| The purpose of this thesis is to discuss problems of solutions for some nonlinear operator equations. And we obtain the conclusion that such equation under certain conditions will have solutions. Moreover, this thesis is also to study the existence of the solutions for some classes nonlinear differential equations by using the fixed point theorem for nonlinear operators. This thesis includes four chapters:In chapter one, a brief introduction and an overview of background materials and recent developments of my research are given. Besides, we introduced some elementary concepts and basic results which will be used in this thesis.In chapter two, we studied the fixed point index of weakly inward 1-set-contraction mappings, and obtained some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax =μx, which extend many famous theorems such as Leray-Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings.In chapter three, the theorem of cone and the techniques of partial order theory are used to discuss the existence of coupled quasi-solutions and solutions for a mixed monotone operator equation Lx = N(x,y) in complete metric space and Banach space, where neither L nor N need to be continuous. We also constructed some new non-symmetric iterative sequences and studied their approximation, then we got some new theorems. Finally, we apply the new results presented in this paper to Hammerstain integral equations.In chapter four, under the condition that the operator is un-continuous, we discussed a class of groups of increasing operator equations in probabilistic metric space, and obtained the existence of its solutions by using semi-ordering method. Moreover, we also proved the convergence of the iterative sequences under proper conditions. Meanwhile, some important conclusions are improved and generalized.
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