Positive solutions of BVPs on the half-line involving functional BCs

,Gennaro Infante,Serena Matucci

doi:10.3934/math.2021285

Abstract

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations.

Highlights

In this manuscript we discuss the existence of multiple non-negative solutions of the boundary value problem (BVP)
The functional formulation of the boundary conditions (BCs) covers, as special cases, the interesting setting of multi-point and integral BCs; there exists a wide literature on this topic, we refer the reader to the recent paper [11] and references therein
Unlike the above cited articles, in which the problem of gluing the solutions is solved with some continuity arguments and an analysis in the phase space, here both the auxiliary problems have the same slope condition in the junction point (namely the condition u′(R) = 0), which simplifies the arguments. This kind of decomposition is some sort of an analogue of one employed by Boucherif and Precup [2] utilized for equations with nonlocal initial conditions, where the associated nonlinear integral

Summary

Introduction

In this manuscript we discuss the existence of multiple non-negative solutions of the boundary value problem (BVP). The problem of the existence and multiplicity of the solutions for the equation in (1.1), which are non-negative in the interval [0, R] and satisfy the functional BCs and the additional assumptions at u′(R) = 0, is considered in Section 2 and is solved by means of the classical fixed point index for compact maps. The approach used in [9] to solve the BVP was based on a combination of the Schauder’s (half)-linearization device, a fixed point result in the Frechet space of continuous functions on [R, +∞), and comparison results for principal and nonprincipal solutions of suitable auxiliary half-linear equations, which allow to find good upper and lower bounds for the solutions of the (half)-linearized problem. Every solution of (3.1) is nonincreasing on the whole half-line

The main result

Methods