Several Questions In Nonlinear Functional Analysis

In this paper, we mainly research several questions in nonlinear functional analy-sis, according to the contents, the paper is divided into four chapters.In the first chapter, we introduce some background knowledge and the main con-tents of this paper.In the second chapter, we give the definition of a class of new operators, namely, convex-power1-set-contraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of this class of operators under the conditions of semi-closed or semi-compact, which generalize fixed point theorems of1-set-contraction operators in Banach spaces. As applications, the existence of solutions in C2[I, E] of nonlinear Sturm-Liouville problems is investigated. In§2.4, we obtain fixed point theorems of convex-power1-set-contraction operators under the conditions of T(aD)(?)D(Rothe type),‖Tx-x‖2≥‖Tx‖2-‖x‖2,(?)x∈aD(Altman type), Leray-Schauder condition and Leray-Schauder priori estimate.In the third chapter, we generalize the conclusions in§2.2into locally convex spaces.In the last chapter, by introducing a suitable partial ordering(?), we obtain theorems in οσ-complete vector lattices which is similar to Caristi’s fixed point theorem.