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Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. (English) Zbl 1208.47071
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.

47J25Iterative procedures (nonlinear operator equations)
47H07Monotone and positive operators on ordered topological linear spaces
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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