- Open Access
Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance
© Nieto; licensee Springer 2013
- Received: 20 February 2013
- Accepted: 30 April 2013
- Published: 20 May 2013
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.
- nonlinear ordinary differential equation
- three-point boundary value problem
- problem at resonance
- existence of solution
The study of multi-point boundary value problems for linear second-order ordinary differential equations goes back to the method of separation of variables . Also, some questions in the theory of elastic stability are related to multi-point problems . In 1987, Il’in and Moiseev [3, 4] studied some nonlocal boundary value problems. Then, for example, Gupta  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.
where , is a continuous function and .
In this paper we consider the resonance case to obtain a new existence result. Although this situation has already been considered in the literature , we point out that our approach and methodology is different.
for a given function .
with , arbitrary constants.
2.1 Nonresonance case
2.2 Resonance case
We note that and for every .
This suggests to introduce the new function . To find a solution u, we have to find v and .
For fixed, we try to solve the integral equation (7).
for every , .
Thus, a solution of (7) is precisely a fixed point of . Note that is a compact operator. For , let .
for any and solution of . This implies that v is bounded independently of , and hence by Schaefer’s fixed point theorem (Theorem 4.3.2 of ), has at least a fixed point, i.e., for given c, equation (7) is solvable.
Now suppose f is Lipschitz continuous.
for every and .
Thus, for small, equation (7) has a unique solution in view of the classical Banach contraction fixed point theorem.
where is the unique solution of (7), and as a consequence of the contraction principle, this map is continuous.
is such that , and therefore is a solution of the original nonlinear problem (1).
uniformly on .
growths asymptotically as c.
This implies that , and there exists with .
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 and, in view of (5), the problem (1) may have no solution.
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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