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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 Applications volume 2011, Article number: 52 (2011)
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.
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:
involving the accretive mapping A.
One popular method of solving equation 0 ∈ 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
where ${J}_{r}^{A}=(I+rA{)}^{1}$, ∀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 infinitedimensional 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:
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 [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):
The equation (1.4) can be written in the following equivalent form:
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 selfmappings 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 selfmapping or nonselfmapping 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
The function
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
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
It is well known that every uniformly convex and uniformly smooth Banach space is reflexive. In what follows, we denote
The function h_{ E } (τ)is nondecreasing. In addition, it is not difficult to show that the estimate
is valid, where L is the Figiel's constant [8–10], 1 < L < 1.7. Indeed, we know that the inequality holds ([8])
It implies that
Taking in (2.7) η = Cτ and ξ = τ, we obtain the inequality:
which implies that (2.5) holds. Similarly, we have
Definition 2.3. A mapping j from E onto E* satisfying the condition
is called the normalized duality mapping of E.
We know that
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 maccretive 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 maccretive 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 maccretive 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:
$$\parallel {Q}_{G}x{Q}_{G}y\parallel \le \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\forall x,y\in E;$$(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
is said to be a semideviation of the set 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(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. [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 $.
Then, {a_{ n } } converges to zero.
Lemma 3.3. [9]Let E be a uniformly smooth Banach space. Then, for all x, y ∈ E,
where c = 48 max(L, x, y).
First, we consider the following problem:
where Fix(T_{ i } ) is the set of fixed points of the nonexpansive mapping T_{ i } : E → E, i = 1, 2, ..., N.
Theorem 3.4. Suppose that E is a uniformly convex and uniformly smooth Banach space which has a weakly sequentially continuous normalized duality mapping j from E to E*. Let T_{ i } : E → E, i = 1, 2, ..., N be nonexpansive mappings with$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
For every x* ∈ S, we have
Therefore,
It gives the inequality as follows:
Consequently, 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 $\stackrel{\u0304}{x}\in E$.
Suppose x_{ n }  ≤ R and x* ≤ R with R > 0. By Lemma 3.1, we have
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 } (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
Next, we have
By the Lemma 3.3 and the above inequality, we conclude that
Consequently, we have
where
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. □
Now, we will give a method to solve more generally following problem:
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.
We consider the iterative sequence {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:
(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
where {δ_{ n } } is a sequence of positive numbers.
(P2) For each set ${C}_{i}^{n}$, there is a nonexpansive selfmapping ${T}_{i}^{n}:{C}_{i}^{n}\to {C}_{i}^{n}$, 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
We establish the convergence and stability of the regularization method (3.10) in the form:
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 maccretive 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
By the accretivity of ${\sum}_{i=1}^{N}{B}_{i}^{n}$ and the equation (3.13), we deduce
For each i ∈ {1, 2, ..., N},
Since {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
By the condition (P2),
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
where M = max{M_{1}, M_{2}, ..., M_{ N } } < +∞ and ${\gamma}_{1,2}=\underset{i=1,2,\dots ,N}{max}\left\{{K}_{1,i}\sqrt{{K}_{2,i}L}\right\}$.
By the above assumption and Lemma 3.2, we conclude that z_{ n }  x_{ n }  → 0. In addition, by Theorem 3.6,
which implies that {z_{ n } } converges strongly to Q_{ S }u. □
Finally, in this article we give a method to solve the following problem:
where T_{ i } : C_{ i } → E, i = 1, 2, ..., N is nonexpansive nonselfmapping 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. □
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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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Authors' contributions
JKK conceived the study and participated in its design and coordination. JKK suggested many good ideas that are useful for achievement this paper and made the revision. TMT and JKK prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Tuyen, T.M. Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2011, 52 (2011). https://doi.org/10.1186/16871812201152
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Keywords
 accretive operators
 uniformly smooth and uniformly convex
 Banach space
 sunny nonexpansive retraction
 weak sequential continuous
 mapping
 regularization