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Discrete fourth-order boundary value problems with four parameters. (English) Zbl 1428.46027

Summary: 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 to Averna and Bonanno, and another one due to Bonanno, we guarantee the existence of two and three solutions for our problem in a special case.

MSC:

46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
34B15 Nonlinear boundary value problems for ordinary differential equations
35A08 Fundamental solutions to PDEs
35B20 Perturbations in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
39A70 Difference operators
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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