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
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.
Here we study the following nonlinear ordinary differential equation of second order subject to the three-point boundary condition:
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.
2 Linear problem
Consider the linear second-order three-point boundary value problem
for a given function .
The general solution is
with , arbitrary constants.
From , we get . From the second boundary condition, we have
2.1 Nonresonance case
If , then
and the linear problem (2) has a unique solution for any . 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 , and . Note that the solution is given by
2.2 Resonance case
If , 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, is a solution of the homogeneous linear equation
satisfying the boundary conditions
We now use that to get
Hence the solution of (2) is given, implicitly, as
We note that and for every .
3 Nonlinear problem
Defining the operators:
the nonlinear problem is equivalent to
We note that (6) can be written as
and the nonlinear problem (1) as
This suggests to introduce the new function . To find a solution u, we have to find v and .
For every constant , we solve
and let 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 , we consider
If , then is a solution of the nonlinear problem (1). We then look for fixed points of the map
For fixed, we try to solve the integral equation (7).
Assume that there exist and such that
for every , .
For , define as
Thus, a solution of (7) is precisely a fixed point of . Note that is a compact operator. For , let .
For , if we have
Hence there exist constants , such that
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.
Then there exists such that
for every and .
Then, for , we have
Thus, for 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 is the unique solution of (7), and as a consequence of the contraction principle, this map is continuous.
Define the map
If there exists such that , then for that c we have , and the function
is such that , and therefore is a solution of the original nonlinear problem (1).
Now, assume that
uniformly on .
Then the growth of is sublinear in view of estimate (9). However, c growths linearly. Hence the norm of the function
growths asymptotically as c.
This implies that , and there exists with .
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 and, in view of (5), the problem (1) may have no solution.
The author declares that he has no competing interests.
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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