Solvability Of Local And Nonlocal Problems In Several Categories Quantum Difference Equations

Quantum calculus,also known as q-calculus,its basic concept is first established by F.Jackson.With the continuous development of q-calculus theory,its applications in physics and other fields have become more and more extensive.Many practical problems can be attributed to the study of quantum difference equation boundary value problems(ie q-difference equation boundary value problems).As far as I know,a lot of research results have been made on the boundary value problems of linear or nonlinear q-difference equations,but there are very few researches on boundary value problems of q-difference equations with p-Laplacian operators.At the same time,with the continuous development of mathematical physics and interdisciplinary subjects,post-quantum calculus(ie(p,q)-calculus)comes into being,and it is applied to both classical mechanics and quantum mechanics.As a generalization of q-difference equations,(p,q)-difference equations are widely used in hypergeometric functions,quantum groups and quantum algebras.However,the study of nonlinear(p,q)-difference boundary value problems is just beginning,so the research of this kind of problems is very important and needs to be further deepened.This plays a vital role in the theoretical promotion and practical application of quantum difference equations.In view of the above research situation,the research content of this article is mainly divided into the following aspects:1.The existence of solutions to boundary value problems of second-order nonlinear q-difference equations on infinite intervals with p-Laplacian operators is studied.Firstly,Green's function of boundary value problem for linear q-difference equation is given,and its compactness and total continuity on infinite interval are proved.Secondly,the existence of three positive solutions for nonlinear q-difference equations boundary value problems with p-Laplacian operator on infinite interval is obtained by using Avery-Peterson fixed point theorem on cone.Finally,the application examples are listed to verify the correctness of the results.2.The solvability of Dirichlet boundary value problems for nonlinear(p,q)-difference equations is studied.Firstly,the exchange integral order formula of double(p,q)-integral is given,and the Green function of the linear(p,q)-difference equation Dirichlet boundary value problem is solved.Secondly,the uniqueness theorem of the solution of the nonlinear(p,q)-difference equation Dirichlet boundary value problem is proved by using the Banach compression mapping principle,and the existence of the solution of the(p,q)-difference boundary value problem is proved by using Leray-Schauder nonlinear choice theorem and Leray-Schauder continuous theorem.Finally,we list two application examples to verify the correctness of the results.3.The existence and uniqueness of solutions for nonlocal problems of second order nonlinear(p,q)-difference equations are studied.Firstly,the uniqueness of the solution to the nonlocal problem of the second order nonlinear(p,q)-difference equation is obtained by using the Banach contraction mapping principle.Secondly,by using Krasnosel'skii fixed point theorem,the existence of the solution of the(p,q)-difference boundary value problem is proved.Finally,the Lyapunov inequality of the problem is given and two application examples are listed.