Eigenvalue Problems Of Fractional Difference Equations With Parameters

Fractional derivative provides an excellent instrument for the description of memory and hereditary properties of varions materials and processes.This is the main advantage of fractional derivative in comparison with classical integer-order models.Fractional calculus has been paied more and more wide attention from scholars at home and abroad.Fractional difference equation is a discrete form of fractional differential equation.Addition to the field of mathematics,fractional difference equation has received increasing attention related to itself development and applications in various fields of science,such as rheology,self-similar dynamics and porous structures,power network,viscoelasticity and chemicophysical.After adding parameters to the discrete system,the stability and structure of the system maybe change with the parameter value.Therefore,it has richer significance and value to study fractional difference system and get what affect on functions,status and dynamical property of system.In addition,the study of discrete fractional boundary value problem with parameters is also an important basis for further study of fractional difference spectrum theory.Fractional difference equation with p-Laclacian operator is an extension of fractional difference equation.Because of p-Laclacian operator is nonlinearity,it has many applications in the fields of dynamical system,molecular structure,internetwork and image processing.If p(28)2,it can be transformed to general fractional difference equation.This paper is devoted to the study of fractional difference equation boundary value problem,which includes boundary value problems with p-Laplacian operator,boundary value problems with parameter,singular fractional difference equation boundary value problems,the smallest eigenvalues for fractional difference equation and Nabla fractional difference equation boundary value problems.The uniqueness of solution,existence and nonexistence of positive solution,comparison theorem for smallest positive eigenvalues and the Lyapunov inequalities are studied and obtained.At last,we present some examples to illustrate the main results.In Chapter 1,we introduce background and significance of research on the boundary value problems of fractional difference equation.We list some basic definitions,relatedlemmas and the main tool used in this paper.In Chapter 2,we study two classes of fractional difference equation boundary value problems with p-Laclacian operator.In first part,by using Banach contraction mapping principle and Brouwer fixed point theorem,the uniqueness and existence of solution are obtained.At last some examples are given to illustrate our results.In second part,by using the properties of Green function and Guo-Krasnosel’skii fixed point theorem,the eigenvalue intervals are obtained.Examples are given to illustrate our results.In Chapter 3,we consider two classes of singular fractional difference equation boundary value problems.By using auxiliary function and Guo-Krasnosel’skii fixed point theorem,some results of existence of positive solution are obtained.Examples are given to illustrate our results.In the first part of Chapter 4,we discuss fractional difference equation with nonlocal boundary conditions.By using the monotone iterative technique and low-upper solution method,we get the existence of positive solutions.The eigenvalue intervals are considered by the properties of Green function and Guo-Krasnosel’skii fixed point theorem.Some examples are given to illustrate our results.In second part,we study a class of fractional difference equation with a forcing term.We establish several Lyapunov inequalities for this problem.Example is given to illustrate our results.In Chapter 5,we study two classes of the smallest eigenvalue problems for fractional difference equation.The existence of smallest positive eigenvalues is established,and a comparison theorem is obtained.In Chapter 6,we study two classes of eigenvalue problems of Nabla fractional difference equation with parameter.The eigenvalue intervals are considered by the properties of Green function and Guo-Krasnosel’skii fixed point theorem.Some examples are given to illustrate our results.In Chapter 7,we summarize the main results in this thesis and point out the innovations of our work.Finally,we prospect some future research work.