Abstract
We present a new existence result for a secondorder nonlinear ordinary differential equation with a threepoint boundary value problem when the linear part is noninvertible.
MSC: 34B10, 34B15.
Keywords:
nonlinear ordinary differential equation; threepoint boundary value problem; problem at resonance; existence of solution1 Introduction
The study of multipoint boundary value problems for linear secondorder 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 multipoint problems [2]. In 1987, Il’in and Moiseev [3,4] studied some nonlocal boundary value problems. Then, for example, Gupta [5] considered a threepoint nonlinear boundary value problem. For some recent works on nonlocal boundary value problems, we refer, for example, to [615] 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 threepoint boundary condition:
where
In this paper we consider the resonance case
2 Linear problem
Consider the linear secondorder threepoint boundary value problem
for a given function
The general solution is
with
From
2.1 Nonresonance case
If
and the linear problem (2) has a unique solution for any
with
For
2.2 Resonance case
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,
satisfying the boundary conditions
Note that
and then
We now use that
and
Hence the solution of (2) is given, implicitly, as
or, equivalently,
where
We note that
3 Nonlinear problem
Defining the operators:
the nonlinear problem is equivalent to
where
We note that (6) can be written as
and the nonlinear problem (1) as
This suggests to introduce the new function
For every constant
and let
and hence
If
For
Assume that there exist
for every
For
Thus, a solution of (7) is precisely a fixed point of
For
and
Hence there exist constants
for any
Now suppose f is Lipschitz continuous.
Then there exists
for every
Then, for
and
Thus, for
Now, under conditions (8) and (10), set
where
Define the map
If there exists
is such that
Now, assume that
uniformly on
Then the growth of
growths asymptotically as c.
This implies that
We have the following result.
Theorem 3.1Suppose thatfsatisfies the growth conditions (8) and (10). If (11) holds, then (1) is solvable forlsufficiently small.
Note that condition (11) is crucial since for
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This research has been partially supported by Ministerio de Economía y Competitividad (Spain), project MTM201015314, and cofinanced by the European Community fund FEDER. The author is thankful to the referees for their useful suggestions.
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