Multiple unbounded solutions for a boundary value problem on infinite intervals

Abstract

This paper is concerned with the existence of multiple unbounded solutions for a Sturm-Liouville boundary value problem on the half-line. By assuming the existence of two pairs of unbounded upper and lower solutions, the existence of at least three solutions is obtained using the degree theories. Nagumo condition plays an important role in the nonlinear term involved in the first-order derivative. It is an interesting point that the method of unbounded upper and lower solutions is extended to obtain conditions for the existence of multiple solutions. Mathematics Subject Classification (2000 34B10, 34B40

The lower and upper solution method for three-point boundary value problems with integral boundary conditions on a half-line

“…These solutions may be unbounded in this paper. Multi-point boundary value problems for second-order differential equations in a finite interval and on an infinite interval included the large amount of priori work and many excellent results are obtained by using Avery-Peterson fixed point theorem, shooting method, lower and upper solution method, Leray-Schauder continuation theorem and so on, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13]15]. Meanwhile, BVPs with integral boundary conditions for ordinary differential equations have been extensively examined by many authors, for example see [11][12][13][14][15][16].…”

“…In [5], Lian and Geng examined Sturm-Liouville boundary value problem on a half-line: u (t) + φ(t) f (t, u(t), u (t)) = 0, t ∈ (0, +∞), u(0) − au (0) = B, u (+∞) = C,…”

Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

“…Boundary value problems on the half-line arise in the study of radially symmetric solutions of nonlinear elliptic equations and various physical phenomena, such as the theory of drain flows and plasma physics; see [1][2][3][4][5][6][7][8][9][10] and the references therein. In 2006, Lian and Ge in [11] investigated the following boundary value problem on the half-line for the second-order differential equation:…”

Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

“…Finally, a three solutions result in terms of couple of lower and upper slopes, in the spirit of Amann's pioneering result for abstract equations in ordered spaces [1], and Dirichlet boundary problems for elliptic equations [2], and of Rachůnková for periodic solutions of ordinary differential equations [17], [16], is stated and proved in Section 7 (Theorem 7.1). See also [13].…”