Positive solutions of singular boundary value problems on the half-line

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Abstract

In this paper, by using the fixed point theory in the cone with a special norm and space, the existence of positive solutions for a class of singular boundary value problems on the half-line is established. Our results improve many known results including singular and non-singular cases.

Introduction

In this paper, we are concerned with the existence of positive solutions for the following nonlinear singular boundary value problems (BVP) on the half-line:(p(t)x(t))+λ(f(t,x(t))-k2x(t))=0,t(0,+),α1x(0)-β1limt0+p(t)x(t)=0,α2limt+x(t)+β2limt+ftyp(t)x(t)=0,where λ > 0 is a parameter, k  (−∞, +∞), f : (0, +∞) × [0, +∞)  [0, +∞) is a continuous function and may be singular at t = 0; p  C[0, +∞)   C1(0, +∞) with p > 0 on (0, +∞), 01p(s)ds<+; αi, βi  0 (i = 1, 2) with ρ = α2β1 + α1β2 + α1α2B(0, ∞) > 0 in which B(t,s)=ts1p(v)dv.

Boundary value problems on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [1], [2], [4], [5]. The boundary value problem, for some special case where f is continuous at t = 0, has been extensively studied by many authors [6], [8], [9], [10], [11], [12]. Zima [6] studied the existence of positive solutions of a BVP on the half-line for the following second-order differential equations with no singularity:x(t)-k2x(t)+f(t,x(t))=0,t(0,+),x(0)=0,limt+x(t)=0,where k > 0, f:[0, +∞) × [0, +∞)  [0, +∞) is a non-negative continuous function, and f(t, x)   a(t) + b(t)x for (t, x)  [0, +∞) × [0, +∞), in which a, b:[0, +∞)  [0, +∞) are continuous functions. Recently, Hao et al. [8] established the existence theorems of positive solutions for the following equations on the half-line:x(t)-k2x(t)+m(t)f(t,x(t))=0,t(0,+),x(0)=0,limt+x(t)=0,where f:[0, +∞) × [0, +∞)  [0, +∞) is continuous and sup{f(t, x):(t, x)  [0, +∞) × [0, +∞)} < +∞, m:(0, +∞)  [0, +∞) is continuous and may be singular at t = 0. Very recently, in [9], with k = 0, Lian and Ge obtained the existence of at least one positive solution for BVP (1.1) by using the Krasnosel’skii fixed point theorem.

Motivated by the above work, we consider the existence of positive solutions to singular BVP (1.1). The problem we discuss is different from those in [6], [8], [9]. Firstly, our study is on singular nonlinear differential equations on the half-line with general boundary conditions. Secondly, BVP (1.1) involves a parameter λ. Finally, the techniques used in this paper are the approximation method, and a special cone in a special space is established in order to overcome the difficulties caused by singularity and infinite interval and to apply the fixed point theorem in cone.

The main purpose of this paper is to obtain the existence of positive solutions for BVP (1.1). Usually, we solve the positive fixed points of an integral operator T instead of the positive solutions of BVP (1.1). In this paper we obtain the positive solutions on [0, +∞), which expands the domain of definition of t from finite interval to infinite interval. The main difficulty is to testify that the operator T is completely continuous, since we can not use the Ascoli-Arzela theorem in infinite interval [0, +∞). Some modification of the compactness criterion in infinite interval [0, +∞) (see Lemma 2.2) can help to resolve this problem.

The rest of the paper is organized as follows. In Section 2, we present some necessary lemmas that will be used to prove our main results. In Section 3, first, we give Lemma 3.1 which is a result about completely continuous operator. Then we discuss the existence of at least one positive solution for the BVP (1.1).

Section snippets

Preliminaries and lemmas

In this section, we present some notations and lemmas that will be used in the proof of our main results.

Lemma 2.1

[9]

Under the condition 01p(s)ds<+ and ρ > 0, BVP(p(t)u(t))+v(t)=0,0<t<+,α1u(0)-β1limt0+p(t)u(t)=0,α2limt+u(t)+β2limt+ftyp(t)u(t)=0,has a unique solution for any v  L(0, +). Moreover, this unique solution can be expressed in the formu(t)=0G(t,s)v(s)ds,where G(t, s) is defined byG(t,s)=1ρ(β1+α1B(0,s))(β2+α2B(t,)),0st<+,(β1+α1B(0,t))(β2+α2B(s,)),0ts<+.

Remark 2.1

From (2.2), it is easy to

Main results

Firstly, we present two basic assumptions which are to be used in our lemma and theorems.

(H1) f:(0, +∞) × [0, +∞)  [0, +∞) is a continuous function and satisfies k2u  f(t, u)   ϕ(t)g(t, u) for any (t, u)  (0, +∞) × [0, +∞), where ϕ:(0, +∞)  [0, +∞) is continuous and singular at t = 0, ϕ(t)  0 on [0, +∞), g:[0, +∞) × [0, +∞)  [0, +∞) is continuous and bounded for 0  t < +∞ and u in any bounded set of [0, +∞).

(H2) 0G(s,s)ϕ(s)ds<+.

From the above assumptions, we can define an integral operator T:K  X by(Tx)(t)=λ0G(t,s)(f(s,x(s)

Acknowledgements

The first and second authors were supported financially by the National Natural Science Foundation of China (10771117) and the State Ministry of Education Doctoral Foundation of China (20060446001). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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