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Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems

Research paper by Wataru Takahashi, Ngai-Ching Wong, Jen-Chih Yao

Indexed on: 17 Oct '12Published on: 17 Oct '12Published in: Fixed Point Theory and Applications



Abstract

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α>0Open image in new window, and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B and W be maximal monotone operators on H such that the domains of B and W are included in C. Let 0<k<1Open image in new window, and let g be a k-contraction of H into itself. Let V be a γ¯Open image in new window-strongly monotone and L-Lipschitzian continuous operator with γ¯>0Open image in new window and L>0Open image in new window. Take μ,γ∈ROpen image in new window as follows:Suppose that F(T)∩(A+B)−10∩W−10≠∅Open image in new window, where F(T)Open image in new window and (A+B)−10Open image in new window, W−10Open image in new window are the set of fixed points of T and the sets of zero points of A+BOpen image in new window and W, respectively. In this paper, we prove a strong convergence theorem for finding a point z0Open image in new window of F(T)∩(A+B)−10∩W−10Open image in new window, where z0Open image in new window is a unique fixed point of PF(T)∩(A+B)−10∩W−10(I−V+γg)Open image in new window. This point z0∈F(T)∩(A+B)−10∩W−10Open image in new window is also a unique solution of the variational inequalityUsing this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73:1562-1568, 2010).MSC:47H05, 47H10, 58E35.