Upper and lower solution method for nth-order BVPs on an infinite interval

Hairong Lian,Ravi P Agarwal,Junfang Zhao

doi:10.1186/1687-2770-2014-100

Abstract

This work is devoted to the study of n th-order ordinary differential equations on a half-line with Sturm-Liouville boundary conditions. The existence results of a solution, and triple solutions, are established by employing a generalized version of the upper and lower solution method, the Schäuder fixed point theorem, and topological degree theory. In our problem the nonlinearity depends on derivatives, and we allow solutions to be unbounded, which is an extra interesting feature.MSC:34B15, 34B40.

Highlights

1 Introduction In this paper, we study nth-order ordinary differential equations on a half-line
Infinite interval problems occur in the study of radially symmetric solutions of nonlinear elliptic equations; see [, ]
Blasius-type equations lead to infinite interval problems

Summary

Introduction

We study nth-order ordinary differential equations on a half-line,. ⎪⎪⎩uu((nn–– ))((+ ∞) –)a=uC(n–, )( ) = B, where q : ( , +∞) → ( , +∞), f : [ , +∞) × Rn → R are continuous, a > , Ai, B, C ∈ R, i = , , . . . , n – , u(+∞) = limt→+∞ u(t). Higher-order boundary value problems (BVPs) have been studied in many papers, such as [ – ] for two-point BVP, [ , ] for multipoint BVP, and [ – ] for infinite interval problem. Most of these works have been done either on finite intervals, or for bounded solutions on an infinite interval. The authors in [ , , , – ] assumed one pair of well-ordered upper and lower solutions, and applied some fixed point theorems or a monotone iterative technique to obtain a solution. When applying the upper and lower solution method to infinite interval problems, the solutions are always assumed to be bounded. An example is included which illustrates the main result

Consider the space X defined by

Consider the modified differential equation with the truncated function