Periodic Third-Order Problems with a Parameter

Feliz Minhós,Nuno Oliveira

doi:10.3390/axioms10030222

Abstract

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters.

Highlights

In this paper we consider a third-order periodic problem composed by the differential equation u (t) + f t, u(t), u (t), u (t) = s g(t), t ∈ [0, 1], (1)
Ambrosetti–Prodi results have been obtained for different types of boundary value problems, such as with separated boundary conditions [2,3,4], Neuman’s type [5], three-point boundary conditions [6], among others
Motivated by the above papers, we present in this work a first approach for third-order periodic fully differential equations, where the existence of periodic solutions depends on a weighted parameter, as in (1)

Summary

Introduction

We underline that the nonlinearity must verify only a local monotone assumption and no periodic, coercivity or super and/or sublinearity conditions are assumed, as usual in the literature Remark that, it will remain open the issue of what are the sufficient conditions on the nonlinearity to have the non-existence and the multiplicity of solutions, depending on s. We consider a reactiondiffusion linear system for the thyroid-pituitary interaction, which is translated by a nonlinear third-order differential equation In this case the role of the parameter s is played by some coefficients with biological and chemical meaning, which ensuring the existence of periodic catatonia phenomena. This application take advantage from the localization part of the main theorem, to show that the periodic solutions are not trivial. Last section discuss the existence of periodic catatonic episodes based on some relations of certain coefficients, considered as parameters

Definitions and a Priori Estimations

Existence Result

Periodic Catatonic Phenomena with a Parameter