Open Access

A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space

Fixed Point Theory and Applications20112011:17

https://doi.org/10.1186/1687-1812-2011-17

Received: 7 January 2011

Accepted: 21 July 2011

Published: 21 July 2011

Abstract

In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inverse-strongly monotone mappings and a system of bifunctions satisfying certain conditions, and the common fixed-point problem (P2) for a family of uniformly quasi-ϕ-asymptotically nonexpansive and locally uniformly Lipschitz continuous or uniformly Hölder continuous mappings are proposed. A new iterative sequence is constructed by using the generalized projection and hybrid method, and a strong convergence theorem is proved on approximating a common solution of (P1) and (P2) in Banach space.

2000 MSC: 26B25, 40A05

Keywords

Common solution Equilibrium problem Fixed-point problem Iterative sequence Strong convergence

1. Introduction

Recently, common solution problems (i.e., to find a common element of the set of solutions of equilibrium problems and/or the set of fixed points of mappings and/or the set of solutions of variational inequalities) with their applications have been discussed. Some authors such as in references [17] presented various iterative schemes and showed some strong or weak convergence theorems on common solution problems in Hilbert spaces. In 2008-2009, Takahashi and Zembayashi [8, 9] introduced several iterative sequences on finding a common solution of an equilibrium problem and a fixed-point problem for a relatively nonexpansive mapping, and established some strong or weak convergence theorems. In 2010, Chang et al. [10] discussed the common solution of a generalized equilibrium problem and a common fixed-point problem for two relatively nonexpansive mappings, and established a strong convergence theorem on the common solution problem. The frameworks of spaces in [810] are the uniformly smooth and uniformly convex Banach spaces. Chang et al. [11] established a strong convergence theorem on solving the common fixed-point problem for a family of uniformly quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Some other problems such as optimization problems (e.g. see [1, 4, 6]) and common zero-point problems (e.g. see [10]) are closely related to common solution problems.

Throughout this paper, unless other stated, and are denoted by the set of the real numbers and the set {1, 2,..., N}, respectively, where N is any given positive integer. Let E be a real Banach space with the norm || · ||, E* be the dual of E, and 〈·,·〉 be the pairing between E and E*. Suppose that C is a nonempty closed convex subset of E.

Let be N mappings and be N bifunctions. For each , the generalized equilibrium problem for f k and A k is to seek such that
(1.1)

The common solution problem (P1) of generalized equilibrium problems for and is to seek an element in , where and G(k) is the set of solutions of (1.1). We write G instead of in the case of N = 1.

Let be a family of mappings. The common fixed-point problem (P2) for is to seek an element in , where and F (S i ) is the set of fixed points of S i .

Motivated by the works in [811], in this paper we will produce a new iterative sequence approximating a common solution of (P1) and (P2) (i.e., some point belonging to ), and show a strong convergence theorem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, where in (P2) is a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings and for each i ≥ 1, S i is locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ i .

2. Preliminaries

Let E be a real Banach space, and {x n } be a sequence in E. We denote by x n x and x n x the strong convergence and weak convergence of {x n }, respectively. The normalized duality mapping J : E → 2E*is defined by

By the Hahn-Banach theorem, Jx for each x E.

A Banach space E is said to be strictly convex if for all x, y U = {u E : ||u|| = 1} with xy; to be uniformly convex if for each ε (0, 2], there exists γ > 0 such that for all x, y U with ||x - y|| ≥ ε; to be smooth if the limit
(2.1)

exists for every x, y U; to be uniformly smooth if the limit (2.1) exists uniformly for all x, y U.

Remark 2.1. The basic properties below hold (see [12]).
  1. (i)

    If E is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

     
  2. (ii)

    If E is a strictly convex reflexive Banach space, then J-1 is hemicontinuous, that is, J-1 is norm-to-weak*-continuous.

     
  3. (iii)

    If E is a smooth and strictly convex reflexive Banach space, then J is single-valued, one-to-one and onto.

     
  4. (iv)

    Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {x n } E, if x n x E and ||x n || → ||x||, then x n x.

     
  5. (v)

    A Banach space E is uniformly smooth if and only if E* is uniformly convex.

     
  6. (vi)
    A Banach space E is strictly convex if and only if J is strictly monotone, that is,
     
  7. (vii)

    Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.

     
Now let E be a smooth and strictly convex reflexive Banach space. As Alber [13] and Kamimura and Takahashi [14] did, the Lyapunov functional ϕ : E × E+ is defined by
It follows from [15] that ϕ(x, y) = 0 if and only if x = y, and that
(2.2)
Further suppose that C is a nonempty closed convex subset of E. The generalized projection (see [13]) Π C : EC is defined by for each x E,
A mapping A : CE* is said to be δ-inverse-strongly monotone, if there exists a constant δ > 0 such that
A mapping S : CC is said to be closed if for each {x n } C, x n x and Sx n y imply Sx = y; to be quasi-ϕ-asymptotically nonexpansive (see [16]) if F(S) ≠ , and there exists a sequence {l n } [1, ∞) with l n → 1 such that

It is easy to see that if A : CE* is δ-inverse-strongly monotone, then A is -Lipschitz continuous. The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of relatively nonexpansive mappings (see [17]) as a subclass.

Definition 2.1 (see [11]). Let be a sequence of mappings. is said to be a family of uniformly quasi- ϕ -asymptotically nonexpansive mappings, if and there exists a sequence {l n } [1, ∞) with l n → 1 such that for each i ≥ 1,

Now we introduce the following concepts.

Definition 2.2. A mapping S : CC is said
  1. (1)
    to be locally uniformly Lipschitz continuous if for any bounded subset D in C, there exists a constant L D > 0 such that
     
  2. (2)
    to be uniformly Hölder continuous with order Θ (Θ > 0) if there exists a constant L > 0 such that
     

Remark 2.2. It is easy to see that any uniformly Lipschitz continuous mapping (see [11]) is locally uniformly Lipschitz continuous, and is also uniformly Hölder continuous with order Θ = 1. However, the converse is not true.

Example 2.1. Suppose that S : is defined by

Then S is locally uniformly Lipschitz continuous. In fact, for any bounded subset D in , setting M = 1 + sup{|x| : x D}, we have |S n x - S n y| ≤ 2M |x - y|, x, y D, n ≥ 1. But S fails to be uniformly Lipschitz continuous.

Example 2.2. Suppose that S : - is defined by

S is uniformly Hölder continuous with order , since , x, y , n ≥ 1. But S fails to be uniformly Lipschitz continuous.

Lemma 2.1 (see [13, 14]). If C is a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, then
  1. (1)

    ϕ(x, Π C (y)) + ϕ C (y), y) ≥ ϕ(x, y), x C, y E;

     
  2. (2)
    for × E and u C, one has
     

Lemma 2.2. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, {x n } and{y n } be two sequences of E, and . If and ϕ(x n , y n ) → 0, then .

Proof. We complete this proof by two steps.

Step 1. Show that there exists a subsequence of {y n } such that .

In fact, since ϕ(x n , y n ) → 0, by (2.2) we have ||x n || - ||y n || → 0. It follows from that
(2.3)
and so
(2.4)
Then {Jy n } is bounded in E*. It follows from Remark 2.1(v) and (vii) that E* is reflexive. Hence there exist a point f0 E* and a subsequence of {Jy n } such that
(2.5)
It follows from Remark 2.1(vii) and (iii) that there exists a point x E such that Jx = f0. By the definition of ϕ, we obtain
By weak lower semicontinuity of norm || · ||, we have

which implies that and . It follows from Remark 2.1(iv) and (v) that E* has the Kadec-Klee property, and so by (2.4) and (2.5). By Remark 2.1(vii) and (ii), we have , which implies that by (2.3) and the Kadec-Klee property of E.

Step 2. Show that .

In fact, suppose that . For some given number ε0 > 0, there exists a positive integer sequence {n k } with n1 < n2 < · · · < n k < · · ·, such that
(2.6)

Replacing {y n } by in Step 1, there exists a subsequence of such that , which contradicts (2.6). □

Lemma 2.3. Let C be a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, and let A : CE* be a δ-inverse-strongly monotone mapping and f : C × C be a bifunction satisfying the following conditions

(B1) f(z, z) = 0, z C;

(B2) ;

(B3) for any z C, the function y α f(z, y) is convex and lower semicontinuous;

(B4) for some β ≥ 0 with βδ,
Then the following conclusions hold:
  1. (1)
    For any r > 0 and u E, there exists a unique point z C such that
    (2.7)
     
  2. (2)
    For any given r > 0, define a mapping K r : EC as follows: u E,
     
We have (i) F(K r ) = G and G is closed convex in C, where

(ii) ϕ(z, K r u) + ϕ(K r u, u) ≤ ϕ(z, u), z F(K r ).

(3) For each n ≥ 1, r n > a > 0 and u n C with , we have
Proof. (1) We consider the bifunction instead of f. It follows from the proof of Lemma 2.5 in [10] that satisfies (B1)-(B3). Since A is δ-inverse-strongly monotone, by (B4), we have
(2.8)
which implies is monotone. By Blum amd Oettli [18], for any r > 0 and u E, there exists z C such that (2.7) holds. Next we show that (2.7) has a unique solution. If for any given r > 0 and u E, z1 and z2 are two solutions of (2.7), then
and
Adding these two inequalities, we have
It follows from (2.8) that

which implies that z1 = z2 by Remark 2.1(vi).

(2) Since satisfies (B1)-(B3) and is monotone, the conclusion (2) follows from Lemmas 2.8 and 2.9 in [9].

(3) Since
we have
(2.9)
by the monotonicity of . It follows from . r n > a > 0 and Remark 2.1(i) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting n → ∞ in (2.9), we have , y C. For any t (0, 1] and y C, setting , we have y t C and , which together with (B1) implies that

Thus f(y t , y) + 〈y - y t , Ay t 〉 ≥ 0, y C, t (0, 1]. Letting t ↓ 0, since z α f(z, y) + 〈y - z, Az〉 satisfies (B2), we have , y C.

Remark 2.3. If β = 0 in (B4), that is, f is monotone, then the conclusions (1) and (2) in Lemma 2.3 reduce to the relating results of Lemmas 2.5 and 2.6 in [10], respectively.

Next we give an example to show that there exist the mapping A and the bifunction f satisfying the conditions of Lemma 2.3. However, f is not monotone.

Example 2.3. Define A : and f : × by x and , (x, y) × , respectively. It is easy to see that A is -inverse-strongly monotone, f satisfies (B1)-(B3), and , (x, y) : × with .

Lemma 2.4 (see [12]). Let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the Kadec-Klee property, S : CC be a closed and quasi- ϕ -asymptotically nonexpansive mapping with a sequence {l n } [1, ∞), l n → 1. Then F(S) is closed convex in C.

Lemma 2.5 (see [11]). Let E be a uniformly convex Banach space, η > 0 be a positive number and B η (0) be a closed ball of E. Then, for any given sequence and for any given with , there exists a continuous, strictly increasing and convex function g : [0, 2η) → [0, ∞) with g(0) = 0 such that for any positive integers i, j with i < j,

3. Strong convergence theorem

In this section, let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.

Theorem 3.1. Suppose that

(C1) for each , the mapping A k : CE* is δ k -inverse-strongly monotone, the bifunction f k : C × C satisfies (B1)-(B3), and for some β k ≥ 0 with β k δ k ,

(C2) is a family of closed and uniformly quasi- ϕ -asymptotically nonexpansive mappings with a sequence {l n } [1, ∞), l n → 1;

(C3) for each i ≥ 1, S i is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ i i > 0), and is bounded in C.

(C4) . Take the sequence generated by

where for each , with some a > 0, , and . If , n ≥ 0 and lim infn→∞αn,0α n, i > 0, i ≥ 1, then .

Proof. We shall complete this proof by seven steps below.

Step 1. Show that , , H n and W n for all n ≥ 0 are closed convex.

In fact, is closed convex since for each i ≥ 1, F(S i ) is closed convex by (C2) and Lemma 2.4. is closed convex since for each , G(k) is closed convex by (C1) and Lemma 2.3(2)(i). H0 = C is closed convex. Since ϕ(v,uN,n) ≤ ϕ(v,x n ) + ξ n is equivalent to

we know that H n (n ≥ 0) are closed convex. Finally, W n is closed convex by its definition. Thus and are well defined.

Step 2. Show that {x n } and are bounded.

From , n ≥ 0 and Lemma 2.1(1), we have
(3.1)
which implies that {ϕ(x n , x0)} is bounded, and so is {x n } by (2.2). It follows from (C2) that for all , i ≥ 1, n ≥ 1,
Hence for all i ≥ 1, is uniformly bounded, and so is by (2.2). Obviously,
(3.2)

Step 3. Show that , n ≥ 0.

Since Banach space E is uniformly smooth, E* is uniformly convex, by Remark 2.1(v). For any given , any n ≥ 1 and any positive integer j, by (C2) and Lemma 2.5, we have
(3.3)
Put , , n ≥ 0. It follows from (3.3) and Lemma 2.3(2)(ii) that
(3.4)
which implies that if , then p H n , n ≥ 0. Hence, , n ≥ 0. By induction, now we prove that , n ≥ 0. In fact, it follows from W0 = C that . Suppose that for some m ≥ 0. By the definition of and Lemma 2.1(2), we have
and so

which shows z Wm+1, so .

Step 4. Show that there exists such that .

Without loss of generalization, we can assume that , since {x n } is bounded and E is reflexive. Moreover, it follows that , n ≥ 0 from Hn+1Wn+1 H n W n and the closeness and convexity of H n W n . Noting that
we have
by (3.1). It follows that
(3.5)
and so by . Hence,
(3.6)
by the Kadec-Klee property of E, and so
(3.7)

by Remark 2.1(i).

Step 5. Show that .

Since xn+1 C, setting u = xn+1in (3.1), we have
By (3.5),
(3.8)
By xn+1 Hn+1, (3.2) and (3.8), we have
which together with (3.6) and Lemma 2.2 implies that
(3.9)
For any j ≥ 1 and any given , it follows from (3.2)-(3.4) and (3.9) that
(3.10)
which implies that
since , i ≥ 1. We obtain
(3.11)
since g(0) = 0 and g is strictly increasing and continuous. By (3.7) and (3.11), we have and for all j ≥ 1. It follows from Remark 2.1(ii) that , which implies
(3.12)
by the uniform boundedness of and the Kadec-Klee property of E. Thus
By (C3) and (3.6), we have
Hence, for each j ≥ 1,

By (3.12) and the closeness of S j , we have for all j ≥ 1 and so .

Step 6. Show that .

In fact, it is easy to see that for each , and , the sequence {ϕ(p, uk,n)} is bounded by (3.2), (3.4) and the boundedness of {x n } and , which implies that {uk,n} is bounded in C by (2.2). Since , by (3.2), (3.3), (3.5) and (3.10), we have
It follows from Lemma 2.2 that
(3.13)
Furthermore, it follows from (3.4) and Lemma 2.3(2)(ii) that for any given ,
which implies
by Remark 2.1(i), (3.9) and (3.13). Then by (3.13) and Lemma 2.2. Similarly, we also obtain . Hence, together with (3.9) and (3.13), for each ,
(3.14)
For each , since , we have

which together with (3.14) and Lemma 2.3(3) implies that , y C. Therefore and so .

Step 7. Show that .

In fact, letting , by and , we have
It follows from (3.6) that

Hence, , and so . □

Setting N = 1, u0, n = y n and uN,n= u n in Theorem 3.1, we can obtain the following result.

Corollary 3.1 Suppose that

(D1) the mapping A : CE* is a mapping with δ -inverse-strongly monotone, the bifunction f : C × C → satisfies (B1)-(B3) and for some β > 0 with βδ,
(D2) both (C2) and (C3) hold, and Take the sequence generated by

where , for some a > 0 and . If , n ≥ 0 and lim infn→∞αn,0αn,i> 0, i ≥ 1, then . □

Furthermore, if S i = S, i ≥ 1 in Corollary 3.1, the following corollary can be obtained immediately.

Corollary 3.2. Suppose that, besides (D1),

(E1) S : CC is closed and quasi- ϕ -asymptotically nonexpansive with {l n } [1, ∞), l n → 1;

(E2) S is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ (Θ > 0), F(S) is bounded in C and F(S) ∩ G. Take the sequence generated by

where , for some a > 0 and ξ = supuF(S)(l n -1)ϕ(u, x n ) . If lim infn→∞α n (1- α n ) > 0, then . □

Declarations

Authors’ Affiliations

(1)
College of Applied Science, Beijing University of Technology
(2)
College of Mathematics, Jilin Normal University

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© Qu and Cheng; licensee Springer. 2011

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