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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 α-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)(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.
MSC:
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
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