Nontrivial solutions for boundary value problems of a fourth order difference equation with sign-changing nonlinearity

Abstract

In this paper, using the topological degree theory, we establish two existence theorems for nontrivial solutions for boundary value problems of a fourth order difference equation with a sign-changing nonlinearity.

Nontrivial solutions of inverse discrete problems with sign-changing nonlinearities

“…The results are based on the topological degree theory and generalize some previous results obtained for this problem. It is important to point out that our results generalize some ones given in that reference, in fact, as we will see in Example 9, Theorems 3.2 and 3.1 in [18] are particular cases (respectively) of Theorems 5 and 6 in this work.…”

“…This section is devoted to proving the existence of a nontrivial solution of Problem (2) which, as we have noted in previous sections, is equivalent to finding a fixed point of operator T defined on (3). The proofs follow similar arguments to the ones developed by Zhang, O'Regan, and Fu in [18] for equation (1). So, we present some assumptions about the nonlinearity f : (H0) f :…”

“…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:…”

Positive solutions for a class of fractional difference systems with coupled boundary conditions

“…They used the Guo-Krasnosel'skii fixed point theorem to obtain the existence of positive solutions for the above two problems, where the nonlinearities in (1.5) can be sign-changing. Motivated by works aforementioned and some results from integer-order equations (including differential and difference equations, see [45][46][47][48][49][50][51][52][53][54][55]), we study the existence of positive solutions for the fractional difference systems (1.1). We use the fixed point index theory to establish our main results based on a priori estimates achieved by utilizing nonnegative matrices (see [10,54,55]) that involve some useful inequalities associated with the Green's functions for (1.1).…”

Multiplicity Results to a Conformable Fractional Differential Equations Involving Integral Boundary Condition

“…In recent years, fractional differential equations have exerted tremendous influence on some mathematical models of research processes and phenomena in many fields such as electrochemistry, heat conduction, underground water flow, and porous media. A growing number of papers deal with the existence or multiplicity of solutions of initial value problem and boundary value problem for fractional differential equations [1][2][3][4][5][6][7][8][9][10][11]. Recently, the authors [12] give an interesting fractional derivative called the "conformal fractional derivative", which depends on the limit definition of the function derivative.…”