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

Open Access Research

Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight

Giselle Antunes Monteiro1 and Milan Tvrdý2*

Author Affiliations

1 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic

2 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic

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Boundary Value Problems 2014, 2014:71  doi:10.1186/1687-2770-2014-71


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/2014/1/71


Received:20 January 2014
Accepted:13 March 2014
Published:26 March 2014

© 2014 Monteiro and Tvrdý; 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

In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) on continuous dependence on a parameter k of solutions to linear integral equations of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4">View MathML</a>, X is a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5">View MathML</a> is the Banach space of linear bounded operators on X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M7">View MathML</a> have bounded variations on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M9">View MathML</a> are regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. The integrals are understood as the abstract Kurzweil-Stieltjes integral and the studied equations are usually called generalized linear differential equations (in the sense of Kurzweil, cf. (Kurzweil in Czechoslov. Math. J. 7(82):418-449, 1957) or (Kurzweil in Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions, 2012)). In particular, we are interested in the situation when the variations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M11">View MathML</a> need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.

MSC: 45A05, 34A30, 34N05.

Keywords:
abstract generalized differential equation; continuous dependence; time scale dynamics

1 Introduction

In the theory of differential equations it is always desirable to ensure that their solutions depend continuously on the input data. In other words to ensure that small changes of the input data causes also small changes of the corresponding solutions. For ordinary differential equations, in some sense a final result on the continuous dependence was delivered by Kurzweil and Vorel in their paper [1] from 1957. In fact, it was a response to the averaging method introduced few years before by Krasnoselskij and Krein [2]. The extension of the averaging method and the problem of the continuous dependence of solutions on input data were the main motivations for Kurzweil to introduce his notion of generalized differential equations in [3].

By generalized linear differential equations we understand linear integral equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M12">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4">View MathML</a>, X is a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5">View MathML</a> is the Banach space of linear bounded operators on X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M16">View MathML</a> has bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M18">View MathML</a> is regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> and the integrals are understood in the Kurzweil-Stieltjes sense. By a solution of (1.1) we understand a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M20">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M21">View MathML</a> exists and (1.1) is true for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M23">View MathML</a>, such equations are special cases of equations introduced in 1957 by Kurzweil (see [3]) in connection with the advanced study of continuous dependence properties of ordinary differential equations (see also [1]). In this connection, we want to highlight the recent monograph [4] bringing a new insight into the topic. Linear equations of the form (1.1) have been in the finite-dimensional case thoroughly treated by Schwabik, Tvrdý and Ashordia (see e.g.[5,6] and [7]).

Basic theory of the abstract Kurzweil-Stieltjes integral (called also abstract Perron-Stieltjes or simply gauge-Stieltjes integral) and generalized linear differential equations in a general Banach space has been established by Schwabik in a series of papers [8-10] written between 1996 and 2000. Some of the needed complements have been added in our paper [11].

Taking into account the closing remark in [9], we can see that the following basic existence result is a particular case of [[9], Proposition 2.10].

Proposition 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M24">View MathML</a>have a bounded variation on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Then, (1.1) possesses a unique solutionxon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15">View MathML</a>and every function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a>regulated on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M30">View MathML</a>

(1.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M31">View MathML</a>stands for the identity operator onX. In such a casexis regulated on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M33">View MathML</a>has a bounded variation on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M35">View MathML</a>

(1.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M36">View MathML</a>.

Primarily we are concerned with the continuous dependence of solutions of generalized linear differential equations on a parameter. In particular, we assume that the given equation (1.1) has a unique solution x for each f regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M15">View MathML</a> and we consider a sequence of equations depending on a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M39">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M40">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M41">View MathML</a> have bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M43">View MathML</a> are regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M45">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. We are looking for conditions ensuring that (1.4) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a> for each k large enough and the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M48">View MathML</a> tends uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> to x, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M50">View MathML</a>

(1.5)

In [12] we proved the following two theorems. The first one deals with the case that the variations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M51">View MathML</a> are uniformly bounded.

Proposition 1.2 [[12], Theorem 3.4]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52">View MathML</a>have bounded variation on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M54">View MathML</a>be regulated on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M56">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Furthermore, assume (1.2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M58">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M59">View MathML</a>

(1.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M60">View MathML</a>

(1.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M61">View MathML</a>

(1.9)

Then (1.1) has a unique solutionxon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Furthermore, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>sufficiently large there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>to (1.4) and (1.5) holds.

The second result from [12], inspired by Opial’s paper [13], concerns the situation when variations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M51">View MathML</a> (1.6) need not be uniformly bounded and (1.1) and (1.4) reduce to homogeneous equations.

Proposition 1.3 [[12], Theorem 4.2]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52">View MathML</a>have bounded variation on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M69">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Furthermore, assume (1.2), (1.9) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M71">View MathML</a>

(1.10)

Then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M72">View MathML</a>

(1.11)

has a unique solutionxon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Moreover, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M74">View MathML</a>sufficiently large, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M75">View MathML</a>

(1.12)

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and (1.5) holds.

Let us recall the following observation.

Lemma 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M52">View MathML</a>have bounded variation on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and let (1.10) be satisfied. Then (1.7) is true as well.

Proof The proof follows from the obvious inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M80">View MathML</a>

 □

The only known result (cf. [[12], Corollary 4.4]) concerning nonhomogeneous equations (1.1), (1.4) and the case when (1.6) is not satisfied requires that X is a finite-dimensional space. The aim of this paper is to fill this gap.

For a more detailed list of related references, see [12].

2 Preliminaries

Throughout these notes X is a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5">View MathML</a> is the Banach space of bounded linear operators on X. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M82">View MathML</a> we denote the norm in X. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M83">View MathML</a> denotes the usual operator norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5">View MathML</a>.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> denotes the corresponding closed interval. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M87">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M88">View MathML</a> is said to be a division of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M90">View MathML</a>. The set of all divisions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M92">View MathML</a>.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a> is called a finite step function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> if there exists a division <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M95">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> such that f is constant on every open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98">View MathML</a>.

For an arbitrary function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a> we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M100">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M101">View MathML</a>

is the variation of f over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M103">View MathML</a>, we say that f is a function of bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M105">View MathML</a> denotes the Banach space of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a> of bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M108">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M109">View MathML</a> is called regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M111">View MathML</a> there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M112">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M113">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M114">View MathML</a> there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M115">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M116">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M117">View MathML</a> we denote the Banach space of regulated functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a> equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M119">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M121">View MathML</a> we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M123">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M124">View MathML</a>cf.e.g. the assertion contained in Section 1.5 of [9].

In what follows, by an integral we mean the Kurzweil-Stieltjes integral. Let us recall its definition. As usual, a partition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> is a tagged system, i.e., a couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M126">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M129">View MathML</a> holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98">View MathML</a>. Furthermore, any positive function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M131">View MathML</a> is called a gauge on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Given a gauge δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, the partition P is called δ-fine if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M134">View MathML</a> holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M98">View MathML</a>. We remark that for an arbitrary gauge δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> there always exists a δ-fine partition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. It is stated by the Cousin lemma (see e.g. [[5], Lemma 1.4]).

For given functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M139">View MathML</a> and a partition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M126">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M143">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M144">View MathML</a>

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M145">View MathML</a> is the Kurzweil-Stieltjes integral (or shortly KS-integral) of g with respect to F on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M147">View MathML</a> if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M148">View MathML</a> there exists a gauge δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M150">View MathML</a>

Analogously, we define the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M151">View MathML</a> using sums of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M152">View MathML</a>

Some basic estimates for the KS-integrals are summarized in the following proposition. For the proofs, see [[12], Proposition 2.1] and [[11], Lemma 2.2].

Proposition 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M139">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M155">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M156">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M157">View MathML</a>exists and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M158">View MathML</a>

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M160">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M157">View MathML</a>exists and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M162">View MathML</a>

For more details concerning the abstract KS-integration and further references, see [8-10,14] and [11].

3 Main result

Our main result is based on the following lemma which is an analog of the assertion formulated for ODEs by Kiguradze in [[15], Lemma 2.5]. Its variant was used also in the study of FDEs by Hakl, Lomtatidze and Stavrolaukis in [[16], Lemma 3.5].

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>and assume that (1.2) and (1.10) hold.

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M165">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M167">View MathML</a>

(3.1)

Proof Assume that (3.1) is not true, i.e. assume that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168">View MathML</a> there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M170">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M171">View MathML</a>

(3.2)

We will prove that (3.2) leads to a contradiction. To this aim, first, rewrite inequality (3.2) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M172">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M173">View MathML</a>

(3.4)

Then, by (3.3) and (3.4) we can immediately see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M176">View MathML</a>

(3.5)

Now, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M177">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M178">View MathML</a>

(3.7)

By (3.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M179">View MathML</a>

(3.8)

and, in particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M180">View MathML</a>

(3.9)

Moreover, the equalities (3.6) and (3.7) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M181">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M182">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M183">View MathML</a>

(3.10)

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168">View MathML</a> be fixed. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M185">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M186">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M187">View MathML</a>

(3.12)

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M188">View MathML</a>

(3.13)

Indeed, by (3.10) and Proposition 2.1(ii) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M189">View MathML</a>

wherefrom

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M190">View MathML</a>

(3.14)

follows due to (1.10). Moreover, using Proposition 2.1(i) and (3.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M191">View MathML</a>

and, hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M192">View MathML</a>

(3.15)

Now, (3.13) follows immediately from (3.14) and (3.15).

Finally, having in mind Proposition 1.1 (cf. (1.3)) and (3.11), (3.5), and (3.13), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M193">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M194">View MathML</a>

(3.16)

This, together with (3.7) and (3.9), implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M195">View MathML</a>, which is impossible as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M196">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M168">View MathML</a>. The assertion of the lemma is true. □

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M200">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M201">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Assume (1.2), (1.9), (1.10), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M203">View MathML</a>

(3.17)

Then (1.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M204">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Moreover, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>sufficiently large, (1.4) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M207">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and (1.5) is true.

Proof First, recall that, by Lemma 1.4 our assumption (1.10) implies that (1.7) is true, as well. Therefore, by [[12], Lemma 4.2], there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M210">View MathML</a>

and (1.4) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M207">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212">View MathML</a> (cf. Proposition 1.1). By Lemma 3.1 we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M214">View MathML</a> in such way that (3.1) holds.

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M215">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M217">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M218">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212">View MathML</a>. Using (3.1) we deduce that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M221">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M212">View MathML</a>. Thus, due to (1.9), (1.10) and (3.17), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M223">View MathML</a>, wherefrom (1.5) immediately follows. The proof of the theorem has been completed. □

Remark 3.3 The proof of Theorem 3.2 could be substantially simplified and also extended to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224">View MathML</a> if the following assertion was true.

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M163">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M227">View MathML</a>

(3.18)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M228">View MathML</a>

(3.19)

holds for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224">View MathML</a>.

Unfortunately, this is in general not true even in the scalar case as shown by the following example that was communicated to us by Ivo Vrkoč.

Example 3.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M230">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a> puta

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M233">View MathML</a>

and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M234">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M235">View MathML</a>

It is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M236">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M237">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. In particular, (3.18) is true. However, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M239">View MathML</a>

(3.21)

then f is regulated, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M240">View MathML</a> and (3.19) is not valid since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M241">View MathML</a>

(3.22)

where the right-hand side evidently tends to ∞ for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M242">View MathML</a>.

Moreover, the functions (3.20) and (3.21) provide us with the argument explaining that the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199">View MathML</a> in Theorem 3.2 cannot be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M224">View MathML</a>. Indeed, consider the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M245">View MathML</a>

(3.23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M246">View MathML</a>

(3.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M247">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M250">View MathML</a> is a solution to (3.23) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> and, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>, (3.24) possesses a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a>. Furthermore, conditions (1.10) and (3.17) are satisfied. However, as we will see, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a> does not converge to x.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a> be fixed. It is not difficult to verify that the solution to (3.24) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M258">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M259">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M260">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M261">View MathML</a>. Furthermore, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M262">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M263">View MathML</a>

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M264">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M265">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M266">View MathML</a>

From these formulas we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M267">View MathML</a>

if m is even, while for m odd and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M268">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M269">View MathML</a>

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M270">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Using the above relations and the definition of f, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M272">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M273">View MathML</a> is even, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M274">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M273">View MathML</a> is odd.

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M276">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M277">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M278">View MathML</a>

On the other hand, like in (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M279">View MathML</a>

where the right-hand side tends to ∞ when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M242">View MathML</a>. Consequently, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M281">View MathML</a> cannot have a finite limit for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M282">View MathML</a>.

Remark 3.5 Reasonable examples of sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M283">View MathML</a> that tend to a function f of bounded variation are provided e.g. by sequences of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M284">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M285">View MathML</a> tends to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M287">View MathML</a> tends to 0.

Remark 3.6 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M289">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M290">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M291">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M292">View MathML</a> is said to be the semi-variation ofF on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> (cf.e.g.[17]).b It is clear that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M155">View MathML</a> then F has bounded semi-variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> while the reversed implication is not true in general (cf. [[18], Theorem 2]). By [8] and [11], the Kurzweil-Stieltjes integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M21">View MathML</a> is well defined when both functions, A and x, are regulated and A has bounded semi-variation. Therefore, the study of generalized linear differential equations has a good sense also when A is regulated and has bounded semi-variation instead of having <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M299">View MathML</a>, cf.[9] and [10]. However, the possible extension of Theorem 3.1 to such a case remains open.

Analogously to operator valued functions, the semi-variation of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M28">View MathML</a> could be defined using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M301">View MathML</a>

However, it may be shown that, in this case, f has a bounded semi-variation if and only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199">View MathML</a>. Therefore, the possible replacement of the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M199">View MathML</a> in Theorem 3.1 by the requirement that f has a bounded semi-variation is not interesting.

4 Some applications

Second-order measure equations

Let Y be a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M305">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M306">View MathML</a>. Consider the following system of generalized linear differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M307">View MathML</a>

(4.1)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M309">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M310">View MathML</a> and define functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M312">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M313">View MathML</a>

(4.2)

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M314">View MathML</a>

and system (4.1) can be reformulated as (1.1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M315">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M316">View MathML</a> is a function with values in X. One can verify that condition (1.2) is satisfied whenever one of the following conditions is true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M317">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M318">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M319">View MathML</a> stands for the identity operator on Y.

Indeed, assume e.g. that (4.3) holds and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M320">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M114">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M323">View MathML</a>

(4.5)

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M324">View MathML</a>

(4.6)

By (4.3) the latter equality can happen only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M325">View MathML</a>. Consequently <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M326">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M327">View MathML</a>, as well. Similarly, we would show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M328">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M327">View MathML</a> also in the case that (4.4) is satisfied. This shows that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330">View MathML</a> is injective.

To prove its surjectivity, assume first (4.3) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M331">View MathML</a> be given. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M332">View MathML</a>

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M333">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M334">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M335">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M316">View MathML</a>. Similarly, we can show that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M331">View MathML</a> there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M338">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M339">View MathML</a> also in the case that (4.4) is satisfied. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330">View MathML</a> is surjective. To summarize, according to the Banach theorem, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M330">View MathML</a> possesses a bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M342">View MathML</a>.

Now, consider the systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M343">View MathML</a>

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M345">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>. Assume that (4.3) or (4.4) is true and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M348">View MathML</a>

(4.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M349">View MathML</a>

(4.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M350">View MathML</a>

(4.10)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M352">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M74">View MathML</a> like A and f in (4.2) (however, replace P, Q, g, and h by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M355">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M357">View MathML</a>, respectively). It is easy to see that then the assumptions of Theorem 3.2 are satisfied. Therefore, we can state the following assertion.

Corollary 4.1Assume that (4.3) or (4.4) holds and that (4.8)-(4.10) are satisfied. Then system (4.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M358">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. Moreover, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>sufficiently large, the system (4.7) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M361">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M363">View MathML</a>

In [19], Meng and Zhang investigated the continuous dependence on a parameter k for second-order linear measure differential equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M364">View MathML</a>

(4.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M365">View MathML</a> are normalized measures on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> (generated by functions of bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> and right-continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M368">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M369">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M370">View MathML</a> stands for the generalized right-derivative of y. The main result of [19] is Theorem 1.1, which states that the weak convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M371">View MathML</a> implies the uniform convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M372">View MathML</a> of the corresponding solutions, the weak convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M373">View MathML</a> and the ending velocity convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M374">View MathML</a>.

Notice that our systems (4.7) reduce to (4.11) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M377">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M378">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248">View MathML</a> and both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M357">View MathML</a> are constant [[19], Definition 3.1]. Similarly, if, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M383">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M248">View MathML</a> and both g and h are constant, then system (4.1) reduces to the second-order linear measure differential equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M385">View MathML</a>

(4.12)

where μ is a normalized measure on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M387">View MathML</a>. Obviously, both existence conditions (4.3) and (4.4) are now satisfied. In view of this, assuming that μ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M365">View MathML</a> have a bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M251">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M390">View MathML</a>

it follows from our Corollary 4.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M391">View MathML</a>

holds for the corresponding solutions of (4.11) and (4.12).

Thus, in comparison with Theorem 1.1 in [19], our convergence assumptions are partially stronger. The reason is that our result includes also the uniform convergence of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M392">View MathML</a>. On the other hand, the weak convergence which appears in [19] includes the uniform boundedness of the variations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M393">View MathML</a> (cf.e.g. [[20], Lemma 2.4] or [[21], Section 26]) which is not required in our case.

Linear dynamic equations on time scales

Let us recall some basics of the theory of dynamic equations on time scales. A nonempty closed subset of ℝ is called time scale. For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M395">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M396">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M397">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M398">View MathML</a>

The point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M397">View MathML</a> is said to be right-dense if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M400">View MathML</a>, while it is left-dense if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M401">View MathML</a>. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M402">View MathML</a> is rd-continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403">View MathML</a> if f is continuous at every right-dense point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M405">View MathML</a> for every left-dense point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M406">View MathML</a> (see e.g.[22]).

Let us consider the linear dynamic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M407">View MathML</a>

(4.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M408">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M410">View MathML</a> are rd-continuous functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M411">View MathML</a> stands for the Δ-derivative. By a solution of (4.13) we understand a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M412">View MathML</a> satisfying the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M413">View MathML</a>

where the integral is the Riemann Δ-integral defined e.g. in [22].

As noticed by Slavík (see [[23], Theorem 5]), the Riemann Δ-integral can be regarded as a special case of the Kurzweil-Stieltjes integral. More precisely:

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M414">View MathML</a>be an rd-continuous function and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M415">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M416">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M417">View MathML</a>holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a>.

As a consequence, a relationship between the solutions of (4.13) and generalized linear differential equations can be deduced.

Proposition 4.2 [[23], Theorem 12]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M412">View MathML</a>is a solution of (4.13) then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M420">View MathML</a>

is a solution of (1.1), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M421">View MathML</a>

(4.14)

Symmetrically, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M422">View MathML</a>is a solution of (1.1), withAandfgiven by (4.14), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M423">View MathML</a>defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M424">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M406">View MathML</a>is a solution of (4.13).

It is important to mention that, thanks to the properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M426">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M427">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M428">View MathML</a> given by (4.14) are well defined, left-continuous and of bounded variation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>.

Using the correspondence stated in Proposition 4.2 and Theorem 3.2 we obtain the following result.

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M430">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M431">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>be rd-continuous functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>, be given. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M436">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M437">View MathML</a>

(4.16)

Then initial value problem (4.13) has a solutiony, the initial value problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M438">View MathML</a>

(4.17)

have solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M439">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M441">View MathML</a>

Proof For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M444">View MathML</a>

(4.18)

It is not difficult to see that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M445">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M446">View MathML</a>

and, consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M447">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M448">View MathML</a>

and, analogously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M449">View MathML</a>

These estimates, together with (4.15) and (4.16) imply that the assumptions of Theorem 3.2 are satisfied. Therefore, the uniform convergence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a> of equation (1.5) to the solution x of (1.1) follows. Since by Proposition 4.2 the solutions of (4.13) and (4.17) are, respectively, obtained as the restriction of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M47">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M403">View MathML</a>, the proof is complete. □

Remark 4.4 It is worth to mention that Theorem 4.3 given above encompasses Theorem 5.5 from [12]. This is due to the fact that the weighted convergence assumptions in [[12], Theorem 5.5] involves not only the supremum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M453">View MathML</a>, but also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M454">View MathML</a>.

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

GA Monteiro has bee supported by the Institutional Research Plan No. AV0Z10190503 and by the Academic Human Resource Program of the Academy of Sciences of the Czech Republic and M Tvrdý has been supported by the grant No. 14-06958S of the Grant Agency of the Czech Republic and by the Institutional Research Plan No. AV0Z10190503. The authors sincerely thank Ivo Vrkoč for his valuable contribution to this paper.

End notes

  1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M232">View MathML</a> stands, as usual, for the integer part of the nonnegative real number x.

  2. Sometimes it is called also the ℬ-variation ofF on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a> (with respect to the bilinear triple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M295">View MathML</a>, cf.e.g.[8]).

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