Existence Of Solutions For Boundary Value Problem Of Fractional Differential Equations

In recent years, with the development of social technology, and the growing demandfor the complicated engineering application, fractional calculus theory not only plays animportant roles in such fields as physics of diffusion model, capacitor electrode,electrolytic chemical etc., but also has highlighted the irreplaceable advantages in suchsocial science as seismic analysis, biological genetic systems, financial, and the robot etc..Fractional calculus and the integer calculus are unified, and it is the extension of thecalculus with integer order, which is of wide practical significance in mathematics model.The existence of solutions for fractional differential equation is the main content in thequalitative theory of fractional calculus. Therefore, studying the existence of solutions forfractional differential equations is of great theoretical significance and practical value.In this thesis, the existence of solutions for boundary value problem of fractionalhigh-order differential equations, and the existence and uniqueness of positive solutions,and the existence of solutions for high-order impulse boundary value problem of fractionaldifferential equations or fractional partial differential equations are studied respectively,which based on the theory of fractional differential equation. Finally, some examples arepresented to illustrate the main results respectively.Firstly, we discuss the existence of solutions for boundary value problem of fractionalhigh-order differential equations. By giving the solutions of corresponding linearfractional high-order differential equations, and using Banach contraction mappingprinciple, Leray-Shauder nonlinear alternative theorem and Guo-Krasnoselskii’s fixedpoint theorem, we obtain sufficient conditions of the uniqueness of (positive) solution andat least one (positive) solution for the boundary value problem.Secondly, we consider the existence of solutions for boundary value problem offractional high-order differential equations with impulsive. By Banach contractionmapping principle and Krasnoselskii’s fixed point theorem, we give the uniqueness ofsolution and at least one solution for boundary value problems.Finally, we study the existence of solutions for impulsive initial boundary value problem for nonlinear fractional partial differential equations. By Banach fixed pointtheorem, Schauer fixed point theorem and Krasnoselskii’s fixed point theorem, we obtainthe uniqueness of solution and at least one solution for impulsive boundary valueproblems.