Impulsive Boundary Value Problems For Caputo Fractional Differential Equation

Fractional order models are of great theoretical significance and practical value in other subjects and the engineering application field,Fractional differential equations have been of great interest and recently that depends on both the intensive development of the theory of fractional calculus itself and the applications in various fields.In this paper we consider the existence of solutions for impulsive boundary value problems of Caputo fractional differential equations and use the methods of Banach contraction mapping principle,Krasnoselekii's fixed point theorem and Mawhin co-incidence degree theory to give some new results.Some examples are presented to illustrate the results.In chapter one we introduces the backgrounds of fractional calculus theory and the main aim of this papers.We also give some preliminary definitions and lemmas on fractional differential equations which are needed in this paper.In chapter two,we study the impulsive hybrid boundary value problems for a class of higher-order Caputo fractional differential equation.In first section,we give the the contraction mapping principle and Krasnoselekii' s fixed point theorem.In second section,we give a formula of solution to the boundary value problem for frac-tional differential equations with impulses.In third section,we give the uniqueness and existence by using the contraction mapping principle and Krasnoselekii' s fixed point theorem.At last,we give some example to demonstrate our main result.In chapter three,we study a class of impulsive periodic boundary values problem-s for higher-order Caputo fractional differential equation.In first section,we give the coincidence degree theory.In second section,we give a formula of solution to the boundary value problem for fractional differential equations with impulses.In third section,we give the existence by using the coincidence degree theory.At last,we give some example to demonstrate our main result.