Solvability of resonance problems for nonlinear q-difference equations on infinite intervals

Abstract

In order to develop the basic theory of resonance boundary value problems for nonlinear quantum difference equations, a class of nonlinear quantum difference equation boundary value problems at resonance on infinite intervals was studied. Firstly, by constructing a suitable Banach space, the Fredholm operator was defined, and its kernel and value domains were obtained. Secondly, by defining other appropriate operators, and using Mawhin′s coincidence degree theory, the existence theorem of solutions to this problem was established. Thirdly, the uniqueness of the solution was obtained by means of proof by contradiction. Finally, an example was given to illustrate the validity of the main results. The results show that under certain growth conditions of nonlinear terms, the boundary value problems for nonlinear quantum difference equation at resonance has at least one solution. The research results enrich the solvability theory of quantum difference equations, and provide important theoretical basis for the application of quantum difference equations in mathematics, physics and other fields.

Keywords

• nonlinear functional analysis; quantum difference equation; mawhin′s coincidence theory; infinite intervals; resonance