Research On Existence Of Solutions For The Boundary Value Problems Of Fractional Differential Equations With P-laplacian Operator

Nonlinear functional analysis is a research field of mathematics with profound theories and extensive applications. It is basis of the study of the nonlinear prob-lems which appeared in mathematics and the natural sciences. It constructs many general theories and methods to deal with nonlinear problems. Its rich theories and advanced methods are widely used in studies of solving many kinds of nonlinear dif-ferential equations, nonlinear integral equations and some other types of equations, and handling many nonlinear problems in computational mathematics, cybernetics, optimized theory, dynamic system, economical mathematics, etc. At present, non-linear functional analysis mainly covers topology degree theory, critical point theory, partial order method, analysis method, monotone mapping theory, and so on.In recent years, the boundary value problems of nonlinear fractional differential equations are important subjects in the theory of differential equations. Fractional differential equations have been widely used in diffusion and transport theory, chaos and turbulence, viscoelastic mechanics, nonnewtonian fluid mechanics etc. As one of the hottest issues in the international research field, it has received highly attention of the domestic and foreign mathematics and natural science field.The theory and method of nonlinear functional analysis has been employed in the present paper, such as fixed point index theory, Leggett-Williams fixed point theorem and the coincidence degree theory, to investigate the existence, multiplicity of solutions of boundary value problems with p- Laplacian and fractional boundary value problems. Having studied thoroughly, some new and deep results under weaker conditions have been obtained.The thesis is organized as follows:The chapter one introduces the historical background of the research questions. Secondly, it introduces some basic concepts and lemmas.In chapter two, by using the fixed point index theorem, a class of singular p-Laplacian fractional differential equations with integral boundary conditions is considered. Two or 2n positive solutions are obtained.In chapter three, constructing a Green function, by the using approxima-tion methods and the Leggett-Williams fixed point theorem, a class of singular p-Laplacian fractional differential equations with Riemann-Stieltjes integral bound-ary conditions is studied. The existence of at least three positive solution to this boundary value system is guaranteed.In chapter four, by using the coincidence degree theory, the existence of so-lutions for a coupled system of fractional p-Laplacian differential equations with multipoint boundary conditions at resonance is guaranteed.In chapter five, the existence of solutions for system of fractional p-Laplacian differential equations with integral boundary conditions at resonance on the half line is studied. The result obtained in this paper extends some known results on finite interval. An example is given to illustrate our result.