# Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces

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## Abstract

We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

**2010 Mathematics Subject Classification**: 47H09; 47J25; 47J30.

### Keywords

accretive operators uniformly smooth and uniformly convex Banach space sunny nonexpansive retraction weak sequential continuous mapping regularization## 1 Introduction

Let *E* be a Banach space with its dual space *E**. For the sake of simplicity, the norms of *E* and *E** are denoted by the symbol || · ||. We write 〈*x*, *x**〉 instead of *x**(*x*) for *x** ∈ *E** and *x* ∈ *E*. We denote as ⇀ and →, the weak convergence and strong convergence, respectively. A Banach space *E* is reflexive if *E* = *E***.

involving the accretive mapping *A*.

*A*(

*x*) is the proximal point algorithm of Rockafellar [1] which is recognized as a powerful and successful algorithm for finding a zero of monotone operators. Starting from any initial guess

*x*

_{0}∈

*H*, this proximal point algorithm generates a sequence {

*x*

_{ n }} given by

*r*> 0 is the resolvent of

*A*in a Hilbert space

*H*. Rockafellar [1] proved the weak convergence of the algorithm (1.2) provided that the regularization sequence {

*c*

_{ n }} remains bounded away from zero, and that the error sequence {

*e*

_{ n }} satisfies the condition ${\sum}_{n=0}^{\infty}\parallel {e}_{n}\parallel <\infty $. However, Güler's example [2] shows that proximal point algorithm (1.2) has only weak convergence in an infinite-dimensional Hilbert space. Recently, several authors proposed modifications of Rockafellar's proximal point algorithm (1.2) for the strong convergence. For example, Solodov and Svaiter [3] and Kamimura and Takahashi [4] studied a modified proximal point algorithm by an additional projection at each step of iteration. Lehdili and Moudafi [5] obtained the convergence of the sequence {

*x*

_{ n }} generated by the algorithm:

*A*

_{ n }=

*μ*

_{ n }

*I*+

*A*is viewed as a Tikhonov regularization of

*A*. When

*A*is maximal monotone in a Hilbert space

*H*, Xu [6], Song and Yang [7] used the technique of nonexpansive mappings to get convergence theorems for {

*x*

_{ n }} defined by the perturbed version of the algorithm (1.3):

In this article, we study a regularization proximal point algorithm to solve the problem of finding a common fixed point of a finite family of nonexpansive self-mappings in a uniformly convex and uniformly smooth Banach space *E*. Moreover, we give some analogue regularization methods for the more general problems, such as: problem of finding a common fixed point of a finite family of nonexpansive mappings *T*_{ i } , *i* = 1, 2, ..., *N*, where *T*_{ i } is self-mapping or nonself-mapping on a closed convex subset of *E*.

## 2 Preliminaries

**Definition 2.1**. A Banach space

*E*is said to be uniformly convex, if for any

*ε*∈ (0, 2] the inequalities ||

*x*|| ≤ 1, ||

*y*|| ≤ 1, ||

*x*-

*y*|| ≥

*ε*imply that there exists a

*δ*=

*δ*(

*ε*) ≥ 0 such that

is called the modulus of convexity of the space *E*. The function *δ*_{ E } (*ε*) defined on the interval [0, 2] is continuous, increasing and *δ*_{ E } (0) = 0. The space *E* is uniformly convex if and only if *δ*_{ E } (*ε*) > 0, ∀*ε* ∈ (0, 2].

is called the modulus of smoothness of the space *E*. The function *ρ*_{ E } (*τ*) defined on the interval [0, +∞) is convex, continuous, increasing and *ρ*_{ E } (0) = 0.

**Definition 2.2**. A Banach space

*E*is said to be uniformly smooth, if

*h*

_{ E }(

*τ*)is nondecreasing. In addition, it is not difficult to show that the estimate

*L*is the Figiel's constant [8, 9, 10], 1 <

*L*< 1.7. Indeed, we know that the inequality holds ([8])

*η*=

*Cτ*and

*ξ*=

*τ*, we obtain the inequality:

**Definition 2.3**. A mapping

*j*from

*E*onto

*E** satisfying the condition

is called the normalized duality mapping of *E*.

in a smooth Banach space, and the normalized duality mapping *J* is the identity operator *I* in a Hilbert space.

**Definition 2.4**. An operator

*A*:

*D*(

*A*) ⊆

*E*→

*E*is called accretive, if for all

*x*,

*y*∈

*D*(

*A*), there exists

*j*(

*x*-

*y*) ∈

*J*(

*x*-

*y*) such that

**Definition 2.5**. An operator *A* : *E* → *E* is called *m*-accretive if it is an accretive operator and the range *R*(*λA* + *I*) = *E* for all *λ* > 0, where *I* is the identity of *E*.

If *A* is an *m*-accretive operator then it is a demiclosed operator, i.e., if the sequence {*x*_{ n } } ⊂ *D*(*A*) satisfies *x*_{ n } ⇀ *x* and *A*(*x*_{ n } ) → *f*, then *A*(*x*) = *f*[10, 11].

**Definition 2.6**. A mapping

*T*:

*C*→

*E*is said to be nonexpansive on a closed convex subset

*C*of Banach space

*E*if

If *T* : *C* → *E* is a nonexpansive then *I* - *T* is an accretive operator. In this case, if the subset *C* coincides *E* then *I* - *T* is an *m*-accretive operator.

**Definition 2.7**. Let

*G*be a nonempty closed convex subset of

*E*. A mapping

*Q*

_{ G }:

*E*→

*G*is said to be

- (i)
a retraction onto

*G*if ${Q}_{G}^{2}={Q}_{G}$; - (ii)a nonexpansive retraction if it also satisfies the inequality:$\parallel {Q}_{G}x-{Q}_{G}y\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\forall x,y\in E;$(2.13)
- (iii)a sunny retraction if for all
*x*∈*E*and for all*t*∈ [0, +∞)${Q}_{G}\left({Q}_{G}x+t\left(x-{Q}_{G}x\right)\right)={Q}_{G}x.$(2.14)

A closed convex subset *C* of *E* is said to be a nonexpansive retract of *E*, if there exists a nonexpansive retraction from *E* onto *C*, and it is said to be a sunny nonexpansive retract of *E*, if there exists a sunny nonexpansive retraction from *E* onto *C*.

**Proposition 2.8**. [9]

*Let G be a nonempty closed convex subset of E. A mapping Q*

_{ G }:

*E*→

*G is a sunny nonexpansive retraction if and only if*

Reich [12] showed that if *E* is uniformly smooth and *D* is the fixed point set of a nonexpansive mapping from *C* into itself, then there is a sunny nonexpansive retraction from *C* onto *D*, and it can be constructed as follows.

**Lemma 2.9**. [12]

*Let E be a uniformly smooth Banach space, and let T*:

*C*→

*C be a nonexpansive mapping with a fixed point. For each u*∈

*C and every t*∈ (0, 1),

*the unique fixed point x*

_{ t }∈

*C of the contraction C*∋

*x*↦

*tu*+ (1 -

*t*)

*Tx converges strongly as t*→ 0

*to a fixed point of T*.

*Define Q*:

*C*→

*Fix*(

*T*)

*by Qu*= lim

_{t→0}

*x*

_{ t }.

*Then*,

*Q is a unique sunny nonexpansive retraction from C onto Fix*(

*T*),

*i.e.*,

*Q satisfies the property:*

**Definition 2.10**. Let

*C*

_{1}and

*C*

_{2}be convex subsets of

*E*. The quantity

*C*

_{1}from the set

*C*

_{2}. The function

is said to be a Hausdorff distance between *C*_{1} and *C*_{2}.

In this article, we will use the following useful lemma:

**Lemma 2.11**. [7]

*If E is a uniformly smooth Banach space, C*

_{1}

*and C*

_{2}

*are closed convex subsets of E such that the Hausdorff distance*$\mathcal{H}\left({C}_{1},{C}_{2}\right)\le \delta $, and ${Q}_{{C}_{1}}$

*and*${Q}_{{C}_{2}}$

*are the sunny nonexpansive retractions onto the subsets C*

_{1}

*and C*

_{2},

*respectively, then*

*where L is Figiel's constant*, *r* = ||*x*||, *d* = max{*d*_{1}, *d*_{2}}, *and R* = 2(2*r* + *d*) + *δ. Here d*_{ i } = *dist*(*θ*, *C*_{ i } ) = *d*(*θ*, *C*_{ i } ), *i* = 1, 2, *and θ is the origin of the space E*.

## 3 Main results

We need the following lemmas in the proof of our results:

**Lemma 3.1**. [9]

*If A*=

*I*-

*T with a nonexpansive mapping T, then for all x*,

*y*∈

*D*(

*T*),

*the domain of T*

*where* ||*x*|| ≤ *R*, ||*y*|| ≤ *R and* 1 < *L* < 1.7 *is Figiel's constant*.

**Lemma 3.2**. [13]

*Let*{

*a*

_{ n }}

*be a sequence of nonnegative real numbers satisfying the property:*

*where*{

*λ*

_{ n }}, {

*β*

_{ n }}

*and*{

*σ*

_{ n }}

*satisfy the following conditions*.

- (i)
${\sum}_{n=0}^{\infty}{\lambda}_{n}=\infty $;

- (ii)
lim sup

_{n→∞}*β*_{ n }≤ 0 or ${\sum}_{n=0}^{\infty}\left|{\lambda}_{n}{\beta}_{n}\right|<\infty $; - (iii)
*σ*_{ n }≥ 0 ∀*n*≥ 0 and ${\sum}_{n=0}^{\infty}{\sigma}_{n}<\infty $.

*Then*, {*a*_{ n } } *converges to zero*.

**Lemma 3.3**. [9]

*Let E be a uniformly smooth Banach space. Then, for all x*,

*y*∈

*E*,

*where c* = 48 max(*L*, ||*x*||, ||*y*||).

*where Fix*(*T*_{ i } ) *is the set of fixed points of the nonexpansive mapping T*_{ i } : *E* → *E*, *i* = 1, 2, ..., *N*.

**Theorem 3.4**.

*Suppose that E is a uniformly convex and uniformly smooth Banach space which has a weakly sequentially continuous normalized duality mapping j from E to E**.

*Let T*

_{ i }:

*E*→

*E*,

*i*= 1, 2, ...,

*N be nonexpansive mappings with*$S={\cap}_{i=1}^{N}Fix\left({T}_{i}\right)\ne \varnothing $.

*If the sequences*{

*r*

_{ n }} ⊂ (0, +∞)

*and*{

*t*

_{ n }} ⊂ (0, 1)

*satisfy*

- (i)
lim

_{n→∞}*t*_{ n }= 0; ${\sum}_{n=0}^{\infty}{t}_{n}=+\infty $; - (ii)
lim

_{n→∞}*r*_{ n }= +∞,

*then the sequence*{

*x*

_{ n }}

*defined by*

*converges strongly to Q*_{ S }*u, where A*_{ i } = *I* - *T*_{ i } , *i* = 1, 2, ..., *N and Q*_{ S } *is a sunny nonexpansive retraction from E onto S*.

*Proof*. First, equation (3.4) defines a unique sequence {

*x*

_{ n }} ⊂

*E*, because for each

*n*, the element

*x*

_{n+1}is a unique fixed point of the contraction mapping

*f*:

*E*→

*E*defined by

*x** ∈

*S*, we have

Therefore, the sequence {*x*_{ n } } is bounded. Every bounded set in a reflexive Banach space is relatively weakly compact. This means that there exists a subsequence $\left\{{x}_{{n}_{k}}\right\}\subseteq \left\{{x}_{n}\right\}$ which converges to a limit $\stackrel{\u0304}{x}\in E$.

*x*

_{ n }|| ≤

*R*and ||

*x**|| ≤

*R*with

*R*> 0. By Lemma 3.1, we have

*i*= 1, 2, ...,

*N*. Since the modulus of convexity

*δ*

_{ E }is continuous and

*E*is a uniformly convex Banach space,

*A*

_{ i }(

*x*

_{n+1}) → 0,

*i*= 1, 2, ...,

*N*. It is clear that $\stackrel{\u0304}{x}\in S$ from the demiclosedness of

*A*

_{ i }. Hence, noting the inequality (2.15), we obtain

Since *E* is a uniformly smooth Banach space, $\frac{{\rho}_{E}\left({t}_{n}\parallel u-{Q}_{S}u\parallel \right)}{{t}_{n}}\to 0,\phantom{\rule{1em}{0ex}}n\to \infty $. By (3.7), we obtain lim sup_{n→∞}*β*_{ n } ≤ 0. Hence, an application of Lemma 3.2 on (3.8) yields the desired result. □

where *T*_{ i } : *C*_{ i } → *C*_{ i } , *i* = 1, 2, ..., *N* is a nonexpansive mapping and *C*_{ i } is a convex closed nonexpansive retract of *E*.

Obviously, we have the following lemma:

**Lemma 3.5**. *Let E be a Banach space, and let C be a closed convex retract of E. Let T* : *C* → *C be a nonexpansive mapping such that Fix*(*T*) ≠ ∅. *Then*, *Fix*(*T*) = *Fix*(*TQ*_{ C } ), *where Q*_{ C } *is a retraction of E onto C*.

*x*

_{ n }} defined by

where ${B}_{i}=I-{T}_{i}{Q}_{{C}_{i}}$, *i* = 1, 2, ..., *N* and ${Q}_{{C}_{i}}$ is a nonexpansive retraction from *E* onto *C*_{ i } , *i* = 1, 2, ..., *N*.

**Theorem 3.6**.

*Suppose that E is a uniformly convex and uniformly smooth Banach space which has a weakly sequentially continuous normalized duality mapping j from E into E**.

*Let C*

_{ i }

*be a convex closed nonexpansive retract of E and let T*

_{ i }:

*C*

_{ i }→

*C*

_{ i },

*i*= 1, 2, ...,

*N be a nonexpansive mapping such that*$S={\cap}_{i=1}^{N}Fix\left({T}_{i}\right)\ne \varnothing $.

*If the sequences*{

*r*

_{ n }} ⊂ (0, +∞)

*and*{

*t*

_{ n }} ⊂ (0, 1)

*satisfy*

- (i)
lim

_{n→∞}*t*_{ n }= 0; ${\sum}_{n=0}^{\infty}{t}_{n}=+\infty $; - (ii)
lim

_{n→∞}*r*_{ n }= +∞,

*then the sequence* {*x*_{ n } } *generated by* (3.10) *converges strongly to Q*_{ S }*u, where Q*_{ S } *is a sunny nonexpansive retraction from E onto S*.

*Proof*. By the Lemma 3.5, we have $S={\cap}_{i=1}^{N}Fix\left({T}_{i}{Q}_{{C}_{i}}\right)$ and applying Theorem 3.4, we obtain the proof of this Theorem. □

Next, we study the stability of the regularization algorithm (3.10) in the case that each *C*_{ i } is a closed convex sunny nonexpansive retract of *E* with respect to perturbations of operators *T*_{ i } and constraints *C*_{ i } , *i* = 1, 2, ..., *N* satisfying following conditions:

*C*

_{ i }, there is a sequence of closed convex sunny nonexpansive retracts ${C}_{i}^{n}\subset E$,

*n*= 1, 2, 3, ... such that

where {*δ*_{ n } } is a sequence of positive numbers.

*i*= 1, 2, ...,

*N*satisfying the conditions: if for all

*t*> 0, there exists the increasing positive functions

*g*(

*t*) and

*ξ*(

*t*) such that

*g*(0) ≥ 0,

*ξ*(0) = 0 and

*x*∈

*C*

_{ i }, $y\in {C}_{i}^{m}$, ||

*x*-

*y*|| ≤

*δ*, then

where ${B}_{i}^{n}=I-{T}_{i}^{n}{Q}_{{C}_{i}^{n}}$, *i* = 1, 2, ..., *N* and ${Q}_{{C}_{i}^{n}}$ is a sunny nonexpansive retraction from *E* onto ${C}_{i}^{n}$, *i* = 1, 2, ..., *N*.

**Theorem 3.7**.

*Suppose that E is a uniformly convex and uniformly smooth Banach space which has a weakly sequentially continuous normalized duality mapping j from E into E**.

*Let C*

_{ i }

*be a convex closed sunny nonexpansive retract of E and let T*

_{ i }:

*C*

_{ i }→

*C*

_{ i },

*i*= 1, 2, ...,

*N be nonexpansive mappings such that*$S={\cap}_{i=1}^{N}Fix\left({T}_{i}\right)\ne \varnothing $.

*If the conditions (P1) and (P2) are fulfilled, and the sequences*{

*r*

_{ n }}, {

*δ*

_{ n }}

*and*{

*t*

_{ n }}

*satisfy*

- (i)
lim

_{n→∞}*t*_{ n }= 0; ${\sum}_{n=0}^{\infty}{t}_{n}=+\infty $; - (ii)
lim

_{n→∞}*r*_{ n }= +∞; - (iii)
${\sum}_{n=0}^{\infty}{r}_{n}\xi \left(a\sqrt{{h}_{E}\left({\delta}_{n}\right)}\right)<+\infty $ for each

*a*> 0,

*then the sequence* {*z*_{ n } } *generated by* (3.12) *converges strongly to Q*_{ S }*u, where Q*_{ S } *is a sunny nonexpansive retraction from E onto S*.

*Proof*. For each

*n*, ${\sum}_{i=1}^{N}{B}_{i}^{n}$ is an

*m*-accretive operator on

*E*, so the equation (3.12) defines a unique element

*z*

_{n+1}∈

*E*. From the equations (3.10) and (3.12), we have

*i*∈ {1, 2, ...,

*N*},

*x*

_{ n }} is bounded and $\mathcal{H}\left({C}_{i},{C}_{i}^{n}\right)\le {\delta}_{n}$, there exist constants

*K*

_{1,i}> 0 and

*K*

_{2,i}> 1 such that

where ${M}_{i}=max\left\{sup\parallel {Q}_{\underset{i}{\overset{n}{C}}}{x}_{n+1}\parallel ,\phantom{\rule{0.3em}{0ex}}sup\parallel {Q}_{{C}_{i}}{x}_{n+1}\parallel \right\}<+\infty $.

where *M* = max{*M*_{1}, *M*_{2}, ..., *M*_{ N } } < +∞ and ${\gamma}_{1,2}=\underset{i=1,2,\dots ,N}{max}\left\{{K}_{1,i}\sqrt{{K}_{2,i}L}\right\}$.

*z*

_{ n }-

*x*

_{ n }|| → 0. In addition, by Theorem 3.6,

which implies that {*z*_{ n } } converges strongly to *Q*_{ S }*u*. □

where *T*_{ i } : *C*_{ i } → *E*, *i* = 1, 2, ..., *N* is nonexpansive nonself-mapping and *C*_{ i } is a closed convex sunny nonexpansive retract of *E*.

**Lemma 3.8**. [14]*Let C be a closed convex subset of a strictly convex Banach space E and let T be a nonexpansive mapping from C into E. Suppose that C is a sunny nonexpansive retract of E. If Fix*(*T*) ≠ ∅, *then Fix*(*T*) = *Fix*(*Q*_{ C }*T*), *where Q*_{ C } *is a sunny nonexpansive retraction from E onto C*.

We have the following result:

**Theorem 3.9**.

*Suppose that E is a uniformly convex and uniformly smooth Banach space which has a weakly sequentially continuous normalized duality mapping j from E into E**.

*Let C*

_{ i }

*be a convex closed sunny nonexpansive retract of E and let T*

_{ i }:

*C*

_{ i }→

*E*,

*i*= 1, 2, ...,

*N be nonexpansive mappings such that*$S={\cap}_{i=1}^{N}Fix\left({T}_{i}\right)\ne \varnothing $.

*If the sequences*{

*r*

_{ n }} ⊂ (0, +∞)

*and*{

*t*

_{ n }} ⊂ (0, 1)

*satisfy*

- (i)
lim

_{n→∞}*t*_{ n }= 0; ${\sum}_{n=0}^{\infty}{t}_{n}=+\infty $; - (ii)
lim

_{n→∞}*r*_{ n }= +∞,

*then the sequence*{

*u*

_{ n }}

*defined by*

*converges strongly to Q*_{ S }*u, where Q*_{ S } *is a sunny nonexpansive retraction from E onto S and*${f}_{i}=I-{Q}_{{C}_{i}}{T}_{i}{Q}_{{C}_{i}}$, *i* = 1, 2, ..., N.

*Proof*. By the Lemma 3.5 and Lemma 3.8, $S={\cap}_{i=1}^{N}Fix\left({T}_{i}\right)={\cap}_{i=1}^{N}Fix\left({f}_{i}\right)$. Applying Theorem 3.4, we obtain the proof of this Theorem. □

## Notes

### Acknowledgements

The authors thank the referees for their valuable comments and suggestions. This work was supported by the Kyungnam University Research Fund, 2010.

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