Some Problems For Solutions Of Equations In Abstract Space

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 boundary value 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 theory of nonlinear impulsive integro-differential equations is a new and important branch of differential equations, which originates from some mathematical model of biology, medicine. Because all the structure of its emergence has deep physical background, the research on nonlinear impulsive integro-differential equations is more meaningful.The present thesis mainly discusses the problems for solutions of nonlinear integro-differential equations 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 nonlinear boundary conditions in Banach space, by using a comparison result and partial method. It generalizes and improves the former corresponding results.In chapter three, we use the fixed point index theory to prove some existence theorems of multiple positive solutions of periodic boundary value problem for the first order impulsive integro-differential equations in Banach space that improved and generalized the results obtained by others.In chapter four, by using the cone theory and lower and upper solutions, we investigate the existence of extremal solutions of nonlinear boundary value problem for second order impulsive integro-differential equations, which involve the derivative x ' and deviating argument x (β(t)) in Banach space.In chapter five, 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.