The Existence And Uniqueness Of Solutions For The Boundary-value Problems Of Fractional Differential Equations

Differential equations with fractional order are generalization of ordinary differenti-ation and integration to arbitrary noninteger order. In recent years, the fractional differe-ntial equations have gained much importance and attention of domestic and foreign mat-hematicians and become a hot issue due to the intensive development of theory of fracti-onal calculus itself as well as their applications in various fields.In this paper, we consider existence and uniqueness of some classes of fractional di-fferential equations with boundary value problem. The main work of this paper divide i-nto the following five parts:In Chapter1, we introduce the research background and present situation of fractio-nal differential equations, and briefly describe other scholars research results of fractio-nal differential equation with boundary value problems.In Chapter2, we study a class of fractional differential equations. By the use of theupper and lower solutions method and monotone iterative techniques, we have someconditions on the existence and uniqueness of solutions for the boundary value pro-blems of fractional differential equations. An example is given to illustrate the results.In Chapter3, we discuss a class of nonlinear fractional differential equations withmulti-point boundary conditions. Using upper and lower solutions method, Krasno-sel skii fixed point theorems and Leray-Schauder fixed point theorems, we establish theexistence of multiple positive solutions, and we obtain some existence criteria for sing-ular boundary value problem by Leray-Schauder nonlinear alternative theorem as well.Finally, examples are given to illustrate the main results.In Chapter4, we consider the systems of nonlinear fractional differential equationsof weighted multi-point boundary value problems. By applying new iterative algorithmcombined with the method of upper and lower solutions, we have conditions on theexistence of extremal solutions. And we present an example to illustrate the result.In Chapter5, we summary the previous work, and point out the innovation of ourpaper.