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

Fixed Point Theory and Applications20112011:52

https://doi.org/10.1186/1687-1812-2011-52

• Accepted: 16 September 2011
• Published:

## 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**.

The problem of finding a fixed point of a nonexpansive mapping is equivalent to the problem of finding a zero of the following operator equation:
$0\in A\left(x\right)$
(1.1)

involving the accretive mapping A.

One popular method of solving equation 0 A(x) is the proximal point algorithm of Rockafellar  which is recognized as a powerful and successful algorithm for finding a zero of monotone operators. Starting from any initial guess x0 H, this proximal point algorithm generates a sequence {x n } given by
${x}_{n+1}={J}_{{c}_{n}}^{A}\left({x}_{n}+{e}_{n}\right),$
(1.2)
where ${J}_{r}^{A}=\left(I+rA{\right)}^{-1}$, r > 0 is the resolvent of A in a Hilbert space H. Rockafellar  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  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  and Kamimura and Takahashi  studied a modified proximal point algorithm by an additional projection at each step of iteration. Lehdili and Moudafi  obtained the convergence of the sequence {x n } generated by the algorithm:
${x}_{n+1}={J}_{{c}_{n}}^{{A}_{n}}\left({x}_{n}\right),$
(1.3)
where 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 , Song and Yang  used the technique of nonexpansive mappings to get convergence theorems for {x n } defined by the perturbed version of the algorithm (1.3):
${x}_{n+1}={J}_{{r}_{n}}^{A}\left({t}_{n}u+\left(1-{t}_{n}\right){x}_{n}\right).$
(1.4)
The equation (1.4) can be written in the following equivalent form:
${r}_{n}A\left({x}_{n+1}\right)+{x}_{n+1}\ni {t}_{n}u+\left(1-{t}_{n}\right){x}_{n}.$
(1.5)

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
$\frac{\parallel x+y\parallel }{2}\le 1-\delta .$
The function
${\delta }_{E}\left(\epsilon \right)=inf\left\{1-{2}^{-1}\parallel x+y\parallel :\parallel x\parallel =\parallel y\parallel =1,\phantom{\rule{2.77695pt}{0ex}}\parallel x-y\parallel =\epsilon \right\}$
(2.1)

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].

The function
${\rho }_{E}\left(\tau \right)=sup\left\{{2}^{-1}\left(\parallel x+y\parallel +\parallel x-y\parallel \right)-1:\parallel x\parallel =1,\phantom{\rule{2.77695pt}{0ex}}\parallel y\parallel =\tau \right\},$
(2.2)

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
$\underset{\tau \to 0}{lim}\frac{{\rho }_{E}\left(\tau \right)}{\tau }=0.$
(2.3)
It is well known that every uniformly convex and uniformly smooth Banach space is reflexive. In what follows, we denote
${h}_{E}\left(\tau \right)=\frac{{\rho }_{E}\left(\tau \right)}{\tau }.$
(2.4)
The function h E (τ)is nondecreasing. In addition, it is not difficult to show that the estimate
${h}_{E}\left(K\tau \right)\le LK{h}_{E}\left(\tau \right),\phantom{\rule{1em}{0ex}}\forall K>1,\phantom{\rule{1em}{0ex}}\tau >0,$
(2.5)
is valid, where L is the Figiel's constant , 1 < L < 1.7. Indeed, we know that the inequality holds ()
$\frac{{\rho }_{E}\left(\eta \right)}{{\eta }^{2}}\le L\frac{{\rho }_{E}\left(\xi \right)}{{\xi }^{2}},\phantom{\rule{1em}{0ex}}\forall \eta \ge \xi >0.$
(2.6)
It implies that
$\xi {h}_{E}\left(\eta \right)\le L\eta {h}_{E}\left(\xi \right),\phantom{\rule{1em}{0ex}}\forall \eta \ge \xi >0.$
(2.7)
Taking in (2.7) η = and ξ = τ, we obtain the inequality:
$\tau {h}_{E}\left(C\tau \right)\le LC\tau {h}_{E}\left(\tau \right),$
(2.8)
which implies that (2.5) holds. Similarly, we have
${\rho }_{E}\left(C\tau \right)\le L{C}^{2}{\rho }_{E}\left(\tau \right),\phantom{\rule{1em}{0ex}}\forall C>1,\phantom{\rule{1em}{0ex}}\tau >0.$
(2.9)
Definition 2.3. A mapping j from E onto E* satisfying the condition
$j\left(x\right)=\left\{f\in {E}^{*}:⟨x,f⟩=\parallel x{\parallel }^{2}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\parallel f\parallel =\parallel x\parallel \right\}$
(2.10)

is called the normalized duality mapping of E.

We know that
$j\left(x\right)={2}^{-1}\mathsf{\text{grad}}\parallel x{\parallel }^{2}.$

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) EE is called accretive, if for all x, y D(A), there exists j(x - y) J (x - y) such that
$⟨A\left(x\right)-A\left(y\right),j\left(x-y\right)⟩\ge 0.$
(2.11)

Definition 2.5. An operator A : EE 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 : CE is said to be nonexpansive on a closed convex subset C of Banach space E if
$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\forall x,y\in C.$
(2.12)

If T : CE 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 : EG is said to be
1. (i)

a retraction onto G if ${Q}_{G}^{2}={Q}_{G}$;

2. (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)

3. (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. Let G be a nonempty closed convex subset of E. A mapping Q G : EG is a sunny nonexpansive retraction if and only if
$⟨x-{Q}_{G}x,J\left(\xi -{Q}_{G}x\right)⟩\le 0,\phantom{\rule{1em}{0ex}}\forall x\in E,\phantom{\rule{2.77695pt}{0ex}}\forall \xi \in G.$
(2.15)

Reich  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. Let E be a uniformly smooth Banach space, and let T : CC 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 : CFix(T) by Qu = limt→0x t . Then, Q is a unique sunny nonexpansive retraction from C onto Fix(T), i.e., Q satisfies the property:
$⟨u-Qu,j\left(z-Qu\right)⟩\le 0,\phantom{\rule{1em}{0ex}}u\in C,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}z\in Fix\left(T\right).$
(2.16)
Definition 2.10. Let C1 and C2 be convex subsets of E. The quantity
$\beta \left({C}_{1},{C}_{2}\right)=\underset{u\in {C}_{1}}{sup}\underset{v\in {C}_{2}}{inf}\parallel u-v\parallel \left(=\underset{u\in {C}_{1}}{sup}d\left(u,{C}_{2}\right)\right)$
is said to be a semideviation of the set C1 from the set C2. The function
$\mathcal{H}\left({C}_{1},{C}_{2}\right)=max\left\{\beta \left({C}_{1},{C}_{2}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\beta \left({C}_{2},{C}_{1}\right)\right\}$

is said to be a Hausdorff distance between C1 and C2.

Lemma 2.11. If E is a uniformly smooth Banach space, C1and C2are 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 C1and C2, respectively, then
$\parallel {Q}_{{C}_{1}}x-{Q}_{{C}_{2}}x{\parallel }^{2}\le 16R\left(2r+d\right){h}_{E}\left(\frac{16L\delta }{R}\right),$
(2.17)

where L is Figiel's constant, r = ||x||, d = max{d1, d2}, and R = 2(2r + 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. If A = I - T with a nonexpansive mapping T, then for all x, y D(T), the domain of T
$⟨Ax-Ay,J\left(x-y\right)⟩\ge {L}^{-1}{R}^{2}{\delta }_{E}\left(\frac{\parallel Ax-Ay\parallel }{4R}\right),$
(3.1)

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

Lemma 3.2. Let {a n } be a sequence of nonnegative real numbers satisfying the property:
${a}_{n+1}\le \left(1-{\lambda }_{n}\right){a}_{n}+{\lambda }_{n}{\beta }_{n}+{\sigma }_{n},\phantom{\rule{1em}{0ex}}\forall n\ge 0$
where {λ n }, {β n } and {σ n } satisfy the following conditions.
1. (i)

${\sum }_{n=0}^{\infty }{\lambda }_{n}=\infty$;

2. (ii)

lim supn→∞ β n ≤ 0 or ${\sum }_{n=0}^{\infty }|{\lambda }_{n}{\beta }_{n}|<\infty$;

3. (iii)

σ n ≥ 0 n ≥ 0 and ${\sum }_{n=0}^{\infty }{\sigma }_{n}<\infty$.

Then, {a n } converges to zero.

Lemma 3.3. Let E be a uniformly smooth Banach space. Then, for all x, y E,
$\parallel x+y{\parallel }^{2}\le \parallel x{\parallel }^{2}+2⟨y,Jx⟩+c{\rho }_{E}\left(\parallel y\parallel \right),$
(3.2)

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

First, we consider the following problem:
(3.3)

where Fix(T i ) is the set of fixed points of the nonexpansive mapping T i : EE, 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 : EE, 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
1. (i)

limn→∞ t n = 0; ${\sum }_{n=0}^{\infty }{t}_{n}=+\infty$;

2. (ii)

limn→∞ r n = +∞,

then the sequence {x n } defined by
${r}_{n}\sum _{i=1}^{N}{A}_{i}\left({x}_{n+1}\right)+{x}_{n+1}={t}_{n}u+\left(1-{t}_{n}\right){x}_{n},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}u,{x}_{0}\in E,\phantom{\rule{1em}{0ex}}n\ge 0$
(3.4)

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 xn+1is a unique fixed point of the contraction mapping f : EE defined by
$f\left(x\right)=\frac{{r}_{n}}{N{r}_{n}+1}\sum _{i=1}^{N}{T}_{i}\left(x\right)+\frac{1}{N{r}_{n}+1}\left[{t}_{n}u+\left(1-{t}_{n}\right){x}_{n}\right],\phantom{\rule{1em}{0ex}}x\in E.$
For every x* S, we have
$⟨{r}_{n}\sum _{i=1}^{N}{A}_{i}\left({x}_{n+1}\right),j\left({x}_{n+1}-{x}^{*}\right)⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall n\ge 0.$
(3.5)
Therefore,
$⟨{t}_{n}u+\left(1-{t}_{n}\right){x}_{n}-{x}_{n+1},j\left({x}_{n+1}-{x}^{*}\right)⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall n\ge 0.$
(3.6)
It gives the inequality as follows:
$\parallel {x}_{n+1}-{x}^{*}{\parallel }^{2}\le \left\{{t}_{n}\parallel u-{x}^{*}\parallel +\left(1-{t}_{n}\right)\parallel {x}_{n}-{x}^{*}\parallel \right\}×\parallel {x}_{n+1}-{x}^{*}\parallel .$
Consequently, we have
$\begin{array}{lll}\hfill \parallel {x}_{n+1}-{x}^{*}\parallel & \le {t}_{n}\parallel u-{x}^{*}\parallel +\left(1-{t}_{n}\right)\parallel {x}_{n}-{x}^{*}\parallel \phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le max\left(\parallel u-{x}^{*}\parallel ,\parallel {x}_{n}-{x}^{*}\parallel \right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}⋮\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \le max\left(\parallel u-{x}^{*}\parallel ,\parallel {x}_{0}-{x}^{*}\parallel \right),\phantom{\rule{1em}{0ex}}\forall n\ge 0.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$

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 $\stackrel{̄}{x}\in E$.

Suppose ||x n || ≤ R and ||x*|| ≤ R with R > 0. By Lemma 3.1, we have
$\begin{array}{lll}\hfill {\delta }_{E}\left(\frac{\parallel {A}_{i}\left({x}_{n+1}\right)\parallel }{4R}\right)& \le \frac{L}{{R}^{2}{r}_{n}}⟨{r}_{n}{A}_{i}\left({x}_{n+1}\right),j\left({x}_{n+1}-{x}^{*}\right)⟩\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le \frac{L}{{R}^{2}{r}_{n}}⟨{r}_{n}\sum _{k=1}^{N}{A}_{k}\left({x}_{n+1}\right),j\left({x}_{n+1}-{x}^{*}\right)⟩\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \le \frac{L}{{R}^{2}{r}_{n}}\parallel {t}_{n}u+\left(1-{t}_{n}\right){x}_{n}-{x}_{n+1}\parallel .\parallel {x}_{n+1}-{x}^{*}\parallel \phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \phantom{\rule{2.77695pt}{0ex}}\to 0,\phantom{\rule{1em}{0ex}}n\to \infty ,\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
for every i = 1, 2, ..., N. Since the modulus of convexity δ E is continuous and E is a uniformly convex Banach space, A i (xn+1) → 0, i = 1, 2, ..., N. It is clear that $\stackrel{̄}{x}\in S$ from the demiclosedness of A i . Hence, noting the inequality (2.15), we obtain
$\begin{array}{lll}\hfill \underset{n\to \infty }{limsup}⟨u-{Q}_{S}u,j\left({x}_{n}-{Q}_{S}u\right)⟩& =\underset{k\to \infty }{lim}⟨u-{Q}_{S}u,j\left({x}_{{n}_{k}}-{Q}_{S}u\right)⟩\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =⟨u-{Q}_{S}u,j\left(\stackrel{̄}{x}-{Q}_{S}u\right)⟩\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \le 0.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(3.7)
Next, we have
By the Lemma 3.3 and the above inequality, we conclude that
Consequently, we have
$\parallel {x}_{n+1}-{Q}_{S}u{\parallel }^{2}\le \left(1-{t}_{n}\right)\parallel {x}_{n}-{Q}_{S}u{\parallel }^{2}+{t}_{n}{\beta }_{n},$
(3.8)
where
${\beta }_{n}=2\left(1-{t}_{n}\right)⟨u-{Q}_{S}u,j\left({x}_{n}-{Q}_{S}u\right)⟩+c\frac{{\rho }_{E}\left({t}_{n}\parallel u-{Q}_{S}u\parallel \right)}{{t}_{n}}.$

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 supn→∞β n ≤ 0. Hence, an application of Lemma 3.2 on (3.8) yields the desired result. □

Now, we will give a method to solve more generally following problem:
(3.9)

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 : CC be a nonexpansive mapping such that Fix(T) ≠ . Then, Fix(T) = Fix(TQ C ), where Q C is a retraction of E onto C.

We consider the iterative sequence {x n } defined by
${r}_{n}\sum _{i=1}^{N}{B}_{i}\left({x}_{n+1}\right)+{x}_{n+1}={t}_{n}u+\left(1-{t}_{n}\right){x}_{n},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{x}_{0}\in E,\phantom{\rule{1em}{0ex}}n\ge 0,$
(3.10)

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
1. (i)

limn→∞ t n = 0; ${\sum }_{n=0}^{\infty }{t}_{n}=+\infty$;

2. (ii)

limn→∞ 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:

(P1) Instead of C i , there is a sequence of closed convex sunny nonexpansive retracts ${C}_{i}^{n}\subset E$, n = 1, 2, 3, ... such that
$\mathcal{H}\left({C}_{i}^{n},{C}_{i}\right)\le {\delta }_{n},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N,$

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

(P2) For each set ${C}_{i}^{n}$, there is a nonexpansive self-mapping ${T}_{i}^{n}:{C}_{i}^{n}\to {C}_{i}^{n}$, 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
$\parallel {T}_{i}x-{T}_{i}^{m}y\parallel \le g\left(max\left\{\parallel x\parallel ,\parallel y\parallel \right\}\right)\xi \left(\delta \right).$
(3.11)
We establish the convergence and stability of the regularization method (3.10) in the form:
${r}_{n}\sum _{i=1}^{N}{B}_{i}^{n}\left({z}_{n+1}\right)+{z}_{n+1}={t}_{n}u+\left(1-{t}_{n}\right){z}_{n},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{z}_{0}\in E,\phantom{\rule{1em}{0ex}}n\ge 0,$
(3.12)

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
1. (i)

limn→∞ t n = 0; ${\sum }_{n=0}^{\infty }{t}_{n}=+\infty$;

2. (ii)

limn→∞ r n = +∞;

3. (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 zn+1 E. From the equations (3.10) and (3.12), we have
$\begin{array}{c}{r}_{n}⟨\sum _{i=1}^{N}{B}_{i}^{n}\left({z}_{n+1}\right)-{B}_{i}^{n}\left({x}_{n+1}\right),j\left({z}_{n+1}-{x}_{n+1}\right)⟩\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{r}_{n}⟨\sum _{i=1}^{N}{B}_{i}^{n}\left({x}_{n+1}\right)-{B}_{i}\left({x}_{n+1}\right),j\left({z}_{n+1}-{x}_{n+1}\right)⟩+\parallel {z}_{n+1}-{x}_{n+1}{\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\left(1-{t}_{n}\right)⟨{z}_{n}-{x}_{n},j\left({z}_{n+1}-{x}_{n+1}\right)⟩.\end{array}$
(3.13)
By the accretivity of ${\sum }_{i=1}^{N}{B}_{i}^{n}$ and the equation (3.13), we deduce
$\parallel {z}_{n+1}-{x}_{n+1}\parallel \le \left(1-{t}_{n}\right)\parallel {z}_{n}-{x}_{n}\parallel +{r}_{n}\sum _{i=1}^{N}\parallel {B}_{i}^{n}\left({x}_{n+1}\right)-{B}_{i}\left({x}_{n+1}\right)\parallel .$
(3.14)
For each i {1, 2, ..., N},
$\parallel {B}_{i}^{n}\left({x}_{n+1}\right)-{B}_{i}\left({x}_{n+1}\right)\parallel =\parallel {T}_{i}^{n}{Q}_{{C}_{i}^{n}}{x}_{n+1}-{T}_{i}{Q}_{{C}_{i}}{x}_{n+1}\parallel .$
(3.15)
Since {x n } is bounded and $\mathcal{H}\left({C}_{i},{C}_{i}^{n}\right)\le {\delta }_{n}$, there exist constants K1,i> 0 and K2,i> 1 such that
$\parallel {Q}_{{C}_{i}^{n}}{x}_{n+1}-{Q}_{{C}_{i}}{x}_{n+1}\parallel \le {K}_{1,i}\sqrt{{h}_{E}\left({K}_{2,i}{\delta }_{n}\right)}\le {K}_{1,i}\sqrt{{K}_{2,i}L}\sqrt{{h}_{E}\left({\delta }_{n}\right)}.$
(3.16)
By the condition (P2),
$\parallel {T}_{i}^{n}{Q}_{{C}_{i}^{n}}{x}_{n+1}-{T}_{i}{Q}_{{C}_{i}}{x}_{n+1}\parallel \le g\left({M}_{i}\right)\xi \left({K}_{1,i}\sqrt{{K}_{2,i}L}\sqrt{{h}_{E}\left({\delta }_{n}\right)}\right),$
(3.17)

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

From (3.14), (3.15) and (3.17), we obtain
$\parallel {z}_{n+1}-{x}_{n+1}\parallel \le \left(1-{t}_{n}\right)\parallel {z}_{n}-{x}_{n}\parallel +Ng\left(M\right){r}_{n}\xi \left({\gamma }_{1,2}\sqrt{{h}_{E}\left({\delta }_{n}\right)}\right),$
(3.18)

where M = max{M1, M2, ..., M N } < +∞ and ${\gamma }_{1,2}=\underset{i=1,2,\dots ,N}{max}\left\{{K}_{1,i}\sqrt{{K}_{2,i}L}\right\}$.

By the above assumption and Lemma 3.2, we conclude that ||z n - x n || → 0. In addition, by Theorem 3.6,
$\parallel {z}_{n}-{Q}_{S}u\parallel \le \parallel {z}_{n}-{x}_{n}\parallel +\parallel {x}_{n}-{Q}_{S}u\parallel \to 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{as}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}n\to \infty ,$
(3.19)

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

Finally, in this article we give a method to solve the following problem:
(3.20)

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. 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
1. (i)

limn→∞ t n = 0; ${\sum }_{n=0}^{\infty }{t}_{n}=+\infty$;

2. (ii)

limn→∞ r n = +∞,

then the sequence {u n } defined by
${r}_{n}\sum _{i=1}^{N}{f}_{i}\left({u}_{n+1}\right)+{u}_{n+1}={t}_{n}u+\left(1-{t}_{n}\right){u}_{n},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{u}_{0}\in E,\phantom{\rule{1em}{0ex}}n\ge 0,$
(3.21)

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. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Department of Mathematics Education, Kyungnam University, Masan, Kyungnam, 631-701, Korea
(2)
College of Sciences, Thainguyen University, Thainguyen, Vietnam

## References 