Researches On The Existence Of Solutions For Some Classes Of Fractional Differential Equations With Boundary Value Conditions

Fractional order models are of more great theoretical significance and practical value than integer order models in really problems.Boundary value problems of fractional differential equations have significantly been discussed by some researchers.Fractional differential equations with various boundary conditions are also studied,because of its practical application in various fields.It is mainly used in:aerodynamics,biology,electrical network,biophysics and image processing,etc.Fractional derivatives are also provide an useful tool for memory characteristics of the processes and hereditary properties of miscellaneous materials.In recent decades years,boundary value problems of fractional-order sup-plemented with classical,nonlocal,multi-point,anti-periodic,integral bound-ary conditions and coupled boundary conditions have been addressed by many researchers.In particular,boundary value problems for impulsive fractional differential equations,sequential fractional differential equations and a coupled system of fractional differential equations also have been attractive to many researchers.In this paper,we will study the existence of solutions for some boundary value problems of fractional differential equations by some functional methods.In this paper,we investigate the existence of solutions for fractional d-ifferential equations with boundary value conditions.The methods used are different and the main ways are Banach contracting mapping principle,Leray-Schauder degree theory,nonlinear contraction,Leray-Schauder nonlinear al-ternative,Leggett-Williams fixed point theorem and coincidence degree.In Chapter one,some background knowledge of boundary value problem for fractional differential equation is introduced,we introduce some necessary definitions and lemmas to prove main results and the preliminary knowledge which is necessary in the paper is given.In Chapter two,we investigate existence of solution for sequential fraction-al differential equation with integral boundary value conditions.By means of Banach contracting mapping principle,Leray-Schauder degree theory,nonlin-ear contraction,some existence results of solutions are obtained.An example is given to illustrate the application of the results.In Chapter three,we investigate a class of boundary value problems of impulsive fractional differential equations with anti-periodic boundary value conditions.By using Leray-Schauder nonlinear alternative and Leray-Schauder degree theory,some new existence results of solution for the boundary value problems are obtained.An example is given to illustrate the results.In Chapter four,we study positive solutions of integral boundary value problem for a class of high-order fractional differential equations.The main job is to discuss existence of positive solutions for the high-order fractional dif-ferential equations with integral boundary value conditions on infinite interval,by using Leggett-Williams fixed point theorem.Finally,an example is given to show the applicability of main result.In Chapter five,By the means of coincidence degree theory,we discuss the existence of solutions for a system of nonlinear fractional differential equations with integral boundary value conditions.Finally,an example is given to show the applicability of main result.