Existence Of Solutions For Two Classes Of Second-order Discrete Boundary Value Problems At Resonance With Spectral Parameters In Boundary Conditions

The existence of solutions to resonance problems is one of the important research topics in recent years,and the research of this kind of problems is closely related to the spectrum of corresponding linear eigenvalue problems.With the development of unbounded operator theory and spectral theory,the study of spectral theory of linear operators with spectral parameters in boundary conditions has attracted the attention of many scholars.However,due to spectral parameters in boundary conditions,it brings many essential difficulties to the study of this kind of problems,and thus delays the development of the corresponding nonlinear problems.In particular,the research results of boundary value problems at resonance with spectral parameters in boundary conditions have not been reported in the literature.Based on this,we consider the existence of solutions for second-order discrete resonance problems with spectral parameters in boundary conditions under different conditions by means of the spectral theory of difference operator with spectral parameters in boundary conditions.The main results of this thesis are described as follows:In the first part,we first transform the resonance problem into its corresponding equivalent system by Lyapunov-Schmidt method.In addition,by using Schauder fixed-point theorem and the connectivity theories of the solution set of compact vector fields,we obtain the existence and multiplicity of solutions for a class of secondorder discrete resonance problems with spectral parameters in boundary conditions under the sign conditions.In the second part,we first transform the resonance problem into its corresponding equivalent system by Lyapunov-Schmidt method.In addition,by using topological degree theory,we obtain the existence of solutions for a class of secondorder discrete resonance problems with spectral parameters in boundary conditions under the local sign condition.