A Fredholm-type 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.
Keywords:functional-differential equations; boundary value problems; existence of solutions
1 Statement of the problem
Consider the system of functional-differential 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 above-stated 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, so-called Fredholm alternative, is well known; for more details see e.g.[1-8] and references therein.
In 1966, Lasota established the Fredholm-type theorem in the case when and are homogeneous operators (see ). Recently, Fredholm-type theorems in the case when and are positively homogeneous operators were established by Kiguradze, Půža, Stavroulakis in  and also by Kiguradze, Šremr in .
In this paper we unify the ideas used in  and  to obtain a new Fredholm-type 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 vector-valued functions with the norm
is the set of absolutely continuous vector-valued 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
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 [, Theorem 41.3, p.325]). We will formulate it in a form suitable for us.
Theorem 2LetXbe a Banach space, be a symmetricabounded domain with . Let, moreover, be a compactbcontinuous operator which has no fixed point on∂Ω. If, in addition,
thenAhas a fixed point in Ω, i.e. there exists such 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 exists such that for any and any , the a priori estimate
Proof Suppose on the contrary that for every there exist and such that
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 [, Corollary IV.8.11] it follows that
Therefore, (14), (15), and (18) imply that the sequences ( ) are uniformly bounded and equicontinuous. Thus, according to Arzelà-Ascoli 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
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
where , i.e.
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). □
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 two-dimensional system of ordinary equations and a particular case of boundary conditions we get the following.
Corollary 3Let , and let
with , have only the trivial solution. Then the problem
has at least one solution for every and .
The particular case of the system discussed in Corollary 3 is so-called second-order differential equation with λ-Laplacian. Therefore, in the case when , Corollary 3 yields the following.
Corollary 4Let the problem
with , , have only the trivial solution. Then the problem
has at least one solution for every and .
The authors declare that they have no competing interests.
RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.
Robert Hakl has been supported by RVO: 67985840. Manuel Zamora has been supported by Ministerio de Educación y Ciencia, Spain, project MTM2011-23652, and by a postdoctoral grant from University of Granada.
Kiguradze, I, Půža, B: On boundary value problems for systems of linear functional differential equations. Czechoslov. Math. J.. 47(2), 341–373 (1997). Publisher Full Text
Kiguradze, I, Šremr, J: Solvability conditions for non-local boundary value problems for two-dimensional half-linear differential systems. Nonlinear Anal. TMA. 74(17), 6537–6552 (2011). Publisher Full Text