Open Access

Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems

Fixed Point Theory and Applications20092009:462489

https://doi.org/10.1155/2009/462489

Received: 21 October 2008

Accepted: 5 March 2009

Published: 17 March 2009

Abstract

The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung (2008) and some others in some respects.

1. Introduction and Preliminaries

Let be a real Hilbert space with inner product and norm and let be a nonempty subset of . A mapping is said to be Lipschitzian if there exists a positive constant such that
(1.1)
In this case, is also said to be -Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is -Lipschitzian with . If , then is known as a nonexpansive mapping. We denote by the set of fixed points of . There are many methods for approximating the fixed points of a nonexpansive mapping. In 1953, Mann [1] introduced the following iteration method:
(1.2)
where the initial guess element is arbitrary, and is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Qin and Su[2]. In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak convergence [3]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [4] proposed the following modification of Mann iteration method(1.2):
(1.3)

where denotes the metric projection from onto a closed convex subset of . The iterative method (1.3) is said to be hybrid method or method. In recent years, the hybrid method (1.3) has been modified by many authors for other nonlinear operators [2, 510].

In 2008, Nilsrakoo and Saejung [11] used the hybrid method to obtain a strong convergence theorem for countable Lipschitzian mappings as follows.

Theorem 1.

Let be a nonempty bounded closed convex subset of a real Hilbert space . Let be a sequence of -Lipschitzian mappings from into itself with and let be nonempty. Assume that is a sequence in with . Let be a sequence in defined as follows:
(1.4)
where
(1.5)

as . Let for any bounded subset of , and let be a mapping of into itself defined by , for all and suppose that , then converges strongly to .

Nilsrakoo and Saejung also apply the aforementioned result to obtain an applied result for equilibrium problems.

The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the Nilsrakoo and Saejung results in some respects.

Recall that given a closed convex subset of a real Hilbert space , the nearest point projection from onto assigns to each its nearest point denoted by in from to ; that is, is the unique point in with the property
(1.6)

The following lemma is well known.

Lemma 1.1.

Let be a closed convex subset of a real Hilbert space . Given and Then if and only if there holds the relation:
(1.7)

Definition 1.2.

Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence of -Lipschitzian mappings from into itself with . is said to satisfy the (SU) condition, if the following conditions hold:

(1)for any strong convergence sequence , the sequence is also strong convergent;

(2)the common fixed points set is nonempty;

(3) , where is defined by , for all , denotes the fixed points set of .

In Section 3 of this paper, we will give an important example of sequence of -Lipschitzian mappings which satisfies the (SU) condition.

Lemma 1.3.

Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence of -Lipschitzian mappings from into itself with . If satisfies the (SU) condition. Then

(1) is bounded implies is uniformly continuous on the ;

(2) implies is -Lipschitzian;

(3) implies is nonexpansive.

Proof.

Observe that, for all , we have
(1.8)

The results (1)–(3) are easy to prove.

2. Main Results

Theorem 2.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of -Lipschitzian mappings from into itself with , as . Assume satisfies the (SU) condition and is a sequence in with . Let be a sequence in defined as follows:
(2.1)
where
(2.2)

Then converges strongly to , where is the metric projection from onto closed convex subset .

Proof.

We first prove that and are closed and convex for each . From the definition of and , it is obvious that is closed and is closed and convex for each . We prove that is convex. Since is equivalent to
(2.3)
it follows that is convex. Next, we show that , for all . For any , we have
(2.4)
so that , therefore, we have , for all . Next, we show that , for all . We prove this by induction. For , we have . Suppose that , then and there exists a unique element such that . By using Lemma 1.1, we have
(2.5)
for all . In particular,
(2.6)

for all . It follows from the definition of that . By using the principle of induction, we claim that , for all . Therefore, we have , for all . Now the sequence is well defined.

It follows from the definition of that . Therefore, we have
(2.7)
for all . This implies that the is bounded. On the other hand, from , we have
(2.8)

for all . Therefore, is nondecreasing and bounded. So that exists.

Note again that , hence for any positive integer , we have which implies that . Therefore, we have
(2.9)

From this inequality, we know that is a Cauchy sequence in , so that there exists a point such that .

Since , then this together with , as , implies that . From , we get
(2.10)
as . Since is nonexpansive, therefore,
(2.11)

as . Then .

Finally, we claim that . If not, we have . There must exists a positive integer , if , then , which leads to
(2.12)

It follows that implies that This is a contradiction, hence . This completes the proof.

The following theorem directly follows from Theorem 2.1.

Theorem 2.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings from into itself. Assume satisfies the (SU) condition and is a sequence in with . Let be a sequence in defined as follows:
(2.13)

Then converges strongly to . Where is the metric projection from onto closed convex subset .

3. Application for Equilibrium and Optimization

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that
(3.1)

The set of solutions of (3.1) is denoted by . Given a mapping , let , for all . Then, if and only if , for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (3.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1216].

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A1) , for all ;

(A2) is monotone, that is, , for all ;

(A3)for each ,
(3.2)

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

We need the following lemmas for the proof of our main results.

Lemma 3.1 (see [1113]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that
(3.3)

Lemma 3.2 (see [1113]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:
(3.4)

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any ,
(3.5)

(3) ;

(4) is closed and convex.

Remark 3.3.

is also nonexpansive, for all .

Now, we prove the following lemma which is very important for the main results of this section.

Lemma 3.4.

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let be a positive real sequence such that . Then the sequence satisfies the (SU) condition.

Proof.
  1. (1)
    Let be a convergent sequence in . Let , for all , then
    (3.6)
     
(3.7)
Putting in (3.6) and in (3.7), we have
(3.8)
So, from (A2) we have
(3.9)
and hence
(3.10)
Thus, we have
(3.11)
which implies that
(3.12)
Therefore, we get
(3.13)
On the other hand, for any , from , we have
(3.14)
so that is bounded. Since , this together with (3.13) implies that the is a Cauchy sequence. Hence is convergent.
  1. (2)
    By using Lemma 3.2, we know that
    (3.15)
     
  1. (3)
    From (1) we know that, exists, for all . So, we can define a mapping from into itself by
    (3.16)
     
It is obvious that the is nonexpansive. It is easy to see that
(3.17)
On the other hand, let , we have
(3.18)
By (A2) we know
(3.19)
Since and from (A4), we have , for all . Then, for and ,
(3.20)
Therefore, we have
(3.21)
Letting and using (A3), we get
(3.22)

and hence . From the aforementioned two respects, we know that . This completes the proof.

Theorem 3.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (A1)–(A4) and . Let and be sequences generated by and
(3.23)

Assume and . Then converges strongly to , where is the metric projection from onto .

Proof.

By using Lemma 3.4, we know that satisfies the (SU) condition. Then . By using Theorem 2.2, we obtain the result of Theorem 3.5

Now, we study a kind of optimization problem by using the aforementioned results of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:

(3.24)
where is a convex and lower semicontinuous functional defined on a closed convex subset of a Hilbert space . We denoted by the set of solutions of (3.24). Let be a bifunction from to defined by . We consider the following equilibrium problem, that is to find such that
(3.25)

It is obvious that , where denotes the set of solutions of equilibrium Problem (3.25). In addition, it is easy to see that satisfies the conditions (A1)–(A4) in Section 2. Therefore, from Theorem 3.5, we can obtain the following theorem.

Theorem 3.6.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex and lower semicontinuous functional defined on . Let and be sequences generated by and

Assume and . Then converges strongly to .

Remark 3.7 ..

It is easy to see that this paper hassome new methods and results than the results of Nilsrakoo and Saejung [11]:

(1)proposed a modified hybrid iterative scheme, so that the new simple method of proof has been used;

(2)removed the bounded restriction for closed convex set ;

(3)relax the conditions of sequence ;

(4)give an application for optimization problem;

(5)the sequence of sets satisfy the following relation:
(3.26)

so that to raise the convergence rate of is possible.

Declarations

acknowledgment

This project is supported by the National Natural Science Foundation of China under grant(10771050).

Authors’ Affiliations

(1)
Department of Fundamental, Tianjin Institute of Urban Construction
(2)
Department of Mathematics, Tianjin Polytechnic University

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Copyright

© L. Yang and Y. Su. 2009

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