Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems
© J. F. Tan and S. S. Chang. 2011
Received: 30 October 2010
Accepted: 9 November 2010
Published: 25 November 2010
The main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.
This problem is called the Hartman-Stampacchia variational inequality (see ). The set of solutions to variational inequality (1.5) is denoted by .
The set of solutions to (1.7) is denoted by EP.
Motivated and inspired by the researches going on in this direction, especially inspired by Zhang et al. , the purpose of this paper is to introduce an explicit iterative algorithm to studying the existence problem and the approximation problem of the solution to the quadratic minimization problem (1.8) and prove some strong convergence theorems for a family of nonexpansive mappings in the setting of Hilbert spaces.
Proposition 2.2 (see ).
Lemma 2.3 (see ).
It is well-known that every continuous mapping must be hemicontinuous.
Lemma 2.7 (see ).
Lemma 2.8 (see ).
Lemma 2.9 (see ).
Lemma 2.10 (see ).
then the following results hold:
- (i)(see ) is a solution of variational inclusion (1.2) if and only if
- (ii)(see ) is a solution of generalized equilibrium problem (1.6) if and only if
(iii) (see ) Let be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. If and , then is a closed convex subset in and GEP is a closed convex subset in .
Lemma 2.12 (see ).
3. Main Results
We divide the proof of Theorem 3.1 into four steps.
This completes the proof of Theorem 3.1.
In Theorem 3.1, if , where is the indicator function of , then the variational inclusion problem (1.2) is equivalent to variational inequality (1.5), that is, to find such that , for all . Since . Consequently, we have the following corollary.
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