Existence Of Solutions For Boundary Value Problems Of Fractional Differential Equations With P-laplacian

Fractional calculus theory is the generalization of the integer calculus theory. It has been300years. Especially in recent decades, fractional differential equations have been widelyused in different areas including physical, chemical, biological, financial mathematics. Theboundary value problem of differential equation with p-Lapalcian operator has been appliedto engineering, physics and other fields for a long time. With the application of fractionaldifferential equations in the actual production and life, the investigation of the boundary valueproblem of fractional differential equation has attracted extensive attention of scholars athome and abroad and gradually becomes a hot topic.The present paper is devoted to the study of the existence of the boundary value problemof fractional differential equation with p-Lapalcian, which includes the general equation,equation with parameters and boundary conditions with parameters and other circumstances.We study the existence, uniqueness, multiplicity and nonexistence of positive solution andgive some new results.In chapter one, we introduce research background, history of development, and presentsituation of fractional calculus, present situation and significance of research of the existenceof solution of the boundary value problem of fractional differential equation with p-Lapalcian.We also give some basic definition of fractional calculus theory and some related lemmas andthe main work of this paper.In chapter two, we investigate the existence of one or more positive solutions of theboundary value problem of nonlinear fractional differential equation with p-Lapalcian. Byusing upper and lower solutions method, Guo-Krasnosel’skii fixed point theorem andLeggett-Williams fixed point theorem, some sufficient conditions of the existence of at leastone or more positive solutions are obtained.In chapter three, we discuss the existence and multiplicity of positive solutions of twotypes of the boundary value problem of p-Lapalcian fractional differential equation with aparameter. By the properties of Green function and Guo-Krasnosel’skii fixed point theorem,some sufficient conditions of nonexistence or the existence of at least one or two positive solutions in terms of different values of the parameter are obtained.In chapter four, we consider the existence and uniqueness of positive solutions of theboundary value problem of p-Lapalcian fractional differential equation with a parameter in theboundary conditions. By the properties of Green function and Schauder fixed point theorem,several existence, nonexistence and uniqueness results for positive solutions in terms of theparameter are obtained.In chapter five, we investigate multiplicity of positive solutions of boundary valueproblems for fractional differential equations with p-Laplacian. By means of the properties ofGreen function, Leggett-Williams fixed point theorems and fixed point index theory, severalnew sufficient conditions for the existence of at least two or at least three positive solutions areobtained.In chapter six, we summarize the main results and the innovations in this paper. Finally,we prospect some future research work based on this paper.