Research On Solvability Of Boundary Value Problems For Some Categories Of Special Quantum Difference Equations

With the development of science and technology,quantum difference equations have been widely used in various fields,such as mathematical physics,cosmic strings and black holes,conformal quantum mechanics,nuclear and high energy physics,numerical theory,combination,orthogonal polynomials,basic hypergeometric functions and other scientific quantum theories,mechanics and relativity,which have attracted more and more scholars' attention.With the development of theoretical research,quantum difference equations provide a suitable mathematical model for solving many practical problems.In practical applications,many problems are attributed to the solution of the boundary value problem of the quantum difference equation.Therefore,it is of great practical significance and application value to study the boundary value problem of the quantum difference equation.In this paper,we mainly study the existence of solutions to boundary value problems of difference equations on infinite intervals,the solvability of second-order Hahn difference equations and the existence of positive solutions to boundary value problems of fractional q-difference equations.In Chapter one,the background and significance of the research on the quantum difference equation are introduced,and the research status at home and abroad and the main contents of this paper are described.In Chapter two,we study the existence of solutions for boundary value problems of difference equations on infinite intervals.Under the condition that the nonlinear term satisfies certain growth conditions,the existence theorem of the solution is obtained by using Leray-Schauder continuation theorem,and the correctness of the result is proved by an example.In Chapter three,the solvability of Sturm-Lioville boundary value problem for the second order nonlinear Hahn difference equation is discussed through using the Banach fixed point theorem and the Leray-Schauder nonlinear alternative.This paper proves the existence and uniqueness theorem of the solution of the boundary value problem when the nonlinear term satisfies certain conditions,and gives a concrete application example.In Chapter four,by using the Guo-Krasnosel'skii fixed point theorem,we study the positive solutions for the fractional q-difference equations eigenvalue problems with ?-Laplacian operators.The conclusion that there is one or two positive solutions andthere is no positive solution in different range is obtained.