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

Open Access
Research
Part of the following topical collections:
1. S. Park's Contribution to the Development of Fixed Point Theory and KKM Theory

## 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 xE. 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:
(1.1)

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 x0H, this proximal point algorithm generates a sequence {x n } given by
(1.2)
where , ∀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 . 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:
(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 [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):
(1.4)
The equation (1.4) can be written in the following equivalent form:
(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
The function
(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
(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
(2.3)
It is well known that every uniformly convex and uniformly smooth Banach space is reflexive. In what follows, we denote
(2.4)
The function h E (τ)is nondecreasing. In addition, it is not difficult to show that the estimate
(2.5)
is valid, where L is the Figiel's constant [8, 9, 10], 1 < L < 1.7. Indeed, we know that the inequality holds ([8])
(2.6)
It implies that
(2.7)
Taking in (2.7) η = and ξ = τ, we obtain the inequality:
(2.8)
which implies that (2.5) holds. Similarly, we have
(2.9)
Definition 2.3. A mapping j from E onto E* satisfying the condition
(2.10)

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) ⊆ EE is called accretive, if for all x, yD(A), there exists j(x - y) ∈ J (x - y) such that
(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
(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 ;

2. (ii)
a nonexpansive retraction if it also satisfies the inequality:
(2.13)

3. (iii)
a sunny retraction if for all xE and for all t ∈ [0, +∞)
(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 : EG is a sunny nonexpansive retraction if and only if
(2.15)

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 : CC be a nonexpansive mapping with a fixed point. For each uC and every t ∈ (0, 1), the unique fixed point x t C of the contraction Cxtu + (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:
(2.16)
Definition 2.10. Let C1 and C2 be convex subsets of E. The quantity
is said to be a semideviation of the set C1 from the set C2. The function

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

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

Lemma 2.11. [7]If E is a uniformly smooth Banach space, C1and C2are closed convex subsets of E such that the Hausdorff distance, and andare the sunny nonexpansive retractions onto the subsets C1and C2, respectively, then
(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. [9]If A = I - T with a nonexpansive mapping T, then for all x, yD(T), the domain of T
(3.1)

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.
1. (i)

;

2. (ii)

lim supn→∞ β n ≤ 0 or ;

3. (iii)

σ n ≥ 0 ∀n ≥ 0 and .

Then, {a n } converges to zero.

Lemma 3.3. [9]Let E be a uniformly smooth Banach space. Then, for all x, yE,
(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. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
1. (i)

limn→∞ t n = 0; ;

2. (ii)

limn→∞ r n = +∞,

then the sequence {x n } defined by
(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
For every x* ∈ S, we have
(3.5)
Therefore,
(3.6)
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 which converges to a limit .

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 (xn+1) → 0, i = 1, 2, ..., N. It is clear that from the demiclosedness of A i . Hence, noting the inequality (2.15), we obtain
(3.7)
Next, we have
By the Lemma 3.3 and the above inequality, we conclude that
Consequently, we have
(3.8)
where

Since E is a uniformly smooth Banach space, . 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
(3.10)

where , i = 1, 2, ..., N and 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. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
1. (i)

limn→∞ t n = 0; ;

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

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

(P2) For each set , there is a nonexpansive self-mapping , 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 xC i , , ||x - y|| ≤ δ, then
(3.11)
We establish the convergence and stability of the regularization method (3.10) in the form:
(3.12)

where , i = 1, 2, ..., N and is a sunny nonexpansive retraction from E onto , 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. 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; ;

2. (ii)

limn→∞ r n = +∞;

3. (iii)

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, is an m-accretive operator on E, so the equation (3.12) defines a unique element zn+1E. From the equations (3.10) and (3.12), we have
(3.13)
By the accretivity of and the equation (3.13), we deduce
(3.14)
For each i ∈ {1, 2, ..., N},
(3.15)
Since {x n } is bounded and , there exist constants K1,i> 0 and K2,i> 1 such that
(3.16)
By the condition (P2),
(3.17)

where .

From (3.14), (3.15) and (3.17), we obtain
(3.18)

where M = max{M1, M2, ..., M N } < +∞ and .

By the above assumption and Lemma 3.2, we conclude that ||z n - x n || → 0. In addition, by Theorem 3.6,
(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. [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. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
1. (i)

limn→∞ t n = 0; ;

2. (ii)

limn→∞ r n = +∞,

then the sequence {u n } defined by
(3.21)

converges strongly to Q S u, where Q S is a sunny nonexpansive retraction from E onto S and, i = 1, 2, ..., N.

Proof. By the Lemma 3.5 and Lemma 3.8, . 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.

### References

1. 1.
Rockaffelar RT: Monotone operators and proximal point algorithm. In J Control Optim. Volume 14. SIAM; 1976:887–897.Google Scholar
2. 2.
Güler O: On the convergence of the proximal point algorithm for convex minimization. In J Control Optim. Volume 29. SIAM; 1991:403–419. 10.1137/0329022Google Scholar
3. 3.
Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iteration in Hilert space. Math Program Ser A 2000, 87: 189–202.
4. 4.
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in Banach spaces. In J Control Optim. Volume 13. SIAM; 2002:938–945.Google Scholar
5. 5.
Lehdili N, Moudafi A: Combining the proximal algorithm and Tikhonov regularization. Optimization 1996, 37: 239–252. 10.1080/02331939608844217
6. 6.
Xu H-K: A regularization method for the proximal point algorithm. J Glob Optim 2006, 36: 115–125. 10.1007/s10898-006-9002-7
7. 7.
Song Y, Yang C: A note on a paper: a regularization method for the proximal point algorithm. J Glob Optim 2009, 43: 171–174. 10.1007/s10898-008-9279-9
8. 8.
Figiel T: On the moduli of convexity and smoothness. Stud Math 1976, 56: 121–155.
9. 9.
Alber Y: On the stability of iterative approximations to fixed points of nonexpansive mappings. J Math Anal Appl 2007, 328: 958–971. 10.1016/j.jmaa.2006.05.063
10. 10.
Alber Y, Ryazantseva I: Nonlinear Ill-Posed Problems of Monotone Type. Dordrecht: Springer; 2006.Google Scholar
11. 11.
Li G, Kim JK: Demiclosed principle and asymptotic behavior for nonexpansive mappings in metric spaces. Appl Math Lett 2001, 14: 645–649. 10.1016/S0893-9659(00)00207-X
12. 12.
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach space. J Math Anal Appl 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6
13. 13.
Xu H-K: Strong convergence of an iterative method for nonexpansive and accretive operators. J Math Anal Appl 2006, 314: 631–643. 10.1016/j.jmaa.2005.04.082
14. 14.
Matsushita S, Takahashi W: Strong convergence theorem for nonexpansive nonself-mappings without boundary conditions. Nonlinear Anal TMA 2008, 68: 412–419. 10.1016/j.na.2006.11.007