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, are continuous operators satisfying Carathéodory conditions, i.e. for every there exists such that
and are continuous functionals which are bounded on every ball by a constant, i.e. for every there exists such that
Furthermore, we assume that and satisfy the following condition: there exist positive real numbers and such that whenever , and for every and we have
Remark 1 From the abovestated assumptions it follows that , for every .
In the case when and are linear bounded operators and , , the relationship between the existence of a solution to problem (1), (2) and the existence of only the trivial solution to its corresponding homogeneous problem, socalled Fredholm alternative, is well known; for more details see e.g.[18] and references therein.
In 1966, Lasota established the Fredholmtype theorem in the case when and are homogeneous operators (see [9]). Recently, Fredholmtype theorems in the case when and are positively homogeneous operators were established by Kiguradze, Půža, Stavroulakis in [10] and also by Kiguradze, Šremr in [11].
In this paper we unify the ideas used in [11] and [9] to obtain a new Fredholmtype theorem for the case when and are positively homogeneous operators. The consequences of the obtained result for particular cases of problem (1), (2) are formulated at the end of the paper.
The following notation is used throughout the paper.
ℕ is the set of all natural numbers;
ℝ is the set of all real numbers, ;
is the linear space of vectors with the elements endowed with the norm
is the Banach space of continuous vectorvalued functions with the norm
is the set of absolutely continuous vectorvalued functions ;
is the Banach space of Lebesgue integrable functions with the norm
if Ω is a set then measΩ, intΩ, , and ∂Ω denotes the measure, interior, closure, and boundary of the set Ω, respectively.
By a solution to (1), (2) we understand a function satisfying (1) almost everywhere in and (2).
Notation 1 Define, for every , the following functions:
2 Main result
Theorem 1Let
If the problem
has only the trivial solution for every, then problem (1), (2) has at least one solution.
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, be a symmetric^{a}bounded domain with. Let, moreover, be a compact^{b}continuous operator which has no fixed point on∂Ω. If, in addition,
thenAhas a fixed point in Ω, i.e. there existssuch that.
Furthermore, to prove Theorem 1 we will need the following lemma.
Lemma 1Let, for every, problem (6), (7) has only the trivial solution. Then there existssuch that for anyand any, the a priori estimate
holds, where
Proof Suppose on the contrary that for every there exist and such that
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 () are uniformly bounded and equicontinuous. Thus, according to ArzelàAscoli theorem, without loss of generality we can assume that there exist and such that
Furthermore, (15)(17) yield and show that it is a solution to (6), (7). However, (14) and (19) result in
which contradicts our assumptions. □
Proof of Theorem 1 Let and for , i.e., define the norm
Then is a Banach space. Let the operators be defined as follows:
and consider the operator equation
It can easily be seen that problem (1), (2), and (23) are equivalent in the following sense: if is a solution to (23), then () and is a solution to (1), (2); and vice versa if is a solution to (1), (2), then is a solution to (23).
Let be such that the conclusion of Lemma 1 is valid. According to (5) we can choose such that
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 and such that
Then from (26), in view of (20)(22) we obtain
Now from (27) and (28) it follows that ,
Moreover, since , on account of (25) and (29) we have
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 and are homogeneous, i.e. if moreover
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 are defined by
where and are measurable functions, we have the following assertion.
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.
with, have only the trivial solution. Then the problem
has at least one solution for everyand.
The particular case of the system discussed in Corollary 3 is socalled secondorder differential equation with λLaplacian. Therefore, in the case when , Corollary 3 yields the following.
Corollary 4Let the problem
with, , have only the trivial solution. Then the problem
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
References

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