Nonlinear problem by using fixed point theory

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We studied some nonlinear problems concerning nonlinear evolution equation, mathematical economics, mathematical programming and image recovery by using nonlinear functional analysis and fixed point theory. We first proved some fixed point theorems for families of nonexpansive mappings in a Banach space. Next, we proved nonlinear ergodic theorems of Baillon's type for nonlinear semigroups of nonexpansive mappings. In particular, we gave an answer to the open problem posed during the Second World Congress on Nonlinear Analysts, Athens, Greece, 1996, by extending Takahashi's result and Rode's result to a Banach space for an amenable semigroup of nonexpansive mappings. Further, we established weak convergence theorems of Mann's type for families of nonexpansive mappings. We also proved strong convergence theorems of Halpern's type for families of nonexpansive mappings. Finally, using these results, we discussed the problem of image recovery by convex combinations of nonexpansive retractions, the problem of finding a common fixed point of a commuting family of nonexpansive mappings, the convex minimization problem and so on.