Abstract
A Fredholmtype theorem for boundary value problems for systems of nonlinear functional differential equations is established. The theorem generalizes results known for the systems with linear or homogeneous operators to the case of systems with positively homogeneous operators.
MSC: 34K10.
Keywords:
functionaldifferential equations; boundary value problems; existence of solutions1 Statement of the problem
Consider the system of functionaldifferential equations
together with the boundary conditions
Here,
and
Furthermore, we assume that
Remark 1 From the abovestated assumptions it follows that
In the case when
In 1966, Lasota established the Fredholmtype theorem in the case when
In this paper we unify the ideas used in [11] and [9] to obtain a new Fredholmtype theorem for the case when
The following notation is used throughout the paper.
ℕ is the set of all natural numbers;
ℝ is the set of all real numbers,
if Ω is a set then measΩ, intΩ,
By a solution to (1), (2) we understand a function
Notation 1 Define, for every
2 Main result
Theorem 1Let
If the problem
has only the trivial solution for every
The proof of Theorem 1 is based on the following result by Krasnosel’skii (see [[12], Theorem 41.3, p.325]). We will formulate it in a form suitable for us.
Theorem 2LetXbe a Banach space,
thenAhas a fixed point in Ω, i.e. there exists
Furthermore, to prove Theorem 1 we will need the following lemma.
Lemma 1Let, for every
holds, where
Proof Suppose on the contrary that for every
where
Put
Then
and from (10) and (11), in view of (3), (4), (12), and (13), we get
On the other hand, from (9) and (12) we have
whence, according to [[13], Corollary IV.8.11] it follows that
Therefore, (14), (15), and (18) imply that the sequences
Furthermore, (15)(17) yield
which contradicts our assumptions. □
Proof of Theorem 1 Let
Then
and consider the operator equation
It can easily be seen that problem (1), (2), and (23) are equivalent in the following
sense: if
Let
Let, moreover,
Now we will show that the operator A has a fixed point in Ω. According to Theorem 2 it is sufficient to show that
Assume on the contrary that there exist
Then from (26), in view of (20)(22) we obtain
where
Now from (27) and (28) it follows that
Moreover, since
Now the equality (32), according to Notation 1, implies
Therefore, in view of Lemma 1, with respect to (30)(34) we obtain
However, the latter inequality contradicts (24). □
3 Corollaries
If the operators
then from Theorem 1 we obtain the following assertion.
Corollary 1Let (5), (35), and (36) be fulfilled. If the problem
has only the trivial solution then problem (1), (2) has at least one solution.
For a particular case when
where
Corollary 2Let (5), (36), (39), and (40) be fulfilled. Let, moreover,
and let problem (37), (38) have only the trivial solution. Then problem (1), (2) has at least one solution.
Namely, for a twodimensional system of ordinary equations and a particular case of boundary conditions we get the following.
Corollary 3Let
with
has at least one solution for every
The particular case of the system discussed in Corollary 3 is socalled secondorder
differential equation with λLaplacian. Therefore, in the case when
Corollary 4Let the problem
with
has at least one solution for every
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.
Acknowledgements
Robert Hakl has been supported by RVO: 67985840. Manuel Zamora has been supported by Ministerio de Educación y Ciencia, Spain, project MTM201123652, and by a postdoctoral grant from University of Granada.
End notes
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