Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations
© Hakl and Zamora; licensee Springer. 2014
Received: 30 January 2014
Accepted: 29 April 2014
Published: 13 May 2014
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.
1 Statement of the problem
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 set of absolutely continuous vector-valued functions ;
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).
2 Main result
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.
then A has a fixed point in Ω, i.e. there exists such that .
Furthermore, to prove Theorem 1 we will need the following lemma.
which contradicts our assumptions. □
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).
However, the latter inequality contradicts (24). □
then from Theorem 1 we obtain the following assertion.
has only the trivial solution then problem (1), (2) has at least one solution.
where and are measurable functions, we have the following assertion.
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.
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.
has at least one solution for every and .
aIf then .
bIt transforms bounded sets into relatively compact sets.
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.
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