Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings
© Wei-You Zeng et al. 2008
Received: 17 July 2008
Accepted: 11 November 2008
Published: 19 November 2008
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see [1–8]. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.
Such a mapping is called the -mapping generated by and , see .
The purpose of this paper is to develop an iterative algorithm for finding a common element of set of solutions of GMEP (1.2) and set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and extends the main result of S. Takahashi and W. Takahashi .
We denote by the set of fixed points of a self-mapping on , that is, . It is well known that if is nonempty, bounded, closed, and convex and is nonexpansive, then is nonempty; see . Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of self-mappings on by (1.4). Then we have the following lemmas which are important to prove our results.
Lemma 2.1 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, for every and the limit exists.
for every . Such a mapping is called the -mapping generated by and Throughout this paper, we will assume that for every . Since is nonexpansive, is also nonexpansive.
Lemma 2.2 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, .
If , then is said to be -convex. In particular, if for all , then is said to be strongly convex.
We denote for weak convergence and for strong convergence. A function is called weakly sequentially continuous at , if as for each sequence in converging weakly to . The function is called weakly sequentially continuous on if it is weakly sequentially continuous at each point of .
Lemma 2.3 (see ).
Let and . Then for , there must exist a point such that .
Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. For , let . Let be a real-valued function satisfying the following:
is skew symmetric;
for each fixed , is convex and upper semicontinuous;
is weakly continuous on .
Let be a differentiable functional with Fréchet derivative at satisfying the following:
is sequentially continuous from the weak topology to the strong topology;
is Lipschitz continuous with Lipschitz constant .
Let be a function satisfying the following:
for all ;
is affine in the first coordinate variable;
for each fixed , is sequentially continuous from the weak topology to the weak topology.
Let us consider the equilibrium-like function which satisfies the following conditions with respect to the multivalued mapping :
for each fixed , is an upper semicontinuous function from to , that is, and imply ;
for each fixed , is a concave function;
for each fixed , is a convex function.
It is easy to see that if , then is a solution of GMEP(1.2).
Lemma 2.4 (see ).
the auxiliary problem (2.8) has a unique solution;
if and for all and all , , it follows that is nonexpansive;
(e) is closed and convex.
We also need the following lemmas for our main results.
Lemma 2.5 (see ).
If , , and , then exists.
If and , then .
Note that Lemma 2.5 implies that exists. Suppose for some . It is obvious that and so inequality (2.12) implies that , which is a contradiction. Thus, . This completes the proof.
Lemma 2.7 (see ).
for all and ;
for all and .
Then is a Cauchy sequence.
Lemma 2.8 (see ).
, and ;
for all and .
3. Iterative Algorithm and Convergence Theorem
Let be a nonempty closed convex subset of a real Hilbert space , a multivalued mapping, a contraction mapping with constant , and an -mapping generated by and , where sequence is nonexpansive. Let be a sequence in and a sequence in . We can develop Algorithm 3.1 for finding a common element of a set of fixed points of -mapping and a set of solutions of GMEP(1.2).
We now prove the strong convergence of iterative sequence , , and generated by Algorithm 3.1.
Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued -Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying conditions and :
for all and , where , and .
Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies conditions and with . Let be an -mapping generated by and and , where sequence is nonexpansive. Let , and be sequences generated by Algorithm 3.1, where is a sequence in and in satisfying the following conditions:
, and ;
Then the sequences and converge strongly to , and converges strongly to , where .
for all and , where , , and . All the conclusions (a)–(e) of Lemma 2.4 hold.
Hence there exists a unique element such that . Noting that and , we get that .
Next, we prove that there exists , such that , , and as , where .
that is, . We conclude that as .
that is, . Thus, .
Then, , , and . It follows from Lemma 2.8 that and so . This completes the proof.
Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi .
We now give some applications of Theorem 3.2. If the set-valued mapping in Theorem 3.2 is single-valued, then we have the following corollary.
Let be a nonempty closed convex bounded subset of a real Hilbert space , a Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying the conditions and :
for all and .
where is a sequence in and in satisfying conditions . Then the sequences and converge strongly to , where .
where is a sequence in and in satisfying conditions and . Then the sequences and converge strongly to , and converges strongly to , where .
Let in Theorem 3.2 for , where is an identity mapping. Then for Thus, the condition is satisfied. Now Corollary 3.5 follows from Theorem 3.2. This completes the proof.
The authors would like to thank the referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005).
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