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
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;
2 Main result
If the problem
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.
Furthermore, to prove Theorem 1 we will need the following lemma.
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
which contradicts our assumptions. □
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).
Now we will show that the operator A has a fixed point in Ω. According to Theorem 2 it is sufficient to show that
Then from (26), in view of (20)-(22) we obtain
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). □
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.
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 4Let the problem
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