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
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 ).
Lemma (Shimizu and Takashi ).
2. Main Results
Our proof is divided into five steps.
So (2.23) holds thank to (2.14).
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.
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