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
x n + 1 = J c n A ( x n + e n ) , Open image in new window
(1.2)
where J r A = ( I + r A ) 1 Open image in new window, ∀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 < Open image in new window. 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 ) , Open image in new window
(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 ) . Open image in new window
(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 . Open image in new window
(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 . Open image in new window
The function
δ E ( ε ) = inf { 1 - 2 - 1 x + y : x = y = 1 , x - y = ε } Open image in new window
(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 } , Open image in new window
(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 . Open image in new window
(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 . Open image in new window
(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 , Open image in new window
(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])
ρ E ( η ) η 2 L ρ E ( ξ ) ξ 2 , η ξ > 0 . Open image in new window
(2.6)
It implies that
ξ h E ( η ) L η h E ( ξ ) , η ξ > 0 . Open image in new window
(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 ) , Open image in new window
(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 . Open image in new window
(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 } Open image in new window
(2.10)

is called the normalized duality mapping of E.

We know that
j ( x ) = 2 - 1 grad x 2 . Open image in new window

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
A ( x ) - A ( y ) , j ( x - y ) 0 . Open image in new window
(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 . Open image in new window
(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 Open image in new window;

     
  2. (ii)
    a nonexpansive retraction if it also satisfies the inequality:
    Q G x - Q G y x - y , x , y E ; Open image in new window
    (2.13)
     
  3. (iii)
    a sunny retraction if for all xE and for all t ∈ [0, +∞)
    Q G ( Q G x + t ( x - Q G x ) ) = Q G x . Open image in new window
    (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 . Open image in new window
(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:
u - Q u , j ( z - Q u ) 0 , u C , z F i x ( T ) . Open image in new window
(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 ) ) Open image in new window
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 ) } Open image in new window

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 Open image in new window, and Q C 1 Open image in new windowand Q C 2 Open image in new windoware 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 , Open image in new window
(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
A x - A y , J ( x - y ) L - 1 R 2 δ E A x - A y 4 R , Open image in new window
(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 Open image in new window
where {λ n }, {β n } and {σ n } satisfy the following conditions.
  1. (i)

    n = 0 λ n = Open image in new window;

     
  2. (ii)

    lim supn→∞ β n ≤ 0 or n = 0 | λ n β n | < Open image in new window;

     
  3. (iii)

    σ n ≥ 0 ∀n ≥ 0 and n = 0 σ n < Open image in new window.

     

Then, {a n } converges to zero.

Lemma 3.3. [9]Let E be a uniformly smooth Banach space. Then, for all x, yE,
x + y 2 x 2 + 2 y , J x + c ρ E ( y ) , Open image in new window
(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 ) , Open image in new window
(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 ) Open image in new window. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + Open image in new window;

     
  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 Open image in new window
(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 . Open image in new window
For every x* ∈ S, we have
r n i = 1 N A i ( x n + 1 ) , j ( x n + 1 - x * ) 0 , n 0 . Open image in new window
(3.5)
Therefore,
t n u + ( 1 - t n ) x n - x n + 1 , j ( x n + 1 - x * ) 0 , n 0 . Open image in new window
(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 * . Open image in new window
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) Open image in new window

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 } Open image in new window which converges to a limit x ̄ E Open image in new window.

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) Open image in new window
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 Open image in new window 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) Open image in new window
(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)  Open image in new window
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)  Open image in new window
Consequently, we have
x n + 1 - Q S u 2 ( 1 - t n ) x n - Q S u 2 + t n β n , Open image in new window
(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 . Open image in new window

Since E is a uniformly smooth Banach space, ρ E ( t n u - Q S u ) t n 0 , n Open image in new window. 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 ) , Open image in new window
(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 , Open image in new window
(3.10)

where B i = I - T i Q C i Open image in new window, i = 1, 2, ..., N and Q C i Open image in new window 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 ) Open image in new window. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + Open image in new window;

     
  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 ) Open image in new window 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 Open image in new window, n = 1, 2, 3, ... such that
H ( C i n , C i ) δ n , i = 1 , 2 , , N , Open image in new window

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

(P2) For each set C i n Open image in new window, there is a nonexpansive self-mapping T i n : C i n C i n Open image in new window, 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 , y C i m Open image in new window, ||x - y|| ≤ δ, then
T i x - T i m y g ( max { x , y } ) ξ ( δ ) . Open image in new window
(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 , Open image in new window
(3.12)

where B i n = I - T i n Q C i n Open image in new window, i = 1, 2, ..., N and Q C i n Open image in new window is a sunny nonexpansive retraction from E onto C i n Open image in new window, 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 ) Open image in new window. 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 = + Open image in new window;

     
  2. (ii)

    limn→∞ r n = +∞;

     
  3. (iii)

    n = 0 r n ξ ( a h E ( δ n ) ) < + Open image in new window 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 Open image in new window 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
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 ) . Open image in new window
(3.13)
By the accretivity of i = 1 N B i n Open image in new window 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 ) . Open image in new window
(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 . Open image in new window
(3.15)
Since {x n } is bounded and H ( C i , C i n ) δ n Open image in new window, 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 ) . Open image in new window
(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 ) ) , Open image in new window
(3.17)

where M i = max { sup Q C i n x n + 1 , sup Q C i x n + 1 } < + Open image in new window.

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 ) ) , Open image in new window
(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 } Open image in new window.

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 , Open image in new window
(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 ) , Open image in new window
(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 ) Open image in new window. If the sequences {r n } ⊂ (0, +∞) and {t n } ⊂ (0, 1) satisfy
  1. (i)

    limn→∞ t n = 0; n = 0 t n = + Open image in new window;

     
  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 , Open image in new window
(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 Open image in new window, 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 ) Open image in new window. 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.MathSciNetGoogle Scholar
  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. 10.1080/02331939608844217MathSciNetCrossRefGoogle Scholar
  6. 6.
    Xu H-K: A regularization method for the proximal point algorithm. J Glob Optim 2006, 36: 115–125. 10.1007/s10898-006-9002-7CrossRefGoogle Scholar
  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/s10898-008-9279-9MathSciNetCrossRefGoogle Scholar
  8. 8.
    Figiel T: On the moduli of convexity and smoothness. Stud Math 1976, 56: 121–155.MathSciNetGoogle Scholar
  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.2006.05.063MathSciNetCrossRefGoogle Scholar
  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-XMathSciNetCrossRefGoogle Scholar
  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)90323-6MathSciNetCrossRefGoogle Scholar
  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.2005.04.082MathSciNetCrossRefGoogle Scholar
  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.2006.11.007MathSciNetCrossRefGoogle Scholar

Copyright information

© Kim and Tuyen; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics EducationKyungnam UniversityMasanKorea
  2. 2.College of SciencesThainguyen UniversityThainguyenVietnam

Personalised recommendations