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# Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance

- Juan J Nieto
^{1, 2}Email author

**2013**:130

https://doi.org/10.1186/1687-2770-2013-130

© Nieto; licensee Springer 2013

**Received:**20 February 2013**Accepted:**30 April 2013**Published:**20 May 2013

## Abstract

We present a new existence result for a second-order nonlinear ordinary differential equation with a three-point boundary value problem when the linear part is noninvertible.

**MSC:**34B10, 34B15.

## Keywords

- nonlinear ordinary differential equation
- three-point boundary value problem
- problem at resonance
- existence of solution

## 1 Introduction

The study of multi-point boundary value problems for linear second-order ordinary differential equations goes back to the method of separation of variables [1]. Also, some questions in the theory of elastic stability are related to multi-point problems [2]. In 1987, Il’in and Moiseev [3, 4] studied some nonlocal boundary value problems. Then, for example, Gupta [5] considered a three-point nonlinear boundary value problem. For some recent works on nonlocal boundary value problems, we refer, for example, to [6–15] and references therein.

As indicated in [16], there has been enormous interest in nonlinear perturbations of linear equations at resonance since the seminal paper of Landesman and Lazer [17]; see [18] for further details.

where $T>0$, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is a continuous function $\alpha \in \mathbb{R}$ and $\eta \in (0,T)$.

In this paper we consider the resonance case $\alpha \eta =T$ to obtain a new existence result. Although this situation has already been considered in the literature [19], we point out that our approach and methodology is different.

## 2 Linear problem

for a given function $\sigma \in C[0,T]$.

with ${c}_{1}$, ${c}_{2}$ arbitrary constants.

### 2.1 Nonresonance case

*i.e.*, when $\sigma =0$, ${c}_{1}={c}_{2}=0$ and $u=0$. Note that the solution is given by

For $T=1$ this is precisely the function given in Lemma 2.3 of [20] or in Remark 12 of [21].

### 2.2 Resonance case

*ct*, $c\in \mathbb{R}$ is a solution of the homogeneous linear equation

We note that $k\in C([0,T]\times [0,T],\mathbb{R})$ and $k(t,s)\ge 0$ for every $(t,s)\in [0,T]\times [0,T]$.

## 3 Nonlinear problem

where $N=K\circ F+L$.

This suggests to introduce the new function $v(t)=u(t)-\frac{t}{T}u(T)$. To find a solution *u*, we have to find *v* and $u(T)$.

For $c\in \mathbb{R}$ fixed, we try to solve the integral equation (7).

for every $t\in [0,T]$, $u\in \mathbb{R}$.

Thus, a solution of (7) is precisely a fixed point of $K\circ {F}_{c}={K}_{c}$. Note that ${K}_{c}$ is a compact operator. For $v\in C[0,T]$, let $\parallel v\parallel ={sup}_{t\in [0,T]}|v(t)|$.

for any $v\in C[0,T]$ and $\lambda \in (0,1)$ solution of $v=\lambda {K}_{c}(v)$. This implies that *v* is bounded independently of $\lambda \in (0,1)$, and hence by Schaefer’s fixed point theorem (Theorem 4.3.2 of [22]), ${K}_{c}$ has at least a fixed point, *i.e.*, for given *c*, equation (7) is solvable.

Now suppose *f* is Lipschitz continuous.

for every $t\in [0,T]$ and $x,y\in \mathbb{R}$.

Thus, for $l>0$ small, equation (7) has a unique solution in view of the classical Banach contraction fixed point theorem.

where ${v}_{c}$ is the unique solution of (7), and as a consequence of the contraction principle, this map is continuous.

*c*we have ${v}_{c}(T)$, and the function

is such that ${u}_{c}(T)=c$, and therefore ${u}_{c}$ is a solution of the original nonlinear problem (1).

uniformly on $t\in [0,T]$.

*c*growths linearly. Hence the norm of the function

growths asymptotically as *c*.

This implies that ${lim}_{c\to \pm \mathrm{\infty}}\phi (c)=\pm \mathrm{\infty}$, and there exists $c\in \mathbb{R}$ with $\phi (c)=0$.

We have the following result.

**Theorem 3.1** *Suppose that* *f* *satisfies the growth conditions* (8) *and* (10). *If* (11) *holds*, *then* (1) *is solvable for* *l* *sufficiently small*.

Note that condition (11) is crucial since for $f(t,u)=\sigma (t)$ and, in view of (5), the problem (1) may have no solution.

## Declarations

### Acknowledgements

This research has been partially supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER. The author is thankful to the referees for their useful suggestions.

## Authors’ Affiliations

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