Feliz Minhós

A discrete fourth-order Lidstone problem with parameters

Various existence, multiplicity, and nonexistence results for nontrivial solutions to a nonlinear discrete fourth-order Lidstone boundary value problem with dependence on two parameters are given, using a symmetric Greens function approach. An existence result is also given for a related semipositone problem, thus relaxing the usual assumption of nonnegativity on the nonlinear term.

Nonlocal boundary value problems

In the last decades, nonlocal boundary value problems have become a rapidly growing area of research. The study of this type of problems is driven not only by a theoretical interest, but also by the fact that several phenomena in engineering, physics and life sciences can be modelled in this way. For example, problems with feedback controls such as the steady-states of a thermostat, where a controller at one of its ends adds or removes heat, depending upon the temperature registered in another point, can be interpreted with a second-order ordinary differential equation subject to a three-point boundary condition.

Sufficient conditions for the existence of heteroclinic solutions for -Laplacian differential equations

In this paper, we consider the second-order discontinuous equation on the real line, with an increasing homeomorphism such that and , with or for an -Carathéodory function and such that We point out that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the non-linearities and f. Moreover, as far as we know, this result is even new when that is, for the equation

Unbounded Solutions for Functional Problems on the Half-Line

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green’s functions, and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.

Higher order functional boundary value problems without monotone assumptions

AbstractIn this paper, given f:[a,b]×(C([a,b]))n−2×R2→R a L1-Carathéodory function, it is considered the functional higher order equation u(n)(x)=f(x,u,u′,…,u(n−2)(x),u(n−1)(x)) together with the nonlinear functional boundary conditions, for i=0,…,n−2 Here, Li, i=0,…,n−1, are continuous functions. It will be proved an existence and location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearity f.

Existence result for impulsive coupled systems on the half-line

This work considers a second order impulsive coupled system of differential equations with generalized jump conditions in half-line, which can depend on the impulses of the unknown functions and their first derivatives. The arguments apply the fixed point theory, Green’s functions technique,

Location results: an under used tool in higher order boundary value problems

L^{1}

Solvability of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels☆

L1-Carathéodory functions and sequences and Schauder’s fixed point theorem. The method is based on Carathéodory concept of functions and sequences, together with the equiconvergence on infinity and on each impulsive moment, and it allows to consider coupled fully nonlinearities and very general impulsive functions.