Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

Lishan Liu,Yonghong Wu,Xinan Hao

doi:10.1155/2010/236560

Lishan Liu, Yonghong Wu + Show 1 more

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https://doi.org/10.1155/2010/236560

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Abstract

This paper investigates the second-order multipoint boundary value problem on the half-line ,, , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

Highlights

In this paper, we consider the following second-order multipoint boundary value problem on the half-line u t f t, u t, u t 0, t ∈ R, n1.1 αu 0 − βu 0 − kiu ξi a ≥ 0, i1 lim u t b > 0, t→ ∞where α > 0, β > 0, ki ≥ 0, 0 < ξ1 < ξ2 < · · · < ξn < ∞, and f : R × R × R → R is continuous, in which R 0, ∞, R −∞, ∞ .The study of multipoint boundary value problems BVPs for second-order differential equations was initiated by Bicadze and Samarskı 1 and later continued by II’in and Boundary Value ProblemsMoiseev 2, 3 and Gupta 4
Many results on the existence of positive solutions for multi-point BVPs have been obtained, and for more details the reader is referred to 5–10 and the references therein
When n 1, β 0, a b 0, BVP 1.1 reduces to the following three-point BVP on the half-line: u t f t, u t, u t 0, t ∈ 0, ∞, 1.2 u 0 αu η, lim u t 0, t→ ∞

Summary

Introduction

We consider the following second-order multipoint boundary value problem on the half-line u t f t, u t , u t 0, t ∈ R , n. Many results on the existence of positive solutions for multi-point BVPs have been obtained, and for more details the reader is referred to 5–10 and the references therein. N, and nonlinearity f is variable separable, BVP 1.1 reduces to the second order two-point BVP on the half-line uΦtft, u, u 0, t ∈ 0, ∞ , 1.3 au 0 − bu 0 u0 ≥ 0, lim u t k > 0. Motivated by the above works, we will study the existence results of unbounded positive solution for second order multi-point BVP 1.1. We formulate two examples to illustrate the main results

Preliminaries and Lemmas

Existence Result

Existence and Uniqueness of Positive Solution

Examples