The Application And Existence Of Solutions Of Some Abstract Equation(s)

Nonlinear functional analysis is an important branch of modern analysis mathematics. It can explain a lot of natural phenomena clearly, so more and more mathematical researchers are devoting their time to it. Among them, the nonlinear problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analysical mathematics.The present thesis mainly discusses the problems for solutions of nonlinear integro-differential equation(s) and nonlinear impulsive integro-differential equations in Banach space. It consists five chapters.In chapter one, we mainly introduce background, research meaning and current situations of this study, and the main conclusions and motive of this thesis.In chapter two, we discuss the existence of solutions of first order integro-differential equations of mixed type with deviating arguments, by using a comparison result and partial method. It generalizes and improves some former corresponding results.In chapter three, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solutions of the initial value problem(IVP) for a class of second order impulsive integro-differential equations on unbounded domain in a Banach space. Our results improve and extend many recent results.In chapter four, by using the cone theory and lower and upper solutions, we investigate the existence of unique solution of the initial value problems for systems of nonlinear second-order integro-differential equations in Banach space. Our results improve and extend some recent results.