Higher-Order Functional Discontinuous Boundary Value Problems on the Half-Line

Feliz Minhós,Infeliz Coxe

doi:10.3390/math9050499

Abstract

In this paper, we consider a discontinuous, fully nonlinear, higher-order equation on the half-line, together with functional boundary conditions, given by general continuous functions with dependence on the several derivatives and asymptotic information on the (n−1)th derivative of the unknown function. These functional conditions generalize the usual boundary data and allow other types of global assumptions on the unknown function and its derivatives, such as nonlocal, integro-differential, infinite multipoint, with maximum or minimum arguments, among others. Considering the half-line as the domain carries on a lack of compactness, which is overcome with the definition of a space of weighted functions and norms, and the equiconvergence at ∞. In the last section, an example illustrates the applicability of our main result.

Highlights

IntroductionLn−1 : C ([0, +∞[) × R → R continuous functions
Ln−1 : C ([0, +∞[) × R → R continuous functions. These types of higher-order boundary value problems have been considered by many authors, with a general higher-order derivative n, and for particular cases of n
They are studied for continuous nonlinearities, and in bounded intervals, with classical boundary conditions, such as [1,2], for linear problems [3,4], for two-point separated and Sturm-Liouville boundary conditions [5,6,7], for multipoint problems [8], and for periodic solutions, among others

Summary

Introduction

Ln−1 : C ([0, +∞[) × R → R continuous functions These types of higher-order boundary value problems have been considered by many authors, with a general higher-order derivative n, and for particular cases of n. For n = 2, we mention an industrial micro-engineering problem to study a membrane MEMS device via an elliptic semilinear 1D model, referred to in [28] Another possible application for higher-order problems defined on unbounded intervals is, for n = 4, the study of the bending of infinite beams with different types of foundations, as can be seen, for example, in [29,30,31,32]. The last section is concerned with a numerical example subject to global boundary conditions

Definitions and Auxiliary Results

Main Result

Example

Conclusions