Existence Of Solutions For Some Boundary Value Problems

Periodic problems of impulsive differential systems come from practical problems in the fields of applied mathematics,physics and engineering science,which are hot issues in differential systems.With the rapid development of modern applied mathematics and physics,there come a large number of nonlinear problems.Periodic problems could explain a variety of phenomena in nature,as a consequence,it has become one of the most popular research topics in nonlinear functional analysis.There are so many scholars have made thorough research on the existence of solutions to periodic problems,and they actually have made remarkable progress in recent years.We mainly discuss the existence of two kinds of differential inclusion solutions in this paper,and he structure of the paper is as follows:In the first chapter,it mainly introduces the research background of differential inclusions and impulsive differential equations,the research status and development trend at home and abroad,as well as the main work of this paper.In the second chapter,there are some preliminary knowledge needed in this paper,including the basic knowledge of functional analysis,the theory of noncompact measures,the definition and some basic properties of multivalued mapping,theory of operators semigroup and some related lemmas used in this paper.In the third chapter,the author mainly studies the existence of periodic solutions for a class of abstract differential inclusion problems,and extends the existence of first order impulsive differential inclusion solutions to second order,which is more difficult to solve the corresponding impulse differential equations.At the beginning,we can solve the corresponding impulse differential equations based on the theory of functional analysis,ordinary differential equation,operator semigroup and so on,transporting the existence of differential inclusion solution into problem on operator fixed point to prove the existence of the solution by defining the upper and lower solutions and combining with Martelli's fixed point theorem.In the fourth chapter,the author mainly studies the existence of second order impulsive differential inclusion solutions with periodic boundary value conditions.Adding the impulse term into the existence of second order impulsive differential inclusion solutions is of great practical significance in the field of mathematical,physics and national defense,which makes it difficult and complicated to solve the corresponding impulse differential equations.We can obtain the corresponding impulse differential equations according to the theories of ordinary differential equations in this chapter,and prove the existence of the solution by defining the upper and lower solutions and combining to the fixed point theorem of multivalued mapping.In the last chapter,the author has summarized the current research status and come up with hypothesis about the future research.