px,J\left({z}_{n}âˆ’Px\right)ã€‰.$
(7)

Step 5. $\left\{{z}_{n}\right\}$ strongly converges to Px.

Indeed, from (6), (7) and $PxâˆˆFix\left(\mathcal{S}\right)$, we conclude

$\underset{n}{limâ€‰sup}{âˆ¥{z}_{n}âˆ’Pxâˆ¥}^{2}â‰¤\frac{2}{1âˆ’\mathrm{Î±}}\underset{n}{limâ€‰sup}ã€ˆxâˆ’Px,J\left({z}_{n}âˆ’Px\right)ã€‰â‰¤0.$

That is, ${z}_{n}â†’Px$.â€ƒâ–¡

Remark 3.2 It would be an interesting problem to prove Theorem 3.1 for continuous representations instead of nonexpansive.

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## Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the referee of the paper for his helpful comments and invaluable suggestions. This research was supported by the Center of Excellence for Mathematics and the Office of Graduate Studies of the Lorestan University and the University of Isfahan.

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Hussain, N., Lashkarizadeh Bami, M. & Soori, E. An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2014, 238 (2014). https://doi.org/10.1186/1687-1812-2014-238