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Zbl 1208.47071
Takahashi, S.; Takahashi, W.; Toyoda, M.
Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. (English)
[J] J. Optim. Theory Appl. 147, No. 1, 27-41 (2010). ISSN 0022-3239; ISSN 1573-2878/e

Authors' abstract: Let $C$ be a closed and convex subset of a real Hilbert space $H$. Let $T$ be a nonexpansive mapping of $C$ into itself, $A$ 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 iteration scheme of finding a point of $F(T)\cap (A + B)^{-l} O$, where $F(T)$ is the set of fixed points of $T$ and $(A + B)^{-l} O$ is the set of zero points of $A + B$. Then, we prove a strong convergence theorem, which is different from the results of Halpern's type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.
[Stepan Agop Tersian (Rousse)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H07 Positive operators on ordered topological linear spaces
47H05 Monotone operators (with respect to duality)
47H09 Mappings defined by "shrinking" properties

Keywords: nonexpansive mapping; maximal monotone operator; inverse strongly-monotone mapping; fixed point; iteration procedure; equilibrium problem

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Zentralblatt MATH Berlin [Germany]