Nonlinear functional analysis and nonlinear problems by using fixed point theory

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We studied nonlinear functional analysis and some nonlinear problems by using fixed point theory. We first studied the existence of zero points of maximal monotone operators in Banach spaces. We proved existence theorems for maximal monotone operators by a new boundary condition. Using one of them, we obtained a generalization of Kakutani's fixed point theorem for multivalued mappings. Then we considered iteration schemes of finding zero points of maximal monotone operators in Banach spaces. We found two new resolvents for maximal monotone operators and then obtained weak and strong convergence theorems for resolvents of maximal monotone operators in Banach spaces with generalized projections. In particular, we obtained two generalizations of Solodov and Svaiter's theorem by using generalized projections and metric projections. Further, we introduced the notion of relatively nonexpansive mappings in a Banach space which generalize nonexpansive mappings in a Hilbert space. Then, we obtained two convergence theorems for relatively nonexpansive mappings in Banach spaces. One of them is a generalization of Nakajo and Takahash's theorem. We also obtained important examples of sunny generalized nonexpansive retractions which are related to one of new resolvents. Next, we introduced an iteration scheme of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an inverse-strongly-monotone operator in a Hilbert space. Further, we extended this iteration scheme to that of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then we proved some weak and strong convergence theorems. Using these results, we improved and extended Wittmann' strong convergence theorem, Reich's weak convergence theorem and Baillon's nonlinear ergodic theorem which are well known in this field.