Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space
© Lihua Li et al. 2008
Received: 19 March 2008
Accepted: 14 August 2008
Published: 3 September 2008
Let be nonempty closed convex subset of real Hilbert space . Consider a nonexpansive semigroup with a common fixed point, a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Let . It is proved that the sequence generated iteratively by converges strongly to a common fixed point which solves the variational inequality for all .
1. Introduction and Preliminaries
Let be a closed convex subset of a Hilbert space , recall that is nonexpansive if for all . Denote by the set of fixed points of , that is, .
for all ;
for all ;
for all and ;
for all is continuous.
We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.
converges strongly to the unique solution of the minimization problem (1.1) provided that the sequence satisfies certain conditions.
where is a potential function for (i.e., , for ).
where , and investigate the problem of approximating common fixed point of nonexpansive semigroup which solves some variational inequality. The results presented in this paper extend and improve the main results in Marino and Xu , and the methods of proof given in this paper are also quite different.
In what follows, we will make use of the following lemmas. Some of them are known; others are not hard to derive.
Lemma 1.1 (Marino and Xu ).
Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .
Lemma (Shimizu and Takashi ).
where denote that converges weakly to , then .
2. Main Results
Let be a Hilbert space, a closed convex subset of , let be a nonexpansive semigroup on , is a sequence, then is monotone.
and the iteration process converges strongly to the unique solution of the variational inequality for all .
Our proof is divided into five steps.
Since , as , we may assume, with no loss of generality, that , for all .
(i) is bounded.
is well defined. Next, we will show that is bounded.
Thus is bounded.
From condition and the boundedness of , we obtain that . Again by boundedness of , we know that there exists a subsequence of such that . Then . From Lemma 1.3 and step (ii), we have that .
where is obtained in step (iii).
So (2.23) holds thank to (2.14).
So . This completes the proof of the Theorem 2.2.
It follows from the above proof that Theorem 2.2 is valid for nonexpansive mappings. Thus, we have that Corollaries 2.3 and 2.4 are two special cases of Theorem 2.2.
then for any , the sequence above converges strongly to the unique solution of the variational inequality for all .
where is a sequence in ( ) satisfying the following condition: , then the sequence converges strongly to the unique solution of the variational inequality for all .
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