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

Open Access Research Article

Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem

Irena Rachůnková* and Jan Tomeček

Author Affiliations

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

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Boundary Value Problems 2014, 2014:118  doi:10.1186/1687-2770-2014-118


Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/118


Received:20 November 2013
Accepted:1 May 2014
Published:14 May 2014

© 2014 Rach¿nková and Tome¿ek; licensee Springer.

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

Abstract

The paper provides an existence principle for a general boundary value problem of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M1">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M4">View MathML</a>, with the state-dependent impulses <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M5">View MathML</a>, where the impulse points t are determined as solutions of the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M10">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>, are Lebesgue integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M14">View MathML</a> satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M15">View MathML</a>. The impulse functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>, and the barrier functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>, are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M22">View MathML</a>, respectively. The functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M24">View MathML</a>, are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> vector functions. Provided the data functions h and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16">View MathML</a> are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M27">View MathML</a> at the unique impulse point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28">View MathML</a> are presented, and consequently the existence of a solution to the problem is proved.

MSC: 34B37, 34B10, 34B15.

Keywords:
nonlinear higher-order ODE; state-dependent impulses; general linear boundary conditions; transversality conditions; fixed point

1 Introduction

In this paper we are interested in the nonlinear ordinary differential equation of the nth-order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M29">View MathML</a>) with state-dependent impulses and general linear boundary conditions on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M30">View MathML</a>. Studies of real-life problems with state-dependent impulses can be found e.g. in [1-6]. Here we consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M31">View MathML</a>

(1)

subject to the impulse conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M32">View MathML</a>

(2)

and the linear boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M33">View MathML</a>

(3)

In what follows we use this notation. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M34">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M35">View MathML</a> we denote the set of all matrices of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M36">View MathML</a> with real valued coefficients. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M37">View MathML</a> denote the transpose of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M38">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M39">View MathML</a> be the set of all n-dimensional column vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M40">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M42">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M43">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M44">View MathML</a> we denote the set of all mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M45">View MathML</a> with continuous components. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M51">View MathML</a>, we denote the sets of all mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M52">View MathML</a> whose components are, respectively, essentially bounded functions, Lebesgue integrable functions, left-continuous regulated functions, absolutely continuous functions, functions with bounded variation and functions with continuous derivatives of the kth order on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M54">View MathML</a> we denote the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M55">View MathML</a> satisfying the Carathéodory conditions on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M56">View MathML</a>. Finally, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M57">View MathML</a> we denote the characteristic function of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M58">View MathML</a>.

Note that a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M59">View MathML</a> is left-continuous regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M61">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M62">View MathML</a> there exist finite limits

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M63">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64">View MathML</a> is a linear space, and equipped with the sup-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M65">View MathML</a> it is a Banach space (see [[7], Theorem 3.6]). In particular, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M66">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M67">View MathML</a> satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M56">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M69">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M71">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72">View MathML</a>,

• for each compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M73">View MathML</a> there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M74">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M75">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M76">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M77">View MathML</a>.

In this paper we provide sufficient conditions for the solvability of problem (1)-(3). This problem is a generalization of problems studied in the papers [8-10] which are devoted to the second-order differential equation. Other types of initial or boundary value problems for the first- or second-order differential equations with state-dependent impulses can be found in [11-19]. To get the existence results for problem (1)-(3), we exploit the paper [20] with fixed-time impulsive problems.

Here we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M78">View MathML</a>

(4)

Remark 1 The integral in formula (4) is the Kurzweil-Stieltjes integral, whose definition and properties can be found in [21]. The fact that each linear bounded functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M79">View MathML</a> can be written uniquely in the form described in (4) is proved in [22]. See also [20].

Now let us define a solution of problem (1)-(3).

Definition 2 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M80">View MathML</a> is said to be a solution of problem (1)-(3) if u satisfies (1) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M76">View MathML</a> and fulfils conditions (2) and (3).

2 Problem with impulses at fixed times

In the paper [20] we have found an operator representation to the special type of problem (1)-(3) having impulses at fixed times. This is the case that the barrier functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19">View MathML</a> in (2) are constant functions, i.e. there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M83">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M85">View MathML</a>

(5)

In this case, each solution of the problem crosses ith barrier at same time instant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M86">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>.

Note that boundary value problems for higher-order differential equations with impulses at fixed times have been studied for example in [23-31] and for delay higher-order impulsive equations in [32,33].

Let us summarize the results of the paper [20] according to our needs. Assume that the linear homogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M88">View MathML</a>

(6)

has only the trivial solution. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M89">View MathML</a> be a fundamental system of solutions of the differential equation from (6), W be their Wronski matrix and w its first row, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M90">View MathML</a>

(7)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M91">View MathML</a>

(8)

From [[20], Lemma 8] (see also Chapter 3 in [34]) it follows that the unique solvability of (6) is equivalent to the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M92">View MathML</a>

(9)

Further assume (9), consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94">View MathML</a>, from (4), and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M95">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M97">View MathML</a>

(10)

If we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M99">View MathML</a> elements of the matrices H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M100">View MathML</a>, respectively, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M101">View MathML</a>

(11)

we can define functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103">View MathML</a>, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M104">View MathML</a>

(12)

For each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M105">View MathML</a> the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M107">View MathML</a>, will be understood as right-continuous extensions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M108">View MathML</a> and left-continuous extensions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M110">View MathML</a>. In this way the Green’s function of problem (6) is built (cf. Remark 6).

Remark 3 In order to state one of the main results of [20] we introduce the set of all functions u continuous on the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M111">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M112">View MathML</a> from (5), having their derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M113">View MathML</a> continuously extendable onto these intervals. This set is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M115">View MathML</a> we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M116">View MathML</a>

Equipped with the standard addition, scalar multiplication, and with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M117">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114">View MathML</a> forms a Banach space.

Now we are ready to state the operator representation theorem for the problem with impulses at fixed times <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M119">View MathML</a> which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M120">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M121">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M122">View MathML</a>

(15)

Theorem 4 [[20], Theorem 17]

Let (4), (9) hold, and letW, w, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94">View MathML</a>be defined in (7), (8), and (12). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M115">View MathML</a>is a fixed point of an operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M127">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M128">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72">View MathML</a>, if and only ifuis a solution of problem (13)-(15). Moreover, the operatoris completely continuous.

Remark 5 Let us note that the row vector

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M130">View MathML</a>

does not depend on the choice of a fundamental system of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M131">View MathML</a>, but only on the data of problem (6).

Remark 6 Let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M132">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M133">View MathML</a>

Then the operator ℋ in Theorem 4 can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M134">View MathML</a>

Theorem 4 implies that u is a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M135">View MathML</a> if and only if u is a solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M136">View MathML</a>

(17)

Therefore a (unique) solution of problem (17) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M137">View MathML</a>

and consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M138">View MathML</a> is the Green’s function of (6).

Remark 7 Under the assumption (9) we are allowed using (11) to define the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M139">View MathML</a>

(18)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103">View MathML</a>. Obviously, due to (12),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M142">View MathML</a>

(19)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103">View MathML</a>. Let us stress that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144">View MathML</a>, as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>, do not depend on the choice of fundamental system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M146">View MathML</a>, but only on the data of problem (6). The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144">View MathML</a> possess crucial properties for our approach. From their definition it follows that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M105">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M149">View MathML</a>

(20)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M152">View MathML</a>. Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M152">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M156">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M157">View MathML</a>

(21)

This follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M144">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M159">View MathML</a>), from the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M160">View MathML</a> and from the boundedness of the matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M100">View MathML</a> and H (cf. (7), (10) and (11)).

3 Transversality conditions

The most results for differential equations with state-dependent impulses concern initial value problems. Theorems about the existence, uniqueness or extension of solutions of initial value problems, and about intersections of such solutions with barriers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19">View MathML</a> can be found for example in [[35], Chapter 5].

A different approach has to be used when boundary value problems with state-dependent impulses are discussed and boundary conditions are imposed on a solution anywhere in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> including unknown points of impulses. This is the case of problem (1)-(3).

Our approach is based on the existence of a fixed point of an operator ℱ in some set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M164">View MathML</a> (cf. Lemma 12), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M165">View MathML</a> is a ball defined in (28). In order to get a fixed point, we need to prove for functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M166">View MathML</a> assertions about their transversality through barriers. Such assertions are contained in Lemmas 9 and 10 and it is important that they are valid for all functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M166">View MathML</a> and not only for solutions of problem (1), (2).

Remark 8 Having the lemmas about the transversality, we will prove in Section 4 the existence of a solution u of problem (1)-(3), which has the following property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M168">View MathML</a>

(22)

Consider real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M170">View MathML</a>, and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M171">View MathML</a>

(23)

Now, we are ready to formulate the following transversality conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M172">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M173">View MathML</a>

(25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M174">View MathML</a>

(26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M175">View MathML</a>

(27)

Let us define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M176">View MathML</a>

(28)

Our current goal is to find a continuous functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17">View MathML</a>, which maps each function u from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M179">View MathML</a> to some time instant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28">View MathML</a> of (2).

Lemma 9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M185">View MathML</a>, be real numbers satisfying (26), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M186">View MathML</a>andbe given by (23) and (28), respectively. Finally, assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17">View MathML</a>, satisfy (24), (25), and choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M189">View MathML</a>. Then the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M190">View MathML</a>

(29)

is continuous and decreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a>and it has a unique root in the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M192">View MathML</a>, i.e. there exists a unique solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M193">View MathML</a>

(30)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195">View MathML</a>. By (24),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M196">View MathML</a>

is valid. This together with the fact that σ is continuous shows that σ has at least one root in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M192">View MathML</a>. Now, we will prove that σ is decreasing, by a contradiction. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M199">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M200">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M201">View MathML</a>

From (25), (26), (28), and the Mean Value Theorem we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M202">View MathML</a>

which is a contradiction.

According to Lemma 9, we can define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M203">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M204">View MathML</a>

(31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M205">View MathML</a> is a solution of (30), i.e. a unique root of the function σ from Lemma 9, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17">View MathML</a>. □

Lemma 10Let the assumptions of Lemma 9 be satisfied. The functionals<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>, are continuous.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M209">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M210">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M211">View MathML</a>

(32)

Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195">View MathML</a> and prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M213">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M214">View MathML</a>. We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M215">View MathML</a>

From Lemma 9 it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M216">View MathML</a> are the unique roots of the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M217">View MathML</a>

and these functions are strictly decreasing. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M219">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M220">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M222">View MathML</a>. According to (32) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M223">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a>, in particular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M226">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M214">View MathML</a>. These facts imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M228">View MathML</a>

From the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M229">View MathML</a> and the Intermediate Value Theorem it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M230">View MathML</a>

which completes the proof. □

Our next step is to define an appropriate operator representation of the BVP with state-dependent impulses. The first idea would be a direct exploitation of the operator ℋ from Theorem 4, putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M231">View MathML</a> in place of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M112">View MathML</a>. This is not possible for many reasons. First, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M177">View MathML</a> acts on the space of functions having continuous derivatives - but we need functions having p discontinuities. Even if we would overcome this difficulty we arrive at a problem of choosing an appropriate Banach space on which ℋ would be acting. According to Remark 8, we search a solution u of problem (1)-(3), which has its jumps (together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M234">View MathML</a>) at the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184">View MathML</a> (see (31)). In general, these points are different for different solutions. Consequently, such solutions have to be searched in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64">View MathML</a>. But then there is a difficulty with the continuity of such operator. In fact the operator ℋ from (16) having <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M238">View MathML</a> in place of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M239">View MathML</a> is not continuous in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M64">View MathML</a> (cf. Remark 6.2 and Example 6.3 in [36]).

Therefore, we choose the way here, which we have developed in our joint papers [8-10]. The main idea of our approach lies in representing the solution u of problem (1)-(3) by an ordered <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M241">View MathML</a>-tuple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M242">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M243">View MathML</a>

(33)

Consequently, we work with the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M244">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M245">View MathML</a>

It is well known that X is a Banach space.

4 Main results

Let us turn our attention to problem (1)-(3) with state-dependent impulses under the assumptions (4) and (9). In our approach we will make use of the tools introduced in the previous sections.

In addition we assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M246">View MathML</a>

(34)

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M247">View MathML</a> from (3), w from (7) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M123">View MathML</a> from (8), and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M249">View MathML</a>

(35)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M250">View MathML</a>

(36)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M251">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M252">View MathML</a> are constants from (21).

Remark 11 Let us note that the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M253">View MathML</a> from (35) do not depend on the choice of the fundamental system of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M254">View MathML</a>, but only on the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M255">View MathML</a> of the differential equation (1) and on the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M256">View MathML</a> from (3) (and, of course, on the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M257">View MathML</a>).

Now, we are ready to construct a convenient operator for a representation of problem (1)-(3). Let us choose its domain as the closure of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M258">View MathML</a>

where ℬ is defined in (28) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M169">View MathML</a> from (36).

Now, we have to modify the operator ℋ from Theorem 4 using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261">View MathML</a> instead of the Green’s functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>, that is, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M263">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M264">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M265">View MathML</a>

(37)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M268">View MathML</a>

and W, w, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M273">View MathML</a> are from (7), (12), (18), and (35), respectively.

Let us compare (16) for the operator ℋ with (37) for the operator ℱ. The first term in (16) expresses a solution of homogeneous boundary value problem without impulses. This term is decomposed in (37) on subintervals which depend on the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M241">View MathML</a>-tuple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M275">View MathML</a>. The second term in (16) caused (according to the discontinuity of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a>) needed impulses of solutions of the fixed-time impulsive problem (13)-(15). We significantly modify this term in (37) in such a way that, instead of discontinuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M102">View MathML</a> which have jumps at the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M278">View MathML</a>, we use smooth functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261">View MathML</a> defined in (18). Due to this modification the operator ℱ maps one tuple of smooth functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M281">View MathML</a> onto another tuple of smooth functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M282">View MathML</a>, and we will be able to prove the compactness of ℱ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283">View MathML</a>.

In the next lemma we arrive at a justification of our definition.

Lemma 12Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M284">View MathML</a>is a fixed point of the operator ℱ, then the functionudefined by (33) is a solution of problem (1)-(3) satisfying (22).

Proof Let ℬ be defined by (28) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M285">View MathML</a>. Further, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M286">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M287">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M288">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M289">View MathML</a>, and hence by Lemma 9 and (31), there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M290">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M291">View MathML</a>. Due to (24), the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M292">View MathML</a> are valid and u can be defined by (33). We will prove that u is a fixed point of the operator ℋ from Theorem 4, taking the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M114">View MathML</a> from Remark 3 with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M294">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M295">View MathML</a>

and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M297">View MathML</a>. Then, according to (33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M298">View MathML</a>

Of course we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M299">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M300">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M301">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M302">View MathML</a> and therefore (19) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M303">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M300">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M305">View MathML</a> (such k exists only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M306">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M307">View MathML</a> and therefore we get by (19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M308">View MathML</a>

These facts imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M309">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M94">View MathML</a>. Consequently, by virtue of (16) and Theorem 4, u is a solution of problem (13)-(15). Clearly u fulfils equation (1) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> and satisfies the boundary conditions (3). In addition, since u fulfils the impulse conditions (14) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M312">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M184">View MathML</a>, we see that u also fulfils the state-dependent impulse conditions (2). According to Remark 8, it remains to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M315">View MathML</a> are the only instants at which the function u crosses the barriers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M316">View MathML</a>, respectively. To this aim, due to (24) and (33), it suffices to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M317">View MathML</a>

(38)

Choose an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M195">View MathML</a> and consider σ from (29). Since u fulfils (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M319">View MathML</a>

Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M320">View MathML</a>

From Lemma 9 it follows that ψ is decreasing. So, by virtue of (38), it suffices to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M321">View MathML</a>

(39)

Using (33), (2), and (27), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M322">View MathML</a>

which yields (39). This completes the proof. □

Lemma 13Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. Then the operatorfrom (37) has a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283">View MathML</a>.

Proof The last term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M324">View MathML</a> in (37) is the same as in (16) for the compact operator ℋ. Therefore it suffices to prove the compactness of the operator ℱ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M326">View MathML</a>. To do it we can use the same arguments as in the proof of Lemma 6 in [9], where the second-order state-dependent impulsive problem is investigated. In particular, the compactness of ℱ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283">View MathML</a> is a consequence of the following properties of functions and functionals contained in (37):

• the first term in (37) can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M328">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M329">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M330">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M332">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M333">View MathML</a> are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M179">View MathML</a> (due to Lemma 10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M335">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M261">View MathML</a> satisfy (20), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M338">View MathML</a> satisfies (19),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M339">View MathML</a> are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M340">View MathML</a>.

For the application of the Schauder Fixed Point Theorem it remains to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M341">View MathML</a>

(40)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M342">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M286">View MathML</a>. Then, by (21), (34), (35), and (37), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M344">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M72">View MathML</a>. From (36) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M348">View MathML</a>

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M349">View MathML</a>. We have proved (40), and consequently there exists at least one fixed point of ℱ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M283">View MathML</a>. □

Theorem 14Let assumptions (4), (9), (23)-(27), (34)-(36) be satisfied. Then there exists at least one solution to problem (1)-(3) satisfying (22).

Proof The assertion follows directly from Lemma 12 and Lemma 13. □

Remark 15 The existence result from Theorem 14 can be extended to unbounded functions h and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16">View MathML</a> by means of the method of a priori estimates. This can be found for the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M352">View MathML</a> in [10].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to the manuscript and read and approved the final draft.

Acknowledgements

The authors were supported by the grant IGA_PrF_2014028. The authors sincerely thank the anonymous referees for their valuable comments and suggestions.

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