The Study of Nonlinear Functional Analysis and Convex Analysis and its Applications Based on Optimization Theory and Fixed Point Theory

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In this study, we obtain many new and important theorems for nonlinear problems in nonlinear functional analysis and convex analysis by using optimization theory and fixed point theory. For example, we solved an open problem in geometry of Banach spaces which has been posed by Ray in 1980 by proving that every nonspreading mapping of a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space into itself has a fixed point if and only if the set is bounded.