Nontrivial solutions of inverse discrete problems with sign-changing nonlinearities

Alberto Cabada,Nikolay D Dimitrov

doi:10.1186/s13662-019-2383-y

Abstract

This paper is concerned with the existence of solutions of an inverse discrete problem with sign-changing nonlinearity. This kind of problems includes, as a particular case, nth order difference equations coupled with suitable conditions on the boundary of the interval of definition. It would be valid for the case in which the related Green’s function is positive on a subset of its rectangle of definition.The existence results follow from spectral theory, as an application of the Krein–Rutman theorem and by means of degree theory.

Highlights

During the last years, many authors discussed the existence of solutions for boundary value problems by using various topological methods
We refer the reader to [9, 11, 12], where the authors used topological methods to deduce, under a similar hypothesis, the existence results to a discrete fractional semipositone boundary value problem
A similar idea can be found in a very recent paper [18], where under suitable conditions concerning the first eigenvalue corresponding to the relevant linear problem, the authors established the existence of nontrivial solutions for boundary value problems of the following fourth order difference equation with a sign-changing nonlinearity:

Summary

Introduction

Many authors discussed the existence of solutions for boundary value problems by using various topological methods. In [2] the authors studied the following problem: We refer the reader to [9, 11, 12], where the authors used topological methods to deduce, under a similar hypothesis, the existence results to a discrete fractional semipositone boundary value problem.

Objectives

Results

Conclusion