Existence Of Solutions For The Vibration Equations Of Beam With Structural Damping

The dissertation mainly deals with the existence of solutions for the vibration equations of beam with structural damping.This thesis consists of four parts.In chapter 1,we wish to recall briefly the research background,the main results of this thesis and some definitions and basic results needed in the sequel.It mainly includes the theory of operator semigroup and the measure of noncompactness,the condensing mappings and its fixed point theorem,the definitions and basic properties about asymptotic periodic function.In chapter 2,we consider the global existence of mild solutions for the initial-value problems for the vibration equations of beam with structural damping in Banach space.Under some conditions on nonlinear term,we prove global existence theorems of mild solutions,our main results are based upon the theory of operator semigroup and LeraySchauder fixed point theorem.Moreover,we obtain the H ¨older regularity of the mild solutions.At last,an example is presented to illustrate our theoretical results.In chapter 3,we consider the existence of mild solutions for nonlocal problem of vibration equations of beam.Under compact semigroup conditions,we obtain the representation of the mild solution for nonlocal problem of vibration equations of beam.By using the measure of noncompactness and fixed point theorem,some existence results of mild solutions are obtained.In chapter 4,we consider the existence and uniqueness of a pseudo S-asymptoticallyω-periodic mild solutions to the vibration equations of beam.We obtain sufficient conditions for the existence of solutions of above equation with the theory of semigroup and a fixed point theorem.At last,we also give an example to illustrate our theoretical results.