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Zbl 1247.47070
Manaka, Hiroko; Takahashi, Wataru
Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.
(English)
[J] Cubo 13, No. 1, 11-24 (2011). ISSN 0716-7776

Summary: Let $C$ be a dosed convex subset of a real Hilbert space $H$. Let $T$ be a nonspreading mapping of $C$ into itself, let $A$ be an $\alpha$-inverse strongly monotone mapping of $C$ into $H$ and let $B$ be a maximal monotone operator on $H$ such that the domain of $B$ is included in $C$. We introduce an iterative sequence of finding a point of $F(T)\cap(A+B)^{-1}0$, where $F(T)$ is the set of fixed points of $T$ and $(A+B)^{-1}0$ is the set of zero points of $A+B$. Then, we obtain the main result which is related to the weak convergence of the sequence. Using this result, we get a weak convergence theorem for finding a common fixed point of a nonspreading mapping and a nonexpansive mapping in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonspreading mapping.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H05 Monotone operators (with respect to duality)
47H09 Mappings defined by "shrinking" properties

Keywords: real Hilbert space; nonspreading mapping; $\alpha$-inverse strongly monotone mapping; maximal monotone operator; iterative sequence; weak convergence; common fixed point; nonspreading mapping; nonexpansive mapping; equilibrium problem

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