Positive Solutions to the Discrete Boundary Value Problem of the Kirchhoff Type

Abstract

The paper aims to study a discrete boundary value problem of the Kirchhoff type based on the critical point theory and the strong maximum principle. Compared to the existing literature, the existence and multiplicity of positive solutions to the problem are considered according to the behavior of the nonlinear term f in some points between the zero and positive infinity, which is a new attempt. Under different assumptions of the nonlinear term f, we obtain the determined open intervals of the parameter λ, such that the problem has at least three positive solutions or at least two positive solutions in different intervals. In the end, two concrete examples are used to illustrate our main conclusions.

Keywords

• discrete boundary value problem
• positive solutions
• critical point theory