ABSTRACT

The purpose of this article is to discuss a modified Halpern-type iteration algorithm for a countable family of uniformly totally quasi--asymptotically nonexpansive multi-valued mappings and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern-type iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Applied Mathematics and Computation, 218, 6489-6497).

1. Introduction

Throughout this paper, we denote the strong convergence and weak convergence of the sequence by and, respectively. We denote by N and R the sets of positive integers and real numbers, respectively. Let be a nonempty closed subset of a real Banach space. A mapping is said to be nonexpansive if, for all. Let and denote the family of nonempty subsets and nonempty bounded closed subsets of, respectively.

Let be a real Banach space with dual. We denote by the normalized duality mapping from to which is defined by

, where

and denotes the generalized duality pairing. The Hausdorff metric on is defined by

, for

, where. The multi-valued mapping is called nonexpansive if for all. An element is called a fixed point of

if. The set of fixed points of is represented by. In the sequel, denote. A Banach space is said to be strictly convex if for all

and. A Banach space is said to be uniformly convex if for any two sequences

and. The norm of Banach space is said to be Gâteaux differentiable if for each, the limit

(1.1)

exists. In this case, is said to be smooth. The norm of Banach space is said to be Fréchet differentiable, if for each, the limit (1.1) is attained uniformly for and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for. In this case, is said to be uniformly smooth.

The following basic properties for Banach space X and for the normalized duality mapping can be found in Cioranescu [1].

(1) is uniformly convex if and only if is uniformly smooth.

(2) If is smooth, then is single-valued and norm-to-weak^* continuous.

(3) If is reflexive, then is onto.

(4) If is strictly convex, then for all.

(5) If has a Fréchet differentiable norm, then is norm-to-norm continuous.

(6) If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of.

(7) Each uniformly convex Banach space has the Kadec-Klee property, i.e., for any sequence, if and, then.

In 1953, Mann [2] introduced the following iterative sequence,

where the initial guess is arbitrary and is a real sequence in. It is known that under appropriate settings the sequence converges weakly to a fixed point of. However, even in a Hilbert space, Mann iteration may fail to converge strongly [3]. Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [4] proposed the following so-called Halpern iteration,

where are arbitrary given and is a real sequence in. Another approach was proposed by Nakajo and Takahashi [5]. They generated a sequence as follows,

(1.2)

where is a real sequence in and denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi--nonexpansive mappings are introduced by Aoyama et al. [6], Chang et al. [7,8], Chidume et al. [9], Matsushita et al. [10-12], Qin et al. [13], Song et al. [14], Wang et al. [15] and others.

Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a countable family of uniformly totally quasi--asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces and some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in [7].

2. Preliminaries

In the sequel, we assume that is a smooth, strictly convex, and reflexive Banach space and is a nonempty closed convex subset of. In the sequel, we always use to denote the Lyapunov bifunction defined by

. (2.1)

It is obvious from the definition of the function that

(2.2)

(2.3)

and

(2.4)

for all and.

Following Alber [16], the generalized projection is defined by

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

Remark 2.1 (see [17]) Let be the generalized projection from a smooth, reflexive and strictly convex Banach space onto a nonempty closed convex subset of, then is a closed and quasi--nonexpansive from onto.

Lemma 2.1 (see [16]) Let be a smooth, strictly convex and reflexive Banach space and be a nonempty closed convex subset of. Then the following conclusions hold(a) if and only if.

(b).

(c) If and, then if and only if.

Lemma 2.2 (see [7]) Let be a real uniformly smooth and strictly convex Banach space with KadecKlee property, and be a nonempty closed convex subset of. Let and be two sequences in such that and where

is the function defined by (1.2), then.

Definition 2.1 A point is said to be an asymptotic fixed point of multi-valued mapping

, if there exists a sequence such that and. Denote the set of all asymptotic fixed points of by.

Definition 2.2

(1) A multi-valued mapping is said to be relatively nonexpansive, if,

, and

.

(2) A multi-valued mapping is said to be closed, if for any sequence with and, then

.

Remark 2.2 If is a real Hilbert space, then and is the metric projection of onto.

Next, We present an example of relatively nonexpansive multi-valued mapping.

Example 2.1 (see [18]) Let be a smooth, strictly convex and reflexive Banach space, be a nonempty closed and convex subset of and be a bifunction satisfying the conditions:

(A1);

(A2);

(A3) for each,

;

(A4) for each given, the function is convex and lower semicontinuous.

The “so-called” equilibrium problem for f is to find a such that. The set of its solutions is denoted by.

Let and define a multi-valued mapping as follows,

(2.5)

then (1) is single-valued, and so; (2) is a relatively nonexpansive mapping, therefore, it is a closed quasi--nonexpansive mapping; (3) .

Definition 2.3

(1) A multi-valued mapping is said to be quasi--nonexpansive, if, and.

(2) A multi-valued mapping is said to be quasi--asymptotically nonexpansive, if and there exists a real sequence such that

(2.6)

(3) A multi-valued mapping is said to be totally quasi--asymptotically nonexpansive, if and there exist nonnegative real sequences, with (as) and a strictly increasing continuous function with such that

(2.7)

Remark 2.3 From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi- -nonexpansive multi-valued mapping, and a quasi-- nonexpansive multi-valued mapping is a quasi--asymptotically nonexpansive multi-valued mapping, and a quasi--asymptotically nonexpansive multi-valued mapping is a total quasi--asymptotically nonexpansive multi-valued mapping, but the converse is not true.

Lemma 2.3 Let and be as in Lemma 2.2. be a closed and totally quasi--asymptotically nonexpansive multi-valued mapping with nonnegative real sequences and a strictly increasing continuous function with ,if (as) and, then is a closed and convex subset of.

Proof. Let be a sequence in, such that. Since is totally quasi--asymptotically nonexpansive multi-valued mapping, we have

for all and for all. Therefore,

By Lemma 2.1(a), we obtain. Hence,. So, we have. This implies is closed.

Let and, and put . Next we prove that. Indeed, in view of the definition of, letting, we have

(2.8)

Since

(2.9)

Substituting (2.8) into (2.9) and simplifying it, we have

By Lemma 2.2, we have. This implies that. Since is closed, we have , i.e.,. This completes the proof of Lemma 2.3.                                   □

Definition 2.4 A mapping is said to be uniformly -Lipschitz continuous, if there exists a constant such that, where.

Definition 2.5

(1) A countable family of mappings is said to be uniformly quasi--nonexpansive, if , and

.

(2) A countable family of mappings is said to be uniformly quasi--asymptotically nonexpansive, if, and there exists a real sequence such that,

(2.10)

(3) A countable family of mappings is said to be totally uniformly quasi--asymptotically nonexpansive multi-valued, if and there exists nonnegative real sequences with (as) and a strictly increasing and continuous function with such that

(2.11)

Remark 2.4 From the definitions, it is obvious that a countable family of uniformly quasi--nonexpansive multi-valued mappings is a countable family of uniformly quasi--asymptotically nonexpansive multi-valued mappings, and a countable family of uniformly quasi--asymptotically nonexpansive multi-valued mappings is a countable family of totally uniformly quasi--asymptotically multi-valued mappings, but the converse is not true.

3. Main Results

Theorem 3.1 Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, D be a nonempty closed convex subset of X, be a closed and uniformly - Lipschitz continuous and a countable family of uniformly totally quasi--asymptotically nonexpansive multi-valued mappings with nonnegative real sequences, (as) and a strictly increasing continuous function with satisfying condition (2.11). Let be a sequence in such that. If is the sequence generated by

(3.1)

where, is the fixed point set of, and is the generalized projection of onto.

If and is bounded and

, then.

Proof. (I) First, we prove that F and are closed and convex subsets in. In fact, it follows from Lemma 2.3 that is a closed and convex subsets in D. therefore F is closed and convex subsets in D. Again by the assumption, is closed and convex. Suppose that is closed and convex for some. In view of the definition of, we have

This shows that is closed and convex. The conclusions are proved.

(II) Next, we prove that, for all.

In fact, it is obvious that. Suppose that. Hence for any, by (2.4), we have

(3.2)

Therefore we have

(3.3)

This shows that and so. The conclusions are proved.

(III) Now we prove that converges strongly to some point.

In fact, since, from Lemma 2.1(c), we have Again since , we have,. It follows from Lemma 2.1(b) that for each and for each,

(3.4)

Therefore, is bounded, and so is. Since and, we have.

This implies that is nondecreasing. Hence

exists. Since X is reflexive, there exists a subsequence such that (some point in). Since is closed and convex and. This implies that is weakly closed and for each. In view of

, we have Since the norm is weakly lower semi-continuous, we have

and so

This shows that and we have. Since, by virtue of KadecKlee property of, we obtain that Since is convergent, this together with

shows that

. If there exists some subsequence such that, then from Lemma 2.1, we have

i.e., and hence

(3.5)

By the way, from (3.4), it is easy to see that

(3.6)

(IV) Now we prove that.

In fact, since, from (3.1), (3.4) and (3.5), we have

(3.7)

Since, it follows from (3.6) and Lemma 2.2 that

(3.8)

Since is bounded and is a countable family of uniformly totally quasi--asymptotically nonexpansive multi-valued mappings, is bounded. In view of, from (3.1), we have

(3.9)

Since, this implies. From Remark 2.1, it yields that

(3.10)

Again since

(3.11)

this together with (3.9) and the Kadec-Klee-property of shows that

(3.12)

On the other hand, by the assumptions that is -Lipschitz continuous for each, we have

(3.13)

From (3.12) and, we have that

. In view of the closeness of, it yields that, which implies that.

(V) Finally we prove that and so .

Let. Since, we have . This implies that

(3.14)

which yields that. Therefore, . The proof of Theorem 3.1 is completed.

By Remark 2.4, the following corollaries are obtained. □

Corollary 3.1 Let X and be as in Theorem 3.1, and a countable family of mappings be a closed and uniformly -Lipschitz continuous a relatively nonexpansive multi-valued mappings. Let in with. Let be the sequence generated by

(3.15)

where is the set of fixed points of, and is the generalized projection of onto, If

and F is bounded, then converges strongly to.

Corollary 3.2 Let and be as in Theorem 3.1, and a countable family of mappings be a closed and uniformly -Lipschitz continuous quasi-phi-asymptotically nonexpansive multivalued mappings with nonnegative real sequences

and satisfying condition (2.1). Let be a sequence in and satisfy

. If is the sequence generated by

(3.16)

where is the set of fixed points of, and is the generalized projection of onto, and

If and F is bounded, then

converges strongly to.

4. Application

We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.

Theorem 4.1 Let, and be the same as in Theorem 3.1. Let be a bifunction satisfying conditions (A1)-(A4) as given in Example 2.6.

Let be a mapping defined by (2.5), i.e.,

Let be the sequence generated by

(4.1)

If, then converges strongly to

which is a common solution of the system of equilibrium problems for.

Proof. In Example 2.6, we have pointed out that, and is a closed quasi--nonexpansive mapping. Hence (4.1) can be rewritten as follows:

Therefore the conclusion of Theorem 4.6 can be obtained from Corollary 3.2.

REFERENCES