Research On The Solutions For Boundary Value Problems Of Coupled System Of Nonlinear Fractional Differential Equations

In recent years,as the theory of fractional calculus is widely used in many fields,the coupled system of fractional differential equations has also become an important tool for describing various practical problems in the fields of natural science and engineering.There-fore,there are a large number of papers dealing with the existence or multiplicity of solutions to the boundary value problems for some nonlinear differential system.In this paper,we use the cone theory in Banach space and some fixed point theorems to study the boundary value problems of two types of nonlinear fractional differential equation coupled systems.We not only get the existence theory of the solution,but also the uniqueness of the solutions.The thesis consists of three sections.Chapter 1,Preface.It mainly describes the domestic and foreign research trends of fractional differential systems,and gives the basic ideas and research methods of the two types boundary value problems of the fractional differential systems studied in this paper.In Chapter 2,we discuss a new coupled system of fractional differential equations(?)where si=xi+?i,xi?(1,2],?i?(1,2],?i?(3,4],zi:[0,1]?[0,+?)is continuous,D0+?i and D0+ ?i are the standard Riemann-Liouville derivatives,?i?(0,1),bi ?(0,?i1-?i),i=1,2,and f,g?C([0,1]×R2,R).We establish the existence and uniqueness of solutions for the problem by a recent fixed point theorem of increasing ?-(h,e)-concave operators defined on ordered sets.Furthermore,the results obtained are well proven by means of a specific example.In Chapter 3,we investigate a coupled system of fractional differential equations with P-Laplacian operators (?) where ?i?(1,2],? i?(3,4],D0+?i ane D0+?i are the standard Riemann-Liouville derivatives,?pi(s)=|s|pi-2s,pi>1,?pi-1=?qi,1/pi+1/qi=1,?i?(0,1),bi?(0,?i1-?i/pi-1),i=1,2,and f,g?C([0,1]×[0,+?)×[0,+?),[0,+?)),? and ? are positive parameters.We establish the existence and uniqueness of positive solutions for the problem by a new fixed point the-orem.Furthermore,an example is given to illustrate our main result.