Global Solutions Of Initial Value Problems For Nonlinear Second Order Integro-differential Equations In Banach Spaces

Nonlinear differential equations is a new and important branch of differential equation, which originates from some mathematical model of biology, medicine. Because all the structure of its emergence has deep physical bachground, the research on the theory of nonlinear impulsive differential equations is very meaningful. This paper is mainly divided into three chapters.In the first chapter, we introduce some important concepts, notations and accept certain basic principles of linear functional analysis and nonlinear functional analysis, which are quoted here for easier references until they are applied in later chapters.In the second chapter, the existence of global solutions of initial value problems for nonlinear second order integro-differential equations in Banach spaces are investigated. Some existence theorems of global solutions are obtained by useing the Schauder fixed point theorem, which improve the related results obtained.In the third chapter, under loose conditions , without the compactness restrictions of impulsive terms, the existence of solutions to initial value problems are studied for second order nonlinear impulsive integro-differential equations with infinite moments of impulse effect on the positive half real axis in Banach spaces . By the use of the locally convex topology , recurrence method , Tonelli method, the new existence theorems are achieved , and we give an sufficient condition of existence of maximal and minimal solutions based on the Kuratowski measure of noncompactness, which essentially improve some related results.