Nonlinear functional analysis and convex analysis problem by using fixed point theory

点击翻译按钮获取中文摘要

We studied some problems concerning nonlinear functional analysis and convex analysis by using fixed point theory. We first considered iteration schemes given by an infinite family of nonexpansive mappings in Hilbert spaces or Banach spaces and then proved strong convergence theorems for the family of nonexpansive mappings. Using these results, we also considered the feasibility problem of finding a common fixed point of infinite nonexpansive mappings. Next, we introduced two proximal point algorithms suggested by the iterative schemes introduced by Solodov and Svaiter in order to find a solution of $v \in T^∧{-1}0$, where $T$ is a maximal monotone operator. Main results were established by using metric projections and generalized projections in the case of the strong convergence. We also applied these results to find a minimizer of a lower semicontinuous convex function in a Banach space. Finally, we introduced iteration schemes of finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequality for inverse-strongly-monotone mappings. Using these results, we considered the problem of finding a common element of the set of zeros of a maximal monotone mapping and the set of zeros of an inverse-strongly-monotone mapping.