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First-order nonlinear differential equations with state-dependent impulses

Lukáš Rachůnek and Irena Rachůnková*

Author Affiliations

Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic

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Boundary Value Problems 2013, 2013:195  doi:10.1186/1687-2770-2013-195

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/195


Received:13 May 2013
Accepted:13 August 2013
Published:28 August 2013

© 2013 Rachůnek and Rachůnková; 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 cited.

Abstract

The paper deals with the state-dependent impulsive problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M3">View MathML</a>, f fulfils the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4">View MathML</a>, the impulse function is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4">View MathML</a>, the barrier function γ has a continuous first derivative on some subset of ℝ and is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a> equipped with the sup-norm. The functional is represented by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of first-order differential equations subject to state-dependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time.

MSC: 34B37, 34B15.

Keywords:
first-order ODE; state-dependent impulses; transversality conditions; general linear boundary conditions; existence; Kurzweil-Stieltjes integral

1 Introduction

The investigation of impulsive differential equations has a long history; see, e.g., the monographs [1-3]. Most papers dealing with impulsive differential equations subject to boundary conditions focus their attention on impulses at fixed moments. But this is a very particular case of a more complicated case with state-dependent impulses. Boundary value problems with state-dependent impulses, where difficulties with an operator representation appear (cf. Remark 6.2), are substantially less developed. We refer to the papers [4-6] and [7] which are devoted to periodic problems, and for problems with other boundary conditions, see [8,9] or [10-12].

Here, in our paper, we present an approach leading to a new existence principle for impulsive boundary value problems. This approach is applicable to each linear boundary condition which is considered with some first-order differential equation subject to state-dependent impulses. The important step is a proof of a transversality (Remark 2.3 and Lemmas 5.1 and 5.2), which makes possible a construction of a continuous operator (Section 6) whose fixed point leads to a solution of our original impulsive problem (Section 7).

Notation

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M11">View MathML</a> is the set of real functions continuous on M.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M12">View MathML</a> is the set of real functions absolutely continuous on M.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M13">View MathML</a> is the set of real functions Lebesgue integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M15">View MathML</a> is the set of real functions essentially bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M17">View MathML</a> is the set of real functions with bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a> is the set of real left-continuous regulated functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M21">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22">View MathML</a>, and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M23">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M24">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M25">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M26">View MathML</a> is the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M27">View MathML</a> such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M28">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M29">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M30">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>,

(iii) for each compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M32">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M33">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M34">View MathML</a>

• The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M15">View MathML</a> equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M36">View MathML</a>

(1.2)

is a Banach space.

• Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M37">View MathML</a>, we equip the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M40">View MathML</a> and get also Banach spaces (cf.[13]). Then (1.2) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M41">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M42">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M43">View MathML</a> is the Banach space of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M46">View MathML</a>, where the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M47">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M48">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M49">View MathML</a> is the characteristic function of a set A, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M50">View MathML</a>.

2 Formulation of problem

We investigate the solvability of the nonlinear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M51">View MathML</a>

(2.1)

subject to the state-dependent impulse condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M52">View MathML</a>

(2.2)

and the general linear boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M53">View MathML</a>

(2.3)

Here we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M54">View MathML</a>

(2.4)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M55">View MathML</a> is a linear bounded functional.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22">View MathML</a> is a solution of problem (2.1), (2.2) if

• there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M58">View MathML</a>;

• the restrictions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M60">View MathML</a> are absolutely continuous;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M61">View MathML</a>;

z satisfies equation (2.1) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>.

Definition 2.2 A graph of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M63">View MathML</a> is called a barrierγ.

Remark 2.3 Let be the set of all solutions of problem (2.1), (2.2). According to Definition 2.1, each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M65">View MathML</a> satisfies a transversality property, which means that the graph of z crosses a barrier γ at a unique point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57">View MathML</a>, where the impulse acts on z. After that (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M68">View MathML</a>) the graph of z lies on the right of the barrier γ. This transversality property follows from transversality conditions (cf. (4.5), (4.6)) and it is proved in Section 5.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M70">View MathML</a>. Then there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M71">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M72">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M74">View MathML</a> can occur. Therefore different functions from can have their discontinuities at different points from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76">View MathML</a>. Our aim in this paper is to prove the existence of a solution of problem (2.1), (2.2) satisfying the general linear boundary condition (2.3). To do this, we need a suitable linear space containing . Due to state-dependent impulses, the Banach space of piece-wise continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a> with the sup-norm cannot be used here. Therefore we choose the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a>. Clearly, by (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M80">View MathML</a>. The operator in the general linear boundary condition (2.3) can be written uniquely in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M81">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M84">View MathML</a> is the Kurzweil-Stieltjes integral (cf.[14], Theorem 3.8). Representation (2.5) is correct on , because for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M21">View MathML</a> the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M87">View MathML</a> exists. Its definition and properties can be found in [15] (see Perron-Stieltjes integral based on the work of Kurzweil).

Definition 2.4 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M22">View MathML</a> is a solution of problem (2.1)-(2.3) if z is a solution of problem (2.1), (2.2) and fulfils (2.3).

3 Green’s function

For further investigation, we will need a linear homogeneous problem corresponding to problem (2.1)-(2.3). Such problem has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M89">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M90">View MathML</a>

(3.2)

because the impulse in (2.2) disappears if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M91">View MathML</a>. We will also work with the non-homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M92">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M93">View MathML</a>.

Definition 3.1 A solution of problem (3.3), (3.2) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M45">View MathML</a> satisfying equation (3.3) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a> and fulfilling condition (3.2).

Remark 3.2 If x is a solution of problem (3.3), (3.2), then x belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M96">View MathML</a>, and consequently condition (3.2) can be written in the form (cf. (2.5))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M97">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M99">View MathML</a> and the Lebesgue integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M100">View MathML</a> is used.

Definition 3.3 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M101">View MathML</a> is the Green’s function of problem (3.1), (3.2) if

(i) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M102">View MathML</a>, the restrictions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M104">View MathML</a> are solutions of equation (3.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M105">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M106">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M107">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>;

(iii) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M93">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M110">View MathML</a>

(3.5)

fulfils condition (3.4).

Lemma 3.4Letbe from (2.5) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83">View MathML</a>.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a>if and only if there exists the Green’s functionGof problem (3.1), (3.2) which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M114">View MathML</a>

(3.6)

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a>if and only if there exists a unique solutionxof problem (3.3), (3.4), which has a form of (3.5) withGfrom (3.6).

Proof Clearly, G given by (3.6) fulfils (i) and (ii) of Definition 3.3 if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a>. A general solution of equation (3.3) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M117">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M118">View MathML</a>. By (3.4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M119">View MathML</a>

The equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M120">View MathML</a>

has a unique solution c if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a>. Then a unique solution x of problem (3.3), (3.4) is written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M122">View MathML</a>

 □

Lemma 3.5LetGbe the Green’s function of problem (3.1), (3.2), whereis from (2.5) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a>. Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M124">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M127">View MathML</a>

(3.7)

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M124">View MathML</a>. By (3.6),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M129">View MathML</a>

Consequently, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a>. This yields that the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M132">View MathML</a> exists for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83">View MathML</a>. Note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125">View MathML</a> is not continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>, formula (3.4) cannot be used for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M125">View MathML</a> in place of x. Instead, we use the properties of the Kurzweil-Stieltjes integral which justify the following computation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M137">View MathML</a>

Hence, by (2.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M138">View MathML</a>

 □

Example 3.6 Consider a solution x of problem (3.3), (3.2), where has a form of the two-point boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M139">View MathML</a>

(3.8)

We will show that can be expressed in a form of (3.4). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M140">View MathML</a>, then k and v can be found from the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M141">View MathML</a>

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M142">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M143">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M145">View MathML</a>. In addition, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M146">View MathML</a>, then (cf. (3.6))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M147">View MathML</a>

Example 3.7 Consider a solution x of problem (3.3), (3.2), where has a form of the multi-point boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M148">View MathML</a>

(3.9)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M149">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M150">View MathML</a>, then k and v of (3.4) can be found from the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M151">View MathML</a>

(3.10)

Assume that v is a piece-wise constant right-continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M153">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M155">View MathML</a>. By (3.10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M156">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M157">View MathML</a>

To summarize, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M158">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M159">View MathML</a>

and further (cf. (3.6))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M160">View MathML</a>

Example 3.8 Consider a solution x of problem (3.3), (3.2), where has a form of the integral condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M161">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M162">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M163">View MathML</a>, then k and v of (3.4) can be found from the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M164">View MathML</a>

(3.11)

Let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M165">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M166">View MathML</a>

and (3.11) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M168">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M169">View MathML</a>

Similarly, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M170">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M163">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M172">View MathML</a>

In both cases, G is written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M173">View MathML</a>

4 Assumptions

An existence result for problem (2.1)-(2.3) will be proved in the next sections under the basic assumption (2.4) and the following additional assumptions imposed on f, , and γ.

(i) Boundedness of f

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M175">View MathML</a>

(4.1)

(ii) Boundedness of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M177">View MathML</a>

(4.2)

(iii) Boundedness of γ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M178">View MathML</a>

(4.3)

(iv) Properties of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M179">View MathML</a>

(4.4)

(v) Transversality conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M180">View MathML</a>

(4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M181">View MathML</a>

(4.6)

where h is from (4.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M183">View MathML</a> are from (4.3).

(vi) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M184">View MathML</a>-continuity of f

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M185">View MathML</a>

(4.7)

Remark 4.1

(a) Boundedness of f and can be replaced by more general conditions, for example, growth or sign ones, if the method of a priori estimates is used. See, e.g., [16,17].

(b) Continuity of v on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M187">View MathML</a> is necessary for the construction of a continuous operator in Section 6. Note that then we need <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M188">View MathML</a> in Example 3.7.

(c) Clearly, if f is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M189">View MathML</a>, then f fulfils (4.7).

(d) Let there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M193">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M194">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M196">View MathML</a>. Then f fulfils (4.7). An example of such a function f is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M197">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M193">View MathML</a>.

5 Transversality

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M203">View MathML</a> and define a set ℬ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M204">View MathML</a>

(5.1)

The following two lemmas for functions from ℬ are the modifications of lemmas in [10] and provide the transversality (cf. Remark 2.3) which will be essential for operator constructions in Section 6.

Lemma 5.1Letγsatisfy (2.4), (4.3) and (4.5). Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205">View MathML</a>, there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M207">View MathML</a>

(5.2)

In addition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M208">View MathML</a>.

Proof Let us take an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205">View MathML</a> and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M210">View MathML</a>

Then, by (2.4) and (5.1), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M211">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M212">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M213">View MathML</a>, condition (4.3) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M214">View MathML</a>

Consequently, there exists at least one zero of σ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M57">View MathML</a> be a zero of σ. By virtue of (4.5) and (5.1), we get, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M218">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M219">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M220">View MathML</a>

(5.3)

Hence τ is a unique zero of σ, and (4.3) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M208">View MathML</a>. □

Due to Lemma 5.1, we can define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M222">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M223">View MathML</a>

(5.4)

where τ fulfils (5.2).

Lemma 5.2Letγsatisfy (2.4), (4.3) and (4.5). Then the functionalis continuous.

Proof Let us choose a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M225">View MathML</a> which is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M43">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M227">View MathML</a>

(5.5)

and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M229">View MathML</a>

(5.6)

So, by virtue of (1.5) and (5.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M230">View MathML</a>

We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M205">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M233">View MathML</a>

By Lemma 5.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M234">View MathML</a>

(5.7)

We need to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M235">View MathML</a>

(5.8)

Conditions (2.4), (1.5) and (5.6) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M236">View MathML</a>

(5.9)

Let us take an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M237">View MathML</a>. By (5.3) and (5.9) we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M240">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M242">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M243">View MathML</a>. By Lemma 5.1 and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M244">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M245">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M243">View MathML</a>, and (5.8) follows. □

6 Fixed point problem

In this section we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M247">View MathML</a>

(6.1)

and we construct a fixed point problem whose solvability leads to a solution of problem (2.1)-(2.3). To this aim, having the set ℬ from (5.1), we define a set Ω by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M248">View MathML</a>

(6.2)

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M249">View MathML</a>, we define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M250">View MathML</a> as follows. We set, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M252">View MathML</a>

(6.3)

where is defined by (5.4) and the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M254">View MathML</a> is uniquely determined due to Lemma 5.1. By (4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M255">View MathML</a>

(6.4)

Now, we can define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M256">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M257">View MathML</a>, where

(6.5)

(6.6)

Here the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M261">View MathML</a> are defined such that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M263">View MathML</a> are continuous at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M264">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M265">View MathML</a>

(6.7)

Differentiating (6.5) and using (3.6) and (6.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M266">View MathML</a>

(6.8)

This together with (4.1) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M267">View MathML</a>

(6.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M83">View MathML</a> (cf. (4.4)), we see that (6.4)-(6.6), (3.6), (4.1) and (4.2) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M269">View MathML</a>

(6.10)

Due to (6.8)-(6.10), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73">View MathML</a>, and the operator ℱ is defined well.

Lemma 6.1Assume that (6.1) holds and that Ω andare given by (6.2) and (6.5), (6.6), respectively. Then the operatoris compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272">View MathML</a>.

ProofStep 1. We show that ℱ is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272">View MathML</a>. Choose a sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M274">View MathML</a>

which is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M275">View MathML</a>, that is, (cf. (1.5)) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M276">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M277">View MathML</a>

(6.11)

Lemma 5.1 and Lemma 5.2 yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M278">View MathML</a>

(6.12)

where is defined by (5.4). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M280">View MathML</a>

(6.13)

We will prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M281">View MathML</a>

(6.14)

By (4.7), (6.8), (6.11) and (6.13),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M282">View MathML</a>

(6.15)

Using (4.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M283">View MathML</a>

(6.16)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M284">View MathML</a>

the Lebesgue dominated convergence theorem and (6.16) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M285">View MathML</a>

(6.17)

Using (6.13) and (6.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M286">View MathML</a>

The continuity and boundedness of , and v (cf. Lemma 5.2, (2.4), (4.2), (4.4) and (6.12)) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M289">View MathML</a>

wherefrom, by the boundedness of G and (6.17),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M290">View MathML</a>

(6.18)

Using (6.13) and integrating (6.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M291">View MathML</a>

and, due to (6.15) and (6.18), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M292">View MathML</a>

(6.19)

Similarly, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M293">View MathML</a>

(6.20)

Properties (6.15), (6.19) and (6.20) yield (6.14).

Step 2. We show that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M294">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M295">View MathML</a>. Choose an arbitrary sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M296">View MathML</a>

We need to prove that there exists a convergent subsequence. Clearly, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M297">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M298">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M299">View MathML</a>. By (5.1) and (6.2), it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M300">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9">View MathML</a>. Therefore, the Arzelà-Ascoli theorem yields that there exists a subsequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M303">View MathML</a>

which converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M304">View MathML</a>. Consequently, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M237">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M306">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M307">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M308">View MathML</a>

Similarly as in Step 1, we prove (cf. (6.15), (6.19), (6.20))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M309">View MathML</a>

which gives by (1.5) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M310">View MathML</a> is convergent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M295">View MathML</a>. □

Remark 6.2 If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M312">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M313">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M314">View MathML</a>, then problem (2.1)-(2.3) has an impulse at fixed time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M315">View MathML</a> and a standard operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316">View MathML</a>, acting on the space of piece-wise continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a> and having the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M318">View MathML</a>

(6.21)

can be used instead of the operator ℱ from (6.5), (6.6). But this is not possible if γ is not constant on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M319">View MathML</a>. The reason is that then an impulse is realized at a state-dependent point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M320">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316">View MathML</a> with τ instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M322">View MathML</a> should be investigated on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a>. But if we write a state-dependent τ instead of a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M322">View MathML</a> in (6.21), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316">View MathML</a> loses its continuity on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M19">View MathML</a>, which we show in the next example.

Example 6.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M328">View MathML</a> and be from (2.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M331">View MathML</a>. Consider the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M332">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M333">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M334">View MathML</a> and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M335">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M9">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M338">View MathML</a>. Assume that the barrier γ is given by the linear function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M339">View MathML</a> on ℝ and the impulse function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M340">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M342">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M343">View MathML</a>

and, according to (6.21), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M341">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M345">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M346">View MathML</a>

due to (3.6). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M347">View MathML</a> and we have also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M348">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M316">View MathML</a> is not continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M350">View MathML</a>.

Lemma 6.1 results in the following theorem.

Theorem 6.4Assume that (6.1) holds and that the set Ω is given by (6.2), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M351">View MathML</a>

(6.22)

Further, let the operatorbe given by (6.5), (6.6). Thenhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272">View MathML</a>.

Proof By Lemma 6.1, ℱ is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272">View MathML</a>. Due to (5.1), (6.2), (6.5), (6.6), (6.10) and (6.22),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M354">View MathML</a>

Therefore, the Schauder fixed point theorem yields a fixed point of ℱ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M272">View MathML</a>. □

7 Main result

The main result, which is contained in Theorem 7.1, guarantees the solvability of problem (2.1)-(2.3) provided the data functions f, and γ are bounded (cf. (4.1)-(4.3)). As it is mentioned in Remark 4.1, Theorem 7.1 serves as an existence principle which, in combination with the method of a priori estimates, can lead to more general existence results for unbounded f and and concrete boundary conditions.

Theorem 7.1Assume that (6.1) and (6.22) hold. Then there exists a solutionzof problem (2.1)-(2.3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M358">View MathML</a>

(7.1)

Proof By Theorem 6.4, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M276">View MathML</a> which is a fixed point of the operator ℱ defined in (6.5) and (6.6). This means that

(7.2)

(7.3)

where G, , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M363">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M365">View MathML</a> are given by (3.6), (5.4), (6.3), (6.7), respectively. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M264">View MathML</a> is a unique point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M76">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M368">View MathML</a>

(7.4)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>, define a function z by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M370">View MathML</a>

(7.5)

Differentiating (7.2), (7.3) and using (3.6) and (6.3), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M371">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M73">View MathML</a>, and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M374">View MathML</a>

By virtue of (7.2)-(7.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M375">View MathML</a>

(7.6)

Let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M376">View MathML</a> is a unique solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M377">View MathML</a>

(7.7)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a>. According to (7.4) and (7.5), it suffices to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M379">View MathML</a>

(7.8)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M380">View MathML</a>, we have (cf. (5.1) and (6.2))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M381">View MathML</a>

Assume that the first condition in (4.6) is fulfilled. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M383">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M314">View MathML</a>. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M385">View MathML</a>

By (7.6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M386">View MathML</a>, and since γ is non-increasing, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M387">View MathML</a>

due to (7.4). Using (4.5), we derive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M388">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M389">View MathML</a>

So, (7.8) is valid. If the second condition in (4.6) is fulfilled, we use the dual arguments.

Finally, let us check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M390">View MathML</a>. By (7.2)-(7.6) and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M391">View MathML</a>

(7.9)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M392">View MathML</a>

(7.10)

Then, according to (iii) of Definition 3.3 and Remark 3.2, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M393">View MathML</a>. Further, using (3.7) from Lemma 3.5, we arrive at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M394">View MathML</a>. Consequently, due to (2.5), (7.9) and (7.10), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M395">View MathML</a> results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M396">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This research was supported by the grant Matematické modely, PrF_2013_013. The authors thank the referees for suggestions which improved the paper.

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