This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations

Robert Hakl1* and Manuel Zamora2

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Žižkova 22, Brno, 61662, Czech Republic

2 Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, 4051381, Chile

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Boundary Value Problems 2014, 2014:113  doi:10.1186/1687-2770-2014-113


Dedicated to Professor Ivan Kiguradze.


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/113


Received:30 January 2014
Accepted:29 April 2014
Published:13 May 2014

© 2014 Hakl and Zamora; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

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.

MSC: 34K10.

Keywords:
functional-differential equations; boundary value problems; existence of solutions

1 Statement of the problem

Consider the system of functional-differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M1">View MathML</a>

(1)

together with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M2">View MathML</a>

(2)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M3">View MathML</a> are continuous operators satisfying Carathéodory conditions, i.e. for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M6">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M7">View MathML</a> are continuous functionals which are bounded on every ball by a constant, i.e. for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M10">View MathML</a>

Furthermore, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> satisfy the following condition: there exist positive real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M14">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M15">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M16">View MathML</a>, and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M18">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M19">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M20">View MathML</a>

(4)

Remark 1 From the above-stated assumptions it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M22">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M23">View MathML</a>.

In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> are linear bounded operators and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M27">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> are homogeneous operators (see [9]). Recently, Fredholm-type theorems in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> 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 Fredholm-type theorem for the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> 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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M34">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M35">View MathML</a> is the linear space of vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M36">View MathML</a> with the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M37">View MathML</a> endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M38">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M39">View MathML</a> is the Banach space of continuous vector-valued functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M40">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M42">View MathML</a> is the set of absolutely continuous vector-valued functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M43">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M44">View MathML</a> is the Banach space of Lebesgue integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M45">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M46">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M47">View MathML</a>;

if Ω is a set then measΩ, intΩ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M48">View MathML</a>, and Ω denotes the measure, interior, closure, and boundary of the set Ω, respectively.

By a solution to (1), (2) we understand a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49">View MathML</a> satisfying (1) almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M50">View MathML</a> and (2).

Notation 1 Define, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M51">View MathML</a>, the following functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M52">View MathML</a>

2 Main result

Theorem 1Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M53">View MathML</a>

(5)

If the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M54">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M55">View MathML</a>

(7)

has only the trivial solution for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56">View MathML</a>, 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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M57">View MathML</a>be a symmetricabounded domain with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M60">View MathML</a>. Let, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M61">View MathML</a>be a compactbcontinuous operator which has no fixed point onΩ. If, in addition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M62">View MathML</a>

thenAhas a fixed point in Ω, i.e. there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M63">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M64">View MathML</a>.

Furthermore, to prove Theorem 1 we will need the following lemma.

Lemma 1Let, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56">View MathML</a>, problem (6), (7) has only the trivial solution. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M56">View MathML</a>, the a priori estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M69">View MathML</a>

(8)

holds, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M70">View MathML</a>

Proof Suppose on the contrary that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M71">View MathML</a> there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M73">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M74">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M75">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M76">View MathML</a>

(11)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M77">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M78">View MathML</a>

(13)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M79">View MathML</a>

(14)

and from (10) and (11), in view of (3), (4), (12), and (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M80">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M81">View MathML</a>

(16)

On the other hand, from (9) and (12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M82">View MathML</a>

(17)

whence, according to [[13], Corollary IV.8.11] it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M83">View MathML</a>

(18)

Therefore, (14), (15), and (18) imply that the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M84">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M85">View MathML</a>) are uniformly bounded and equicontinuous. Thus, according to Arzelà-Ascoli theorem, without loss of generality we can assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M87">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M88">View MathML</a>

(19)

Furthermore, (15)-(17) yield <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M89">View MathML</a> and show that it is a solution to (6), (7). However, (14) and (19) result in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M90">View MathML</a>

which contradicts our assumptions. □

Proof of Theorem 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M91">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M92">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M93">View MathML</a>, define the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M94">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M95">View MathML</a> is a Banach space. Let the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M96">View MathML</a> be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M97">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M98">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M99">View MathML</a>

(22)

and consider the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M100">View MathML</a>

(23)

It can easily be seen that problem (1), (2), and (23) are equivalent in the following sense: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M101">View MathML</a> is a solution to (23), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M102">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M85">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M104">View MathML</a> is a solution to (1), (2); and vice versa if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M104">View MathML</a> is a solution to (1), (2), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M106">View MathML</a> is a solution to (23).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M4">View MathML</a> be such that the conclusion of Lemma 1 is valid. According to (5) we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M108">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M109">View MathML</a>

(24)

Let, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M110">View MathML</a>

(25)

Now we will show that the operator A has a fixed point in Ω. According to Theorem 2 it is sufficient to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M111">View MathML</a>

Assume on the contrary that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M113">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M114">View MathML</a>

(26)

Then from (26), in view of (20)-(22) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M115">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M116">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M117">View MathML</a>

(27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M118">View MathML</a>

(28)

Now from (27) and (28) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M120">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M121">View MathML</a>

(30)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M122">View MathML</a>

(31)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M123">View MathML</a>, on account of (25) and (29) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M124">View MathML</a>

(32)

Now the equality (32), according to Notation 1, implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M125">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M126">View MathML</a>

(34)

Therefore, in view of Lemma 1, with respect to (30)-(34) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M127">View MathML</a>

However, the latter inequality contradicts (24). □

3 Corollaries

If the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M12">View MathML</a> are homogeneous, i.e. if moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M130">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M131">View MathML</a>

(36)

then from Theorem 1 we obtain the following assertion.

Corollary 1Let (5), (35), and (36) be fulfilled. If the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M132">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M133">View MathML</a>

(38)

has only the trivial solution then problem (1), (2) has at least one solution.

For a particular case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M11">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M135">View MathML</a>

(39)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M136">View MathML</a>

(40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M138">View MathML</a> are measurable functions, we have the following assertion.

Corollary 2Let (5), (36), (39), and (40) be fulfilled. Let, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M139">View MathML</a>

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M140">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M141">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M143">View MathML</a>have only the trivial solution. Then the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M144">View MathML</a>

has at least one solution for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M145">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M146">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M147">View MathML</a>, Corollary 3 yields the following.

Corollary 4Let the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M148">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M143">View MathML</a>have only the trivial solution. Then the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M152">View MathML</a>

has at least one solution for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M153">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M146">View MathML</a>.

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 MTM2011-23652, and by a postdoctoral grant from University of Granada.

End notes

  1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M58">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/113/mathml/M59">View MathML</a>.

  2. It transforms bounded sets into relatively compact sets.

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