Coupled systems with Ambrosetti-Prodi-type differential equations

Abstract

<abstract><p>In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.</p></abstract>

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

“…The first theorem is an existence and localization result that is a particular case of Theorem 3.1 of [22]. In short, it guarantees the existence of a solution for the values of µ and λ such that there are lower and upper solutions of Problems (1) and (2).…”

“…The a priori estimate for the first derivatives is given by the next lemma following the arguments of [22].…”

“…By Lemma 1, and applying the technique suggested in the proof of Theorem 1 (see [22], Theorem 3.1) adapted to strict lower and upper solutions γ i (t), and φ i (t), there are positive real numbers M i , where i = 1, 2 such that u 1 < M 1 , and u 2 < M 2 , independently of the parameters θ and ϑ.…”

“…Recently, in [22], the authors presented a technique to discuss the existence of coupled systems of two Ambrosetti-Prodi-type second-order fully differential equations and proved the existence of solutions for the values of the parameters for which there are lower and upper solutions for the system. This paper extends, for the first time, as far as we know, the Ambrosetti-Prodi alternative to coupled systems of differential equations with two parameters.…”

“…This paper extends, for the first time, as far as we know, the Ambrosetti-Prodi alternative to coupled systems of differential equations with two parameters. The existence and nonexistence of solutions are obtained for Problems (1) and ( 2), and a discussion of multiplicity for particular cases of the boundary conditions are considered in [22].…”