Existence Of Solutions For The Vibration Equations Of Beam With Structural Damping And Infinite Delay

In this thesis,we study the existence of solutions for vibration equations of beam with structural damping and infinite time delay.This thesis consists of five chapters.In chapter 1,some research background of the subject and the main work of this paper will be presented.Meanwhile,we also introduce the definitions and basic results to be used in the research institute.It main includes the theory of operator semigroup,phase space theory,the measure of noncompactness,condensing mapping etc.In chapter 2,we mainly consider the existence of the mild solutions of the initial-value problems for the vibration equations of beam with nonlinear term is f(t,ut)in Banach space.The particularity of this kind of initial value problem is that it considers the factor of infinite time delay.We obtain the existence theorem of local and global of the mild solutions,when the nonlinear term satisfies different constraints,the theory of operator semigroup,phase space theory and Banach compression mapping principle are established.In chapter 3,we mainly consider the existence of the mild solution of the vibration equations of beam with nonlinear term is f(t,u(t),ut)in Banach space.In the case of the compact semigroup,the problem is transformed into two non-homogeneous initial value problems and the representation of the mild solution is given.We combine the operator semigroup theory and the phase space theory Banach.Compressed mapping original,Schauder fixed point theorem proves the existence of the mild solution,and further uses the non-compactness measure as a tool,using the condensed map definition and the fixed point theorem of condensed mapping,get the existence of the mild solution.The chapter 4,we considers the existence of the mild solutions for nonlocal problem of the vibration equations of beam.Combining the theory of operator semigroup and the phase space theory,using the Monch fixed point theorem in the measure of noncompactness,Darbo fixed point theorem obtained the existence of the mild solutions for nonlocal problem of the vibration equations of beam.The chapter 5,it is a summary of this paper and the prospect of the vibration equations of beam.