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

*Boundary Value Problems*
**volume 2013**, Article number: 130 (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.

## 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.

Here we study the following nonlinear ordinary differential equation of second order subject to the three-point boundary condition:

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

Consider the linear second-order three-point boundary value problem

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

The general solution is

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

From u(0)=0, we get {c}_{1}=0. From the second boundary condition, we have

### 2.1 Nonresonance case

If \alpha \eta \ne T, then

and the linear problem (2) has a unique solution for any \sigma \in C[0,T]. In this case, we say that (2) is a nonresonant problem since the homogeneous problem has only the trivial solution as a solution, *i.e.*, when \sigma =0, {c}_{1}={c}_{2}=0 and u=0. Note that the solution is given by

with

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

If T=\alpha \eta, then (3) is solvable if and only if

and then (2) has a solution if and only if (5) holds. In such a case, (2) has an infinite number of solutions given by

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

satisfying the boundary conditions

Note that

and then

We now use that u(T)=\frac{T}{\eta}u(\eta ) to get

and

Hence the solution of (2) is given, implicitly, as

or, equivalently,

where

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

Defining the operators:

the nonlinear problem is equivalent to

where N=K\circ F+L.

We note that (6) can be written as

and the nonlinear problem (1) as

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 every constant c\in \mathbb{R}, we solve

and let \phi (c) be the set of solutions of (7). This set may be empty (no solution), a singleton (unique solution) or with more than one element (multiple solutions). For every {v}_{c}\in \phi (c), we consider

and hence

If c={u}_{c}(T), then {u}_{c} is a solution of the nonlinear problem (1). We then look for fixed points of the map

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

Assume that there exist a,b\in C[0,T] and \alpha \in [0,1) such that

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

For v\in C[0,T], define {F}_{c}v\in C[0,T] as

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 \lambda \in (0,1), if v=\lambda {K}_{c}(v) we have

and

Hence there exist constants {a}_{0}, {b}_{0} such that

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.

Then there exists l>0 such that

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

Then, for v,w\in C[0,T], we have

and

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

Now, under conditions (8) and (10), set

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

Define the map

If there exists c\in \mathbb{R} such that \phi (c)=0, then for that *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).

Now, assume that

uniformly on t\in [0,T].

Then the growth of \parallel v\parallel is sublinear in view of estimate (9). However, *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.

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## 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.

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Nieto, J.J. Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance.
*Bound Value Probl* **2013**, 130 (2013). https://doi.org/10.1186/1687-2770-2013-130

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