Boundary Value Problems For Discrete Kirchhoff Type Equations

This article mainly studies the existence and multiplicity of solutions for boundary value problems of discrete Kirchhoff type equations.Firstly,we constructed the basic function space and established the variational functional corresponding to the boundary value problem.By using critical point theory of G.Bonanno,we discuss the existence of critical points for this variational functional,and thus prove the existence of solutions to the original boundary value problem.Secondly,by establishing the strong maximum principle corresponding to the boundary value problem,we obtain sufficient conditions for the existence of infinite positive solutions to the boundary value problem.This article consists of three chapters as follows:In the first chapter,we briefly introduce the background and significance of the research and expound the current relevant research situation and development.For the convenience of reading,we have also listed the preparatory knowledge.In the second chapter,we investigate the existence and multiplicity of nontrivial solutions for Robin boundary value problems of a special class of discrete Kirchhoff type equations.By using the oscillation behavior of the nonlinear term at point at infinity and the origin,the sufficient conditions for the existence of infinitely many nontrivial solutions of the boundary value problem are obtained respectively.In addition,we construct the corresponding strong maximum principle,and obtain the sufficient conditions for the existence of infinite positive solutions of the corresponding problem.In the third chapter,we consider the Dirichlet boundary value problem for general discrete Kirchhoff type equations.We prove that when the nonlinear term satisfies appropriate conditions and the parameters are within the appropriate range,there are infinitely many nontrivial solutions to the problem.At the same time,using the strong maximum principle,we obtain a sufficient condition for the existence of infinite positive solutions to the problem.