Asymptotic solutions for a second-order differential equation with three-point boundary conditions

Abstract

In this paper, we are concerned with the following singularly perturbed second-order three-point boundary value problem

(1.1)$$y^{″}+({\lambda }^2{q}_1(x)+q_2(x))y=0\text{,}\lambda \gg 1\text{,}$$

(1.2)$$y(0)=A\text{,}y(1)=\alpha y(\eta )\text{,}$$

where 0 < η < 1,A, α ∈ R, λ is a parameter, q₁ ∈ C²[0, 1],q₂ ∈ C[0, 1]. The asymptotic solutions of the problem are given by using the method of nonlinear analysis and Liouville–Green transform.

Introduction

This paper deals with the asymptotic solutions to the following singularly perturbed second-order three-point boundary value problem (BVP, for short)

$$y^{″}+({\lambda }^2{q}_1(x)+q_2(x))y=0\text{,}\lambda \gg 1\text{,}$$

$$y(0)=A\text{,}y(1)=\alpha y(\eta )\text{,}$$

where 0 < η < 1,A, α ∈ R, λ is a parameter, q₁ ∈ C²[0, 1],q₂ ∈ C[0, 1].

For the past few years multi-point boundary value problem has received a wide attention, for example, to see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and the reference therein. Ma [1] employed fixed-point index theorems, Leray–Schauder degree and upper and lower solutions to consider second-order three-point boundary value problem

$$u^{″}(t)+\lambda h(t)f(u(t))=0\text{,}t\in (0\text{,}1)\text{,}$$

$$u(0)=0\text{,}u(1)=\xi u(\eta )\text{,}$$

where 0 < η < 1, 0 < ξη < 1. In the case λ = 1, Ma [2], Anderson [3], Liu [4], respectively, studied the existence of positive solutions of BVP (1.3), (1.4) by using Krasnoselskii’s fixed-point theorems, Leggett–Williams fixed-point theorems and fixed-point index theorem. In [5], by using Krasnoselskii’s fixed-point theorem, Liu considered the existence of single and multiple positive solutions to (1.3) for the case λ = 1 with boundary condition y′(0) = 0, y(1) = βy(η). In [6], some conditions on f and two pairs of lower and upper solutions were assumed to ensure the existence of at least three solutions of the following second-order three-point boundary value problem

$$\begin{array}{cc}\hfill & u^{″}(t)+f(t\text{,}u(t)\text{,}{u}^{\prime }(t))=0\text{,}t\in (0\text{,}1)\text{,}\hfill \\ \hfill & u(0)=0\text{,}u(1)=\xi u(\eta )\text{.}\hfill \end{array}$$

In [7], by applying Leray–Schauder degree theory and differential inequalities theory, we studied the existence, uniqueness and asymptotic estimates of solutions for a third-order multi-point singularly perturbed boundary value problem. In [8], Elias and Gingold studied periodic solutions of the following autonomous singularly perturbed differential equations with initial values u″ + f(x, p) = 0, u(0) = α, u′(0) = β, where p → ∞ is a parameter.

However, in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], the authors obtained the existence of solutions. By so far, very few asymptotic solutions were established for multi-point boundary value problem. In this paper, we mainly apply the method of nonlinear analysis and Liouville–Green transform [14], [15] to study BVP (1.1), (1.2). Under the very simple restricted conditions q₁ ∈ C²[0, 1], q₂ ∈ C[0, 1], the asymptotic solutions of BVP (1.1), (1.2) are obtained.