Discrete fourth-order boundary value problems with four parameters

Abstract

The theory of nonlinear difference equations and discrete boundary value problems has been widely used to study discrete models in many fields such as computer science, economics, mechanical engineering, control systems, artificial or biological neural networks, ecology, cybernetics, and so on. Fourth-order difference equations derived from various discrete elastic beam problems. In this paper, we seek further study of the multiplicity results for discrete fourth-order boundary value problems with four parameters. In fact, using a consequence of the local minimum theorem due Bonanno we look for the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for our problem in a special case.