Open Access

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

Received: 22 November 2010

Accepted: 16 September 2011

Published: 16 September 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.

Keywords

accretive operatorsuniformly smooth and uniformly convexBanach spacesunny nonexpansive retractionweak sequential continuousmappingregularization

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 A ( x )
(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 x0 H, this proximal point algorithm generates a sequence {x n } given by
x n + 1 = J c n A ( x n + e n ) ,
(1.2)
where J r A = ( I + r A ) 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 n = 0 e n < . 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:
x n + 1 = J c n A n ( x n ) ,
(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):
x n + 1 = J r n A ( t n u + ( 1 - t n ) x n ) .
(1.4)
The equation (1.4) can be written in the following equivalent form:
r n A ( x n + 1 ) + x n + 1 t n u + ( 1 - t n ) 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
x + y 2 1 - δ .
The function
δ E ( ε ) = inf { 1 - 2 - 1 x + y : x = y = 1 , x - y = ε }
(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
ρ E ( τ ) = sup { 2 - 1 ( x + y + x - y ) - 1 : x = 1 , y = τ } ,
(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
lim τ 0 ρ E ( τ ) τ = 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 ( τ ) = ρ E ( τ ) τ .
(2.4)
The function h E (τ)is nondecreasing. In addition, it is not difficult to show that the estimate
h E ( K τ ) L K h E ( τ ) , K > 1 , τ > 0 ,
(2.5)
is valid, where L is the Figiel's constant [810], 1 < L < 1.7. Indeed, we know that the inequality holds ([8])
ρ E ( η ) η 2 L ρ E ( ξ ) ξ 2 , η ξ > 0 .
(2.6)
It implies that
ξ h E ( η ) L η h E ( ξ ) , η ξ > 0 .
(2.7)
Taking in (2.7) η = and ξ = τ, we obtain the inequality:
τ h E ( C τ ) L C τ h E ( τ ) ,
(2.8)
which implies that (2.5) holds. Similarly, we have
ρ E ( C τ ) L C 2 ρ E ( τ ) , C > 1 , τ > 0 .
(2.9)
Definition 2.3. A mapping j from E onto E* satisfying the condition
j ( x ) = { f E * : x , f = x 2 and f = x }
(2.10)

is called the normalized duality mapping of E.

We know that
j ( x ) = 2 - 1 grad x 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 ( x ) - A ( y ) , j ( x - y ) 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
T x - T y x - y , x , y 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:
    Q G x - Q G y x - y , x , y E ;
    (2.13)
     
  3. (iii)
    a sunny retraction if for all x E and for all t [0, +∞)
    Q G ( Q G x + t ( x - Q G x ) ) = 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 : EG is a sunny nonexpansive retraction if and only if
x - Q G x , J ( ξ - Q G x ) 0 , x E , ξ G .
(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 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 - Q u , j ( z - Q u ) 0 , u C , z F i x ( T ) .
(2.16)
Definition 2.10. Let C1 and C2 be convex subsets of E. The quantity
β ( C 1 , C 2 ) = sup u C 1 inf v C 2 u - v ( = sup u C 1 d ( u , C 2 ) )
is said to be a semideviation of the set C1 from the set C2. The function
H ( C 1 , C 2 ) = max { β ( C 1 , C 2 ) , β ( C 2 , C 1 ) }

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 H ( C 1 , C 2 ) δ , and Q C 1 and Q C 2 are the sunny nonexpansive retractions onto the subsets C1and C2, respectively, then
Q C 1 x - Q C 2 x 2 1 6 R ( 2 r + d ) h E 1 6 L δ R ,
(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, y D(T), the domain of T
A x - A y , J ( x - y ) L - 1 R 2 δ E A x - A y 4 R ,
(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:
a n + 1 ( 1 - λ n ) a n + λ n β n + σ n , n 0
where {λ n }, {β n } and {σ n } satisfy the following conditions.
  1. (i)

    n = 0 λ n = ;

     
  2. (ii)

    lim supn→∞ β n ≤ 0 or n = 0 | λ n β n | < ;

     
  3. (iii)

    σ n ≥ 0 n ≥ 0 and n = 0 σ n < .

     

Then, {a n } converges to zero.

Lemma 3.3. [9]Let E be a uniformly smooth Banach space. Then, for all x, y E,
x + y 2 x 2 + 2 y , J x + c ρ E ( y ) ,
(3.2)

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

First, we consider the following problem:
Finding an element x * S = i = 1 N F i x ( T i ) ,
(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 = i = 1 N F i x ( T i ) . If the sequences {r n } (0, +∞) and {t n } (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + ;

     
  2. (ii)

    limn→∞ r n = +∞,

     
then the sequence {x n } defined by
r n i = 1 N A i ( x n + 1 ) + x n + 1 = t n u + ( 1 - t n ) x n , u , x 0 E , n 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 ( x ) = r n N r n + 1 i = 1 N T i ( x ) + 1 N r n + 1 [ t n u + ( 1 - t n ) x n ] , x E .
For every x* S, we have
r n i = 1 N A i ( x n + 1 ) , j ( x n + 1 - x * ) 0 , n 0 .
(3.5)
Therefore,
t n u + ( 1 - t n ) x n - x n + 1 , j ( x n + 1 - x * ) 0 , n 0 .
(3.6)
It gives the inequality as follows:
x n + 1 - x * 2 { t n u - x * + ( 1 - t n ) x n - x * } × x n + 1 - x * .
Consequently, we have
x n + 1 - x * t n u - x * + ( 1 - t n ) x n - x * (1) max ( u - x * , x n - x * ) (2) (3) max ( u - x * , x 0 - x * ) , n 0 . (4) (5)

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 { x n k } { x n } which converges to a limit x ̄ E .

Suppose ||x n || ≤ R and ||x*|| ≤ R with R > 0. By Lemma 3.1, we have
δ E A i ( x n + 1 ) 4 R L R 2 r n r n A i ( x n + 1 ) , j ( x n + 1 - x * ) (1) L R 2 r n r n k = 1 N A k ( x n + 1 ) , j ( x n + 1 - x * ) (2) L R 2 r n t n u + ( 1 - t n ) x n - x n + 1 . x n + 1 - x * (3) 0 , n , (4) (5)
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 x ̄ S from the demiclosedness of A i . Hence, noting the inequality (2.15), we obtain
limsup n u - Q S u , j ( x n - Q S u ) = lim k u - Q S u , j ( x n k - Q S u ) (1) = u - Q S u , j ( x ̄ - Q S u ) (2) 0 . (3) (4)
(3.7)
Next, we have
x n + 1 - Q S u 2 = - r n i = 1 N A i ( x n + 1 ) + t n u + ( 1 - t n ) x n - Q S u , J ( x n + 1 - Q S u ) (1) = - r n i = 1 N A i ( x n + 1 ) , J ( x n + 1 - Q S u ) (2) + t n u + ( 1 - t n ) x n - Q S u , J ( x n + 1 - Q S u ) (3) t n ( u - Q S u ) + ( 1 - t n ) ( x n - Q S u ) , J ( x n + 1 - Q S u ) (4) 1 2 { t n ( u - Q S u ) + ( 1 - t n ) ( x n - Q S u ) 2 + x n + 1 - Q S u 2 } . (5) (6) 
By the Lemma 3.3 and the above inequality, we conclude that
| | x n + 1 - Q S u 2 | | t n ( u - Q S u ) + ( 1 - t n ) ( x n - Q S u ) 2 (1) ( 1 - t n ) 2 x n - Q S u 2 + 2 t n ( 1 - t n ) u - Q S u , j ( x n - Q S u ) (2) + c ρ E ( t n u - Q S u ) . (3) (4) 
Consequently, we have
x n + 1 - Q S u 2 ( 1 - t n ) x n - Q S u 2 + t n β n ,
(3.8)
where
β n = 2 ( 1 - t n ) u - Q S u , j ( x n - Q S u ) + c ρ E ( t n u - Q S u ) t n .

Since E is a uniformly smooth Banach space, ρ E ( t n u - Q S u ) t n 0 , n . 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:
Finding an element  x * S = i = 1 N F i x ( T i ) ,
(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 i = 1 N B i ( x n + 1 ) + x n + 1 = t n u + ( 1 - t n ) x n , u , x 0 E , n 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 = i = 1 N F i x ( T i ) . If the sequences {r n } (0, +∞) and {t n } (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + ;

     
  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 = i = 1 N F i x ( T i Q C i ) 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 E , n = 1, 2, 3, ... such that
H ( C i n , C i ) δ n , i = 1 , 2 , , 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 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 C i m , ||x - y|| ≤ δ, then
T i x - T i m y g ( max { x , y } ) ξ ( δ ) .
(3.11)
We establish the convergence and stability of the regularization method (3.10) in the form:
r n i = 1 N B i n ( z n + 1 ) + z n + 1 = t n u + ( 1 - t n ) z n , u , z 0 E , n 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 = i = 1 N F i x ( T i ) . 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; n = 0 t n = + ;

     
  2. (ii)

    limn→∞ r n = +∞;

     
  3. (iii)

    n = 0 r n ξ ( a h E ( δ n ) ) < + 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, 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
r n i = 1 N B i n ( z n + 1 ) - B i n ( x n + 1 ) , j ( z n + 1 - x n + 1 ) + r n i = 1 N B i n ( x n + 1 ) - B i ( x n + 1 ) , j ( z n + 1 - x n + 1 ) + z n + 1 - x n + 1 2 = ( 1 - t n ) z n - x n , j ( z n + 1 - x n + 1 ) .
(3.13)
By the accretivity of i = 1 N B i n and the equation (3.13), we deduce
z n + 1 - x n + 1 ( 1 - t n ) z n - x n + r n i = 1 N B i n ( x n + 1 ) - B i ( x n + 1 ) .
(3.14)
For each i {1, 2, ..., N},
B i n ( x n + 1 ) - B i ( x n + 1 ) = T i n Q C i n x n + 1 - T i Q C i x n + 1 .
(3.15)
Since {x n } is bounded and H ( C i , C i n ) δ n , there exist constants K1,i> 0 and K2,i> 1 such that
Q C i n x n + 1 - Q C i x n + 1 K 1 , i h E ( K 2 , i δ n ) K 1 , i K 2 , i L h E ( δ n ) .
(3.16)
By the condition (P2),
T i n Q C i n x n + 1 - T i Q C i x n + 1 g ( M i ) ξ ( K 1 , i K 2 , i L h E ( δ n ) ) ,
(3.17)

where M i = max { sup Q C i n x n + 1 , sup Q C i x n + 1 } < + .

From (3.14), (3.15) and (3.17), we obtain
z n + 1 - x n + 1 ( 1 - t n ) z n - x n + N g ( M ) r n ξ ( γ 1 , 2 h E ( δ n ) ) ,
(3.18)

where M = max{M1, M2, ..., M N } < +∞ and γ 1 , 2 = max i = 1 , 2 , , N { K 1 , i K 2 , i L } .

By the above assumption and Lemma 3.2, we conclude that ||z n - x n || → 0. In addition, by Theorem 3.6,
z n - Q S u z n - x n + x n - Q S u 0 , as n ,
(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:
Finding an element  x * S = i = 1 N F i x ( T i ) ,
(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 S = i = 1 N F i x ( T i ) . If the sequences {r n } (0, +∞) and {t n } (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + ;

     
  2. (ii)

    limn→∞ r n = +∞,

     
then the sequence {u n } defined by
r n i = 1 N f i ( u n + 1 ) + u n + 1 = t n u + ( 1 - t n ) u n , u , u 0 E , n 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 = i = 1 N F i x ( T i ) = i = 1 N F i x ( f i ) . 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
(2)
College of Sciences, Thainguyen University

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© Kim and Tuyen; licensee Springer. 2011

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