Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 242965 | 26 pages | https://doi.org/10.1155/2011/242965

Linear Hyperbolic Functional-Differential Equations with Essentially Bounded Right-Hand Side

Academic Editor: Stefan Siegmund
Received14 Mar 2011
Revised18 May 2011
Accepted19 May 2011
Published18 Aug 2011

Abstract

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem ๐‘ข(1,1)(๐‘ก,๐‘ฅ)=โ„“0(๐‘ข)(๐‘ก,๐‘ฅ)+โ„“1(๐‘ข(1,0))(๐‘ก,๐‘ฅ)+โ„“2(๐‘ข(0,1))(๐‘ก,๐‘ฅ)+๐‘ž(๐‘ก,๐‘ฅ),๐‘ข(๐‘ก,๐‘)=๐›ผ(๐‘ก) for ๐‘กโˆˆ[๐‘Ž,๐‘],๐‘ข(๐‘Ž,๐‘ฅ)=๐›ฝ(๐‘ฅ)for๐‘ฅโˆˆ[๐‘,๐‘‘] given on the rectangle [๐‘Ž,๐‘]ร—[๐‘,๐‘‘] are established, where the linear operators โ„“0, โ„“1, โ„“2 map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations.

1. Introduction

On the rectangle ๐’Ÿ=[๐‘Ž,๐‘]ร—[๐‘,๐‘‘], we consider the linear partial functional-differential equation๐‘ข(1,1)(๐‘ก,๐‘ฅ)=โ„“0(๐‘ข)(๐‘ก,๐‘ฅ)+โ„“1๎€ท๐‘ข(1,0)๎€ธ(๐‘ก,๐‘ฅ)+โ„“2๎€ท๐‘ข(0,1)๎€ธ(๐‘ก,๐‘ฅ)+๐‘ž(๐‘ก,๐‘ฅ),(1.1) where ๐‘ข(1,0) and ๐‘ข(0,1) (resp., ๐‘ข(1,1)) denote the first-order (resp., the second-order mixed) partial derivatives. The operators โ„“0, โ„“1, and โ„“2 are supposed to be linear and acting from suitable function spaces (see Section 3) to the space of Lebesgue measurable and essentially bounded functions. By a solution to (1.1), we mean a function ๐‘ขโˆถ๐’Ÿโ†’โ„ absolutely continuous in the sense of Carathรฉodory possessing some additional properties (namely, inclusions (2.20)) which satisfies equality (1.1) almost everywhere on ๐’Ÿ.

Three main initial value problems for the hyperbolic equations are studied in the literatureโ€”Darboux, Cauchy, and Goursat problems. In this paper, we consider the Darboux problem in which case the values of a solution ๐‘ข to (1.1) are prescribed on both characteristics ๐‘ก=๐‘Ž and ๐‘ฅ=๐‘, that is, the initial conditions are[][].๐‘ข(๐‘ก,๐‘)=๐›ผ(๐‘ก)for๐‘กโˆˆ๐‘Ž,๐‘,๐‘ข(๐‘Ž,๐‘ฅ)=๐›ฝ(๐‘ฅ)for๐‘ฅโˆˆ๐‘,๐‘‘(1.2) Properties of the initial functions ๐›ผ and ๐›ฝ will be specified in Section 3. It is worth to remember here that various initial and boundary value problems for the hyperbolic equation ๐‘ข๐‘ก๐‘ฅ๎€ท=๐‘“๐‘ก,๐‘ฅ,๐‘ข,๐‘ข๐‘ก,๐‘ข๐‘ฅ๎€ธ(1.3) with continuous as well as discontinuous right-hand sides but without argument deviations have been studied in detail (see, e.g., [1โ€“13] and references therein). As for the hyperbolic functional-differential equations, we can mention for example the works [14โ€“16] (see also references cited therein) but, as far as the authors know, there is still a broad field for further investigation. We have made the first steps in the papers [17, 18] where the Darboux problem for (1.1) with โ„“1=0 and โ„“2=0 is considered.

2. Notation and Definitions

The following notation is used throughout the paper.(i)โ„•, โ„š, and โ„ are the sets of all natural, rational, and real numbers, respectively, โ„+=[0,+โˆž[.(ii)๐’Ÿ=[๐‘Ž,๐‘]ร—[๐‘,๐‘‘], where โˆ’โˆž<๐‘Ž<๐‘<+โˆž and โˆ’โˆž<๐‘<๐‘‘<+โˆž.(iii)The first-order partial derivatives of the function ๐‘ขโˆถ๐’Ÿโ†’โ„ at the point (๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ are denoted by ๐‘ข(1,0)(๐‘ก,๐‘ฅ) (or ๐‘ข๐‘ก(๐‘ก,๐‘ฅ)) and ๐‘ข(0,1)(๐‘ก,๐‘ฅ) (or ๐‘ข๐‘ฅ(๐‘ก,๐‘ฅ)). The second-order mixed partial derivatives of the function ๐‘ขโˆถ๐’Ÿโ†’โ„ at the point (๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ are denoted by ๐‘ข๐‘ก๐‘ฅ(๐‘ก,๐‘ฅ) and ๐‘ข๐‘ฅ๐‘ก(๐‘ก,๐‘ฅ) whereas we use ๐‘ข(1,1)(๐‘ก,๐‘ฅ) if ๐‘ข๐‘ก๐‘ฅ(๐‘ก,๐‘ฅ)=๐‘ข๐‘ฅ๐‘ก(๐‘ก,๐‘ฅ).(iv)๐ถ(๐’Ÿ;โ„) is the Banach space of continuous functions ๐‘ขโˆถ๐’Ÿโ†’โ„ equipped with the norm โ€–๐‘ขโ€–๐ถ๎€ฝ||๐‘ข||โˆถ๎€พ=max(๐‘ก,๐‘ฅ)(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.1)(v)๐ถ([๐›ผ,๐›ฝ];โ„), where โˆ’โˆž<๐›ผ<๐›ฝ<+โˆž, is the linear space of continuous functions ๐‘ฃโˆถ[๐›ผ,๐›ฝ]โ†’โ„.(vi)๐ด๐ถ([๐›ผ,๐›ฝ];โ„), where โˆ’โˆž<๐›ผ<๐›ฝ<+โˆž, is the linear space of absolutely continuous functions ๐‘ฃโˆถ[๐›ผ,๐›ฝ]โ†’โ„.(vii)๐ฟโˆž(๐’Ÿ;โ„) is the Banach space of Lebesgue measurable and essentially bounded functions ๐‘โˆถ๐’Ÿโ†’โ„ equipped with the norm โ€–๐‘โ€–๐ฟโˆž๎€ฝ||๐‘||โˆถ๎€พ=esssup(๐‘ก,๐‘ฅ)(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.2)(viii)๐ฟโˆž(๐’Ÿ;โ„+)={๐‘โˆˆ๐ฟโˆž(๐’Ÿ;โ„)โˆถ๐‘(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ}.(ix)For any ๐‘ง1,๐‘ง2โˆˆ๐ฟโˆž(๐’Ÿ;โ„), we put ๐‘ง2โ‰ฅ๐‘ง1โŸบ๐‘ง2(๐‘ก,๐‘ฅ)โˆ’๐‘ง1๐‘ง(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,2โ‰ซ๐‘ง1โŸบ๐‘ง2(๐‘ก,๐‘ฅ)โˆ’๐‘ง1(๐‘ก,๐‘ฅ)โ‰ฅ๐œ€fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿwithsome๐œ€>0.(2.3)(x)๐ฟโˆž([๐›ผ,๐›ฝ];โ„), where โˆ’โˆž<๐›ผ<๐›ฝ<+โˆž, is the linear space of Lebesgue measurable and essentially bounded functions ๐‘“โˆถ[๐›ผ,๐›ฝ]โ†’โ„.(xi)measโ€‰๐ด denotes the Lebesgue measure of the set ๐ดโŠ‚๐‘…๐‘š, ๐‘š=1,2.(xii)If ๐‘‹, ๐‘Œ are Banach spaces and ๐‘‡โˆถ๐‘‹โ†’๐‘Œ is a linear bounded operator then โ€–๐‘‡โ€– denotes the norm of the operator ๐‘‡, that is, ๎€ฝโ€–๐‘‡โ€–=supโ€–๐‘‡(๐‘ง)โ€–๐‘Œโˆถ๐‘งโˆˆ๐‘‹,โ€–๐‘งโ€–๐‘‹๎€พโ‰ค1.(2.4)

Two subsections below contain a number of definitions used in the sequel.

2.1. Spaces ๐‘[1](๐’Ÿ;โ„), ๐‘[2](๐’Ÿ;โ„), and Set ๐ถโˆ—(๐’Ÿ;โ„)

Motivated by [19, Section 2], the authors introduce the following assertions and definitions.

Lemma 2.1 (see [19, Section 1, Lemma 1]). Let the function ๐‘ขโˆถ๐’Ÿโ†’โ„ be such that [][],[][].๐‘ข(โ‹…,๐‘ฅ)โˆถ๐‘Ž,๐‘โŸถโ„iscontinuousfora.e.๐‘ฅโˆˆ๐‘,๐‘‘๐‘ข(๐‘ก,โ‹…)โˆถ๐‘,๐‘‘โŸถโ„ismeasurableforall๐‘กโˆˆ๐‘Ž,๐‘(2.5) Then the function max{|๐‘ข(๐‘ก,โ‹…)|โˆถ๐‘กโˆˆ[๐‘Ž,๐‘]}โˆถ[๐‘,๐‘‘]โ†’โ„ is measurable.

Notation 1. ๐‘[1](๐’Ÿ;โ„) denotes the linear space of all functions ๐‘ขโˆถ๐’Ÿโ†’โ„ satisfying conditions (2.5), and ๎€ฝ๎€ฝ||๐‘ข||[]๎€พ[]๎€พesssupmax(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘โˆถ๐‘ฅโˆˆ๐‘,๐‘‘<+โˆž.(2.6) If one identifies functions ๐‘ข1, ๐‘ข2 from ๐‘[1](๐’Ÿ;โ„) such that ๐‘ข1(โ‹…,๐‘ฅ)โ‰ก๐‘ข2(โ‹…,๐‘ฅ) for a.e. ๐‘ฅโˆˆ[๐‘,๐‘‘] then โ€–๐‘ขโ€–๐‘[1]๎€ฝ๎€ฝ||๐‘ข||[]๎€พ[]๎€พ=esssupmax(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘โˆถ๐‘ฅโˆˆ๐‘,๐‘‘(2.7) defines a norm in the space ๐‘[1](๐’Ÿ;โ„).
Analogously, we introduce the space ๐‘[2](๐’Ÿ;โ„) of functions which are โ€œmeasurable in the first variable and continuous in the second oneโ€ and define the norm โ€–โ‹…โ€–๐‘[2] there.

The proof of the following proposition is similar to those presented in [19, Section 2, Lemma 1]. For the sake of completeness we prove the proposition here in detail.

Proposition 2.2. ๐‘[1](๐’Ÿ;โ„) and ๐‘[2](๐’Ÿ;โ„) are Banach spaces.

Proof. We only prove the assertion for the space ๐‘[1](๐’Ÿ;โ„), the assertion of the lemma concerning the space ๐‘[2](๐’Ÿ;โ„) can be proven analogously by exchanging the roles of the variables ๐‘ก and ๐‘ฅ.
Let {๐‘ข๐‘˜}+โˆž๐‘˜=1 be an arbitrary Cauchy sequence in ๐‘[1](๐’Ÿ;โ„). For a decreasing sequence of positive numbers {๐œ€๐‘–}+โˆž๐‘–=1 with โˆ‘+โˆž๐‘–=1๐œ€๐‘–<+โˆž there exists an increasing sequence {๐‘˜๐‘–}+โˆž๐‘–=1 such that๎€ฝ๎€ฝ||๐‘ขesssupmax๐‘›(๐‘ก,๐‘ฅ)โˆ’๐‘ข๐‘˜||[]๎€พ[]๎€พ(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘โˆถ๐‘ฅโˆˆ๐‘,๐‘‘<๐œ€๐‘–,(2.8) for every ๐‘›,๐‘˜โ‰ฅ๐‘˜๐‘–, ๐‘–โˆˆโ„•. Let ๐‘ฃ๐‘–=๐‘ข๐‘˜๐‘–(๐‘–=1,2,โ€ฆ). Then, for any ๐‘–โˆˆโ„•, there is a set ๐ธ๐‘–โŠ†[๐‘,๐‘‘], measโ€‰๐ธ๐‘–=๐‘‘โˆ’๐‘, such that ๎€ฝ||๐‘ฃmax๐‘–+1(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ๐‘–||[]๎€พ(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘<๐œ€๐‘–for๐‘ฅโˆˆ๐ธ๐‘–,๐‘–โˆˆโ„•.(2.9) Put ๐ธ=โˆฉ+โˆž๐‘–=1๐ธ๐‘–. Then, clearly, we have measโ€‰๐ธ=๐‘‘โˆ’๐‘ and ๎€ฝ||๐‘ฃmax๐‘›(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ๐‘˜||[]๎€พโ‰ค(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘๐‘›โˆ’1๎“๐‘š=๐‘˜๎€ฝ||๐‘ฃmax๐‘š+1(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ๐‘š||[]๎€พโ‰ค(๐‘ก,๐‘ฅ)โˆถ๐‘กโˆˆ๐‘Ž,๐‘+โˆž๎“๐‘š=๐‘˜๐œ€๐‘šfor๐‘ฅโˆˆ๐ธ,๐‘›>๐‘˜.(2.10) Consequently, for any fixed ๐‘ฅโˆˆ๐ธ, the sequence {๐‘ฃ๐‘–(โ‹…,๐‘ฅ)}+โˆž๐‘–=1 converges uniformly on [๐‘Ž,๐‘], say to ๐‘ข(โ‹…,๐‘ฅ). Hence, {๐‘ฃ๐‘–(๐‘ก,โ‹…)}+โˆž๐‘–=1 converges point-wise on ๐ธ to ๐‘ข(๐‘ก,โ‹…) for every fixed ๐‘กโˆˆ[๐‘Ž,๐‘]. Therefore, the function ๐‘ข satisfies conditions (2.5). Since ๐‘ข๐‘˜(๐‘ก,๐‘ฅ)โˆ’๐‘ข(๐‘ก,๐‘ฅ)=๐‘ข๐‘˜(๐‘ก,๐‘ฅ)โˆ’๐‘ข๐‘˜๐‘–(๐‘ก,๐‘ฅ)+lim๐‘›โ†’+โˆž๎€บ๐‘ฃ๐‘–(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ๐‘›๎€ป(๐‘ก,๐‘ฅ)=๐‘ข๐‘˜(๐‘ก,๐‘ฅ)โˆ’๐‘ข๐‘˜๐‘–(๐‘ก,๐‘ฅ)+lim๐‘›โ†’+โˆž๐‘›โˆ’1๎“๐‘š=๐‘–๎€บ๐‘ฃ๐‘š(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ๐‘š+1(๎€ป๐‘ก,๐‘ฅ)(2.11) holds for ๐‘–,๐‘˜โˆˆโ„•, all ๐‘กโˆˆ[๐‘Ž,๐‘] and a.e. ๐‘ฅโˆˆ[๐‘,๐‘‘], in view of (2.8) and (2.9), we obtain โ€–โ€–๐‘ข๐‘˜โ€–โ€–โˆ’๐‘ข๐‘[1]โ‰ค๐œ€๐‘–++โˆž๎“๐‘š=๐‘–๐œ€๐‘šfor๐‘˜โ‰ฅ๐‘˜๐‘–,๐‘–โˆˆโ„•.(2.12) Hence, ๐‘ขโˆˆ๐‘[1](๐’Ÿ;โ„) and ๐‘ข๐‘›โ†’๐‘ข in ๐‘[1](๐’Ÿ;โ„), that is, the space ๐‘[1](๐’Ÿ;โ„) is complete.

For the investigation of hyperbolic differential equations with discontinuous right-hand side, the concept of a Carathรฉodory solution is usually used (see, e.g., [7, 10, 20, 21]), that is, solutions are considered in the class of absolutely continuous functions. One possible definition of absolute continuity of functions of two variables was given by Carathรฉodory in his monograph [22]. It is also known that such functions admit a certain integral representation. Following the concept mentioned, we introduce the following.

Notation 2. ๐ถโˆ—(๐’Ÿ;โ„) stands for the set of functions ๐‘ขโˆถ๐’Ÿโ†’โ„ admitting the integral representation ๎€œ๐‘ข(๐‘ก,๐‘ฅ)=๐‘ง+๐‘ก๐‘Ž๎€œ๐‘“(๐‘ )d๐‘ +๐‘ฅ๐‘๎€œ๐‘”(๐œ‚)d๐œ‚+๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘โ„Ž(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(2.13) where ๐‘งโˆˆโ„, ๐‘“โˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐‘”โˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and โ„Žโˆˆ๐ฟโˆž(๐’Ÿ;โ„).

The next lemma on differentiating of an indefinite double integral plays a crucial role in our investigation.

Lemma 2.3 (see [23, Proposition 3.5]). Let โ„Žโˆถ๐’Ÿโ†’โ„ be a Lebesgue integrable function and ๎€œ๐‘ฃ(๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘โ„Ž(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.14) Then (1)there exists a set ๐ธโŠ†[๐‘Ž,๐‘] such that meas๐ธ=๐‘โˆ’๐‘Ž and ๐‘ฃ(1,0)๎€œ(๐‘ก,๐‘ฅ)=๐‘ฅ๐‘[],โ„Ž(๐‘ก,๐œ‚)d๐œ‚for๐‘กโˆˆ๐ธ,๐‘ฅโˆˆ๐‘,๐‘‘(2.15)(2)there exists a set ๐นโŠ†[๐‘,๐‘‘] such that meas๐น=๐‘‘โˆ’๐‘ and ๐‘ฃ(0,1)(๎€œ๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž[]โ„Ž(๐‘ ,๐‘ฅ)d๐‘ for๐‘กโˆˆ๐‘Ž,๐‘and๐‘ฅโˆˆ๐น,(2.16)(3)there exists a set ๐บโŠ†๐’Ÿ such that meas๐บ=(๐‘โˆ’๐‘Ž)(๐‘‘โˆ’๐‘) and ๐‘ฃ(1,1)(๐‘ก,๐‘ฅ)=โ„Ž(๐‘ก,๐‘ฅ)for(๐‘ก,๐‘ฅ)โˆˆ๐บ.(2.17)

Remark 2.4. If ๐‘ขโˆˆ๐ถโˆ—(๐’Ÿ;โ„), that is, the function ๐‘ข admits integral representation (2.13), then by using Lemma 2.3 we get ๐‘ข(1,0)๎€œ(๐‘ก,๐‘ฅ)=๐‘“(๐‘ก)+๐‘ฅ๐‘[][],๐‘ขโ„Ž(๐‘ก,๐œ‚)d๐œ‚fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘(0,1)๎€œ(๐‘ก,๐‘ฅ)=๐‘”(๐‘ฅ)+๐‘ก๐‘Ž[][],๐‘ขโ„Ž(๐‘ ,๐‘ฅ)d๐‘ forall๐‘กโˆˆ๐‘Ž,๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘(1,1)(๐‘ก,๐‘ฅ)=โ„Ž(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.18) Consequently, for any ๐‘ขโˆˆ๐ถโˆ—(๐’Ÿ;โ„), we have ๐‘ข(1,0)โˆˆ๐‘[2](๐’Ÿ;โ„),๐‘ข(0,1)โˆˆ๐‘[1](๐’Ÿ;โ„),๐‘ข(1,1)โˆˆ๐ฟโˆž(๐’Ÿ;โ„).(2.19)

Remark 2.5. It follows from Remark 2.4 and [22, Satz 1, page 654] that ๐‘ขโˆˆ๐ถโˆ—(๐’Ÿ;โ„) if and only if ๐‘ขโˆถ๐’Ÿโ†’โ„ is absolutely continuous in the sense of Carathรฉodory with the properties ๐‘ข(1,0)(โ‹…,๐‘)โˆˆ๐ฟโˆž([]๐‘Ž,๐‘;โ„),๐‘ข(0,1)(๐‘Ž,โ‹…)โˆˆ๐ฟโˆž([]๐‘,๐‘‘;โ„),๐‘ข(1,1)โˆˆ๐ฟโˆž(๐’Ÿ;โ„).(2.20)

2.2. Positive and Volterra-Type Operators

We recall here some definitions from the theory of linear operators. We start with the operators acting on the space ๐ถ(๐’Ÿ;โ„).

Definition 2.6. A linear operator โ„“โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) is said to be positive if the relation โ„“(๐‘ข)(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ(2.21) holds whenever the function ๐‘ขโˆˆ๐ถ(๐’Ÿ;โ„) is such that ๐‘ข(๐‘ก,๐‘ฅ)โ‰ฅ0for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.22)

Example 2.7. For any ๐‘ฃโˆˆ๐ถ(๐’Ÿ;โ„), we put โ„“0(๐‘ฃ)(๐‘ก,๐‘ฅ)=๐‘0๎€ท๐œ(๐‘ก,๐‘ฅ)๐‘ฃ0(๐‘ก,๐‘ฅ),๐œ‡0๎€ธ(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(2.23) where ๐‘0โˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐œ0โˆถ๐’Ÿโ†’[๐‘Ž,๐‘], ๐œ‡0โˆถ๐’Ÿโ†’[๐‘,๐‘‘] are measurable functions. Then the operator โ„“0โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) is linear and bounded. Moreover, โ„“0 is positive if and only if ๐‘0(๐‘ก,๐‘ฅ)โ‰ฅ0 for a.e. (๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.

Definition 2.8. A linear operator โ„“โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) is called (๐‘Ž,๐‘)-Volterra operator if, for any (๐‘ก0,๐‘ฅ0)โˆˆ๐’Ÿ and ๐‘ขโˆˆ๐ถ(๐’Ÿ;โ„) such that ๐‘ข๎€บ(๐‘ก,๐‘ฅ)=0for(๐‘ก,๐‘ฅ)โˆˆ๐‘Ž,๐‘ก0๎€ปร—๎€บ๐‘,๐‘ฅ0๎€ป,(2.24) we have โ„“๎€บ(๐‘ข)(๐‘ก,๐‘ฅ)=0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐‘Ž,๐‘ก0๎€ปร—๎€บ๐‘,๐‘ฅ0๎€ป.(2.25)

Remark 2.9. It can be shown by using Lemma 5.8 stated below that the operator โ„“0 given by formula (2.23) is an (๐‘Ž,๐‘)-Volterra one if and only if ||๐‘0||๎€ท๐œ(๐‘ก,๐‘ฅ)0๎€ธ||๐‘(๐‘ก,๐‘ฅ)โˆ’๐‘กโ‰ค0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,0(||๎€ท๐œ‡๐‘ก,๐‘ฅ)0(๎€ธ๐‘ก,๐‘ฅ)โˆ’๐‘ฅโ‰ค0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.26)

Now we introduce analogous notions for linear operators defined on the spaces ๐‘[1](๐’Ÿ;โ„) and ๐‘[2](๐’Ÿ;โ„).

Definition 2.10. We say that a linear operator โ„“โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) (resp., โ„“โˆถ๐‘[2](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„)) is positive if relation (2.21) is satisfied for every function ๐‘ขโˆˆ๐‘[1](๐’Ÿ;โ„) (resp., ๐‘ขโˆˆ๐‘[2](๐’Ÿ;โ„)) such that [][][][]๐‘ข(๐‘ก,๐‘ฅ)โ‰ฅ0for๐‘กโˆˆ๐‘Ž,๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘(resp.,๐‘ข(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘).(2.27)

Example 2.11. For any ๐‘ฃโˆˆ๐‘[2](๐’Ÿ;โ„) (resp., ๐‘ฃโˆˆ๐‘[1](๐’Ÿ;โ„)), we put โ„“1(๐‘ฃ)(๐‘ก,๐‘ฅ)=๐‘1๎€ท(๐‘ก,๐‘ฅ)๐‘ฃ๐‘ก,๐œ‡1๎€ธ(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(2.28) respectively, โ„“2(๐‘ฃ)(๐‘ก,๐‘ฅ)=๐‘2๎€ท๐œ(๐‘ก,๐‘ฅ)๐‘ฃ2๎€ธ(๐‘ก,๐‘ฅ),๐‘ฅfora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(2.29) where ๐‘1,๐‘2โˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐œ‡1โˆถ๐’Ÿโ†’[๐‘,๐‘‘], ๐œ2โˆถ๐’Ÿโ†’[๐‘Ž,๐‘] are measurable functions. Then the operators โ„“1โˆถ๐‘[2](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) and โ„“2โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) are linear and bounded. Moreover, โ„“1 (resp., โ„“2) is positive if and only if ๐‘1(๐‘ก,๐‘ฅ)โ‰ฅ0 (resp., ๐‘2(๐‘ก,๐‘ฅ)โ‰ฅ0) for a.e. (๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.

Definition 2.12. We say that a linear operator โ„“โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) (resp., โ„“โˆถ๐‘[2](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„)) is an ๐‘Ž-Volterra operator (resp., a ๐‘-Volterra operator) if, for any ๐‘ก0โˆˆ[๐‘Ž,๐‘] (resp., ๐‘ฅ0โˆˆ[๐‘,๐‘‘]) and ๐‘ขโˆˆ๐‘[1](๐’Ÿ;โ„) (resp., ๐‘ขโˆˆ๐‘[2](๐’Ÿ;โ„)) such that ๐‘ข๎€บ(๐‘ก,๐‘ฅ)=0for๐‘กโˆˆ๐‘Ž,๐‘ก0๎€ป[]๎€ท[]๎€บanda.e.๐‘ฅโˆˆ๐‘,๐‘‘resp.,๐‘ข(๐‘ก,๐‘ฅ)=0fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘ฅ0,๎€ป๎€ธ(2.30) we have โ„“๎€บ(๐‘ข)(๐‘ก,๐‘ฅ)=0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐‘Ž,๐‘ก0๎€ปร—[]๎€ท[]ร—๎€บ๐‘,๐‘‘resp.,โ„“(๐‘ข)(๐‘ก,๐‘ฅ)=0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐‘Ž,๐‘๐‘,๐‘ฅ0.๎€ป๎€ธ(2.31)

Remark 2.13. One can show by using Lemma 5.9 (resp., Lemma 5.10) stated below that the operator โ„“1 (resp., โ„“2) given by formula (2.28) (resp., (2.29)) is a ๐‘-Volterra one (resp., an ๐‘Ž-Volterra one) if and only if ||๐‘1||๎€ท๐œ‡(๐‘ก,๐‘ฅ)1๎€ธ(๐‘ก,๐‘ฅ)โˆ’๐‘ฅโ‰ค0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(2.32) respectively, ||๐‘2||๎€ท๐œ(๐‘ก,๐‘ฅ)2๎€ธ(๐‘ก,๐‘ฅ)โˆ’๐‘กโ‰ค0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(2.33)

3. Statement of Problem

On the rectangle ๐’Ÿ, we consider the linear nonhomogeneous Darboux problem (1.1), (1.2) in which โ„“0โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„), โ„“1โˆถ๐‘[2](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„), and โ„“2โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) are linear bounded operators, ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„), and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) are such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„),๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘). By a solution to problem (1.1), (1.2), we mean a function ๐‘ขโˆˆ๐ถโˆ—(๐’Ÿ;โ„) possessing property (1.2) and satisfying equality (1.1) almost everywhere on ๐’Ÿ. Let us mention that, in view of Remark 2.4, the definition of a solution to the problem considered is meaningful.

We are interested in question on the unique solvability of problem (1.1), (1.2), and nonnegativity of its solutions. Clearly, the second-order hyperbolic differential equation๐‘ข๐‘ก๐‘ฅ=๐‘0(๐‘ก,๐‘ฅ)๐‘ข+๐‘1(๐‘ก,๐‘ฅ)๐‘ข๐‘ก+๐‘2(๐‘ก,๐‘ฅ)๐‘ข๐‘ฅ+๐‘ž(๐‘ก,๐‘ฅ),(3.1) where ๐‘0,๐‘1,๐‘2,๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„), is a particular case of (1.1). It follows from the results due to Deimling (see [20, 21]) that, among others, problem (3.1), (1.2) has a unique solution without any additional assumptions imposed on the coefficients ๐‘0, ๐‘1, and ๐‘2. We would like to get solvability conditions for general problem (1.1), (1.2) which conform to those well known for (3.1), (1.2).

The main results (namely, Theorems 4.1 and 4.4) will be illustrated on the hyperbolic differential equation with argument deviations๐‘ข(1,1)(๐‘ก,๐‘ฅ)=๐‘0๎€ท๐œ(๐‘ก,๐‘ฅ)๐‘ข0(๐‘ก,๐‘ฅ),๐œ‡0๎€ธ(๐‘ก,๐‘ฅ)+๐‘1(๐‘ก,๐‘ฅ)๐‘ข(1,0)๎€ท๐‘ก,๐œ‡1๎€ธ(๐‘ก,๐‘ฅ)+๐‘2(๐‘ก,๐‘ฅ)๐‘ข(0,1)๎€ท๐œ2๎€ธ(๐‘ก,๐‘ฅ),๐‘ฅ+๐‘ž(๐‘ก,๐‘ฅ),(3.2) in which coefficients ๐‘0,๐‘1,๐‘2,๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and argument deviations ๐œ0,๐œ2โˆถ๐’Ÿโ†’[๐‘Ž,๐‘], ๐œ‡0,๐œ‡1โˆถ๐’Ÿโ†’[๐‘,๐‘‘] are measurable functions. We obtain this equation from (1.1) if the operators โ„“0, โ„“1, and โ„“2 are defined by formulas (2.23), (2.28), and (2.29), respectively. Let us also mention that in the case, where๐œ๐‘˜(๐‘ก,๐‘ฅ)=๐‘ก,๐œ‡๐‘—(๐‘ก,๐‘ฅ)=๐‘ฅfora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ(๐‘˜=0,2,๐‘—=0,1),(3.3) equation (3.2) takes form (3.1).

4. Main Results

At first, we put๐ด๐‘˜(๐‘ง)=โ„“๐‘˜๎€ท๐œ‘๐‘˜๎€ธ(๐‘ง)for๐‘งโˆˆ๐ฟโˆž(๐’Ÿ;โ„),๐‘˜=0,1,2,(4.1) where ๐œ‘0(๎€œ๐‘ง)(๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘๐œ‘๐‘ง(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,1๎€œ(๐‘ง)(๐‘ก,๐‘ฅ)=๐‘ฅ๐‘[][],๐œ‘๐‘ง(๐‘ก,๐œ‚)d๐œ‚fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘2๎€œ(๐‘ง)(๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž[][].๐‘ง(๐‘ ,๐‘ฅ)d๐‘ for๐‘กโˆˆ๐‘Ž,๐‘,a.e.๐‘ฅโˆˆ๐‘,๐‘‘(4.2)

Clearly, ๐œ‘0โˆถ๐ฟโˆž(๐’Ÿ;โ„)โ†’๐ถ(๐’Ÿ;โ„), ๐œ‘1โˆถ๐ฟโˆž(๐’Ÿ;โ„)โ†’๐‘[2](๐’Ÿ;โ„), ๐œ‘2โˆถ๐ฟโˆž(๐’Ÿ;โ„)โ†’๐‘[1](๐’Ÿ;โ„) and thus the operators ๐ด0, ๐ด1, ๐ด2 mapping the space ๐ฟโˆž(๐’Ÿ;โ„) into itself are linear and bounded.

Theorem 4.1. Let ๐ด=๐ด0+๐ด1+๐ด2, where the operators ๐ด0, ๐ด1, ๐ด2 are defined by relations (4.1), (4.2). If the spectral radius of the operator ๐ด is less than one then problem (1.1), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„),๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘).

Theorem 4.1 implies the following.

Corollary 4.2. If the inequality โ€–โ€–โ„“(๐‘โˆ’๐‘Ž)(๐‘‘โˆ’๐‘)0โ€–โ€–โ€–โ€–โ„“+(๐‘‘โˆ’๐‘)1โ€–โ€–โ€–โ€–โ„“+(๐‘โˆ’๐‘Ž)2โ€–โ€–<1(4.3) holds then problem (1.1), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„),๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘).

Remark 4.3. On the rectangle [๐‘Ž,๐‘]ร—[๐‘,๐‘‘], we consider the equation ๐‘ข(1,1)(๐‘ก,๐‘ฅ)=๐‘0๐‘ข(๐‘,๐‘‘)+๐‘1๐‘ข(1,0)(๐‘ก,๐‘‘)+๐‘2๐‘ข(0,1)(๐‘,๐‘ฅ)(4.4) subjected to the initial conditions [][],๐‘ข(๐‘ก,๐‘)=0for๐‘กโˆˆ๐‘Ž,๐‘,๐‘ข(๐‘Ž,๐‘ฅ)=0for๐‘ฅโˆˆ๐‘,๐‘‘(4.5) where ๐‘0=๐‘š0(๐‘โˆ’๐‘Ž)(๐‘‘โˆ’๐‘),๐‘1=๐‘š1๐‘‘โˆ’๐‘,๐‘2=๐‘š2.๐‘โˆ’๐‘Ž(4.6) Clearly (4.4) is a particular case of (1.1). If ๐‘š0+๐‘š1+๐‘š2=1, then problem (4.4), (4.5) has the trivial solution ๐‘ข(๐‘ก,๐‘ฅ)โ‰ก0 and the nontrivial solution ๐‘ข(๐‘ก,๐‘ฅ)โ‰ก(๐‘กโˆ’๐‘Ž)(๐‘ฅโˆ’๐‘). It justifies that the strict inequality (4.3) in the previous corollary is essential and cannot be replaced by the nonstrict one. On the other hand, it is worth to mention that the inequality indicated is very restrictive and thus it is far from being optimal for a wide class of equations (1.1).

If the operators โ„“0, โ„“1, and โ„“2 on the right-hand side of (1.1) are positive then we can estimate the spectral radius of the operator ๐ด by using the well-known results due to Krasnosel'skij and we thus obtain the following.

Theorem 4.4. Let the operators โ„“0, โ„“1, โ„“2 be positive and ๐ด=๐ด0+๐ด1+๐ด2, where the operators ๐ด0, ๐ด1, ๐ด2 are defined by relations (4.1), (4.2). Then the following four assertions are equivalent. (1)There exists a function ๐‘ง0โˆˆ๐ฟโˆž(๐’Ÿ;โ„+) such that ๐‘ง0โ‰ซ๐ด(๐‘ง0).(2)The spectral radius of the operator ๐ด is less than one.(3)Problem (1.1), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„),๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘).If, in addition, the initial functions ๐›ผ,๐›ฝand the forcing term ๐‘žare such that ๐›ผ(๐‘Ž)โ‰ฅ0,๐›ผ๎…ž(๐‘ก)โ‰ฅ0,๐›ฝ๎…ž(๐‘ฅ)โ‰ฅ0,๐‘ž(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(4.7) then the solution ๐‘ขto problem (1.1), (1.2) satisfies ๐‘ข๐‘ข(๐‘ก,๐‘ฅ)โ‰ฅ0for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(1,0)[][],๐‘ข(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.๐‘กโˆˆa,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘(0,1)[][].(๐‘ก,๐‘ฅ)โ‰ฅ0for๐‘กโˆˆ๐‘Ž,๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘(4.8)(4)There exists a function ๐›พโˆˆ๐ถโˆ—(๐’Ÿ;โ„) such that๐›พ(1,1)โ‰ซโ„“0(๐›พ)+โ„“1๎€ท๐›พ(1,0)๎€ธ+โ„“2๎€ท๐›พ(0,1)๎€ธ,๐›พ(4.9)๐›พ(๐‘Ž,๐‘)โ‰ฅ0,(4.10)(1,0)[](๐‘ก,๐‘)โ‰ฅ0fora.e.๐‘กโˆˆ๐‘Ž,๐‘,๐›พ(0,1)[]๐›พ(๐‘Ž,๐‘ฅ)โ‰ฅ0fora.e.๐‘ฅโˆˆ๐‘,๐‘‘,(4.11)(1,1)(๐‘ก,๐‘ฅ)โ‰ฅ0fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(4.12)

For Volterra-type operators โ„“0, โ„“1, and โ„“2, we derive from the previous theorem the following.

Corollary 4.5. Let โ„“0, โ„“1, and โ„“2 be positive (๐‘Ž,๐‘)-Volterra, ๐‘-Volterra, and ๐‘Ž-Volterra operators, respectively, such that the inequalities โ„“1(๐‘ฆ)(๐‘ก,๐‘ฅ)โ‰ค๐‘ฆ(๐‘ก)โ„“1(1)(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ(4.13)(here,โ„“1(๐‘ฆ) means โ„“1(๐‘ฆ) in which ๐‘ฆ(๐‘ก,๐‘ฅ)=๐‘ฆ(๐‘ก) for ๐‘Ž.๐‘’.๐‘กโˆˆ[๐‘Ž,๐‘] and all ๐‘ฅโˆˆ[๐‘,๐‘‘]) and โ„“2(๐‘ง)(๐‘ก,๐‘ฅ)โ‰ค๐‘ง(๐‘ฅ)โ„“2(1)(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ(4.14)(byโ„“2(๐‘ง) we mean โ„“2(๐‘ง), where ๐‘ง(๐‘ก,๐‘ฅ)=๐‘ง(๐‘ฅ)forall๐‘กโˆˆ[๐‘Ž,๐‘] and a.e. ๐‘ฅโˆˆ[๐‘,๐‘‘]) hold for every ๐‘ฆโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„) and ๐‘งโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„).
Then problem (1.1), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„),๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘). If, in addition, the initial functions ๐›ผ, ๐›ฝ and the forcing term ๐‘ž are such that relations (4.7) hold, then the solution ๐‘ข to problem (1.1), (1.2) satisfies inequalities (4.8).

Following our previous results concerning the case, where โ„“1=0 and โ„“2=0 (see [18]), we can introduce the following.

Definition 4.6. Let โ„“0โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„), โ„“1โˆถ๐‘[2](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„), and โ„“2โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„). We say that the triplet (โ„“0,โ„“1,โ„“2) belongs to the set ๐’ฎ๎…ž๐‘Ž๐‘ if the implication ๐‘ขโˆˆ๐ถโˆ—๐‘ข(๐’Ÿ;โ„),(1,1)(๐‘ก,๐‘ฅ)โ‰ฅโ„“0(๐‘ข)(๐‘ก,๐‘ฅ)+โ„“1๎€ท๐‘ข(1,0)๎€ธ(๐‘ก,๐‘ฅ)+โ„“2๎€ท๐‘ข(0,1)๎€ธ๐‘ข(๐‘ก,๐‘ฅ),fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,๐‘ข(๐‘Ž,๐‘)โ‰ฅ0,(1,0)[],๐‘ข(๐‘ก,๐‘)โ‰ฅ0fora.e.๐‘กโˆˆ๐‘Ž,๐‘(0,1)[],(๐‘Ž,๐‘ฅ)โ‰ฅ0fora.e.๐‘ฅโˆˆ๐‘,๐‘‘โŸน๐‘ขsatis๏ฌes(4.8)(4.15) holds.

Remark 4.7. If (โ„“0,โ„“1,โ„“2)โˆˆ๐’ฎ๎…ž๐‘Ž๐‘, we usually say that a certain theorem on differential inequalities holds for (1.1). It should be noted here that there is another terminology which says that a certain maximum principle holds for (1.1) if the inclusion (โ„“0,โ„“1,โ„“2)โˆˆ๐’ฎ๎…ž๐‘Ž๐‘ is fulfilled.

Theorem 4.4 immediately yields the following.

Corollary 4.8. If one of assertions (1)โ€“(4) stated in Theorem 4.4 holds then (โ„“0,โ„“1,โ„“2)โˆˆ๐’ฎ๎…žac.

Remark 4.9. The inclusion (โ„“0,โ„“1,โ„“2)โˆˆ๐’ฎ๎…ž๐‘Ž๐‘ ensures that every solution ๐‘ข to problem (1.1), (1.2) with (4.7) satisfies relations (4.8). However, we do not know whether this inclusion also guarantees the unique solvability of problem (1.1), (1.2) for arbitrary ๐‘ž, ๐›ผ, and ๐›ฝ. Consequently, we cannot reverse the assertion of the previous corollary.
The reason lays in the question whether the Fredholm alternative holds for problem (1.1), (1.2) or not. In fact, we are not able to prove compactness of the operator ๐ด appearing in Theorem 4.4 which plays a crucial role in the proofs of the Fredholm alternative for problem (1.1), (1.2) as well as a continuous dependence of its solutions on the initial data and parameters.

Now we apply general results to (3.2) with argument deviations in which coefficients ๐‘0,๐‘1,๐‘2,๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and argument deviations ๐œ0,๐œ2โˆถ๐’Ÿโ†’[๐‘Ž,๐‘], ๐œ‡0,๐œ‡1โˆถ๐’Ÿโ†’[๐‘,๐‘‘] are measurable functions.

As a consequence of Corollary 4.2 we obtain the following.

Corollary 4.10. If the inequality โ€–โ€–๐‘(bโˆ’๐‘Ž)(๐‘‘โˆ’๐‘)0โ€–โ€–๐ฟโˆžโ€–โ€–๐‘+(๐‘‘โˆ’๐‘)1โ€–โ€–๐ฟโˆžโ€–โ€–๐‘+(๐‘โˆ’๐‘Ž)2โ€–โ€–๐ฟโˆž<1(4.16) holds, then problem (3.2), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘).

If the coefficients ๐‘0, ๐‘1, ๐‘2 in the previous corollary are non-negative then the assertion of the corollary follows also from implication (4)โ‡’(3) of Theorem 4.4. More precisely, the following statement holds.

Corollary 4.11. Let ๐‘0,๐‘1,๐‘2โˆˆ๐ฟโˆž(๐’Ÿ;โ„+) and ๎€ฝ๐‘esssup0๎€ท๐œ(๐‘ก,๐‘ฅ)0๐œ‡(๐‘ก,๐‘ฅ)โˆ’๐‘Ž๎€ธ๎€ท0๎€ธ(๐‘ก,๐‘ฅ)โˆ’๐‘+๐‘1๎€ท๐œ‡(๐‘ก,๐‘ฅ)1๎€ธ(๐‘ก,๐‘ฅ)โˆ’๐‘+๐‘2(๎€ท๐œ๐‘ก,๐‘ฅ)2(๎€ธ๎€พ๐‘ก,๐‘ฅ)โˆ’๐‘Žโˆถ(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ<1.(4.17) Then problem (3.2), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘). If, in addition, the initial functions ๐›ผ, ๐›ฝ, and the forcing term ๐‘ž are such that relations (4.7) hold, then the solution ๐‘ข to problem (3.2), (1.2) satisfies inequalities (4.8).

Finally, Corollary 4.5 implies the following.

Corollary 4.12. Let ๐‘0,๐‘1,๐‘2โˆˆ๐ฟโˆž(๐’Ÿ;โ„+) and argument deviations ๐œ0, ๐œ‡0, ๐œ‡1, and ๐œ2 satisfy inequalities (2.26), (2.32), and (2.33). Then problem (3.2), (1.2) is uniquely solvable for arbitrary ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘). If, in addition, the initial functions ๐›ผ, ๐›ฝ and the forcing term ๐‘ž are such that relations (4.7) hold, then the solution ๐‘ข to problem (3.2), (1.2) satisfies inequalities (4.8).

The assumptions of the previous corollary require, in fact, that (3.2) is delayed in all its deviating arguments. Observe that in the case, where (3.3) holds, the inequalities (2.26), (2.32), and (2.33) are satisfied trivially and Corollary 4.12 thus conform to the results well known for (3.1). The following statements show that the assertion of Corollary 4.12 remains true if the deviations ๐œ0, ๐œ‡0, ๐œ‡1, and ๐œ2 are not necessarily delays but the differences ๐œ๐‘˜(๐‘ก,๐‘ฅ)โˆ’๐‘ก,๐œ‡๐‘—(๐‘ก,๐‘ฅ)โˆ’๐‘ฅ(๐‘˜=0,2,๐‘—=0,1)(4.18) are small enough, that is, if (3.2) with deviating arguments is โ€œcloseโ€ to (3.1).

Corollary 4.13. Let ๐‘0,๐‘1,๐‘2โˆˆ๐ฟโˆž(๐’Ÿ;โ„+),๐‘๐‘˜โ‰ข0(๐‘˜=1,2), and ๎‚ป๎€œesssup๐œ0๐‘ก(๐‘ก,๐‘ฅ)๎€œ๐œ‡0๐‘(๐‘ก,๐‘ฅ)๐‘0๎€œ(๐‘ ,๐œ‚)d๐œ‚d๐‘ +๐‘ก๐‘Ž๎€œ๐œ‡0๐‘ฅ(๐‘ก,๐‘ฅ)๐‘0โ€–โ€–๐‘(๐‘ ,๐œ‚)d๐œ‚d๐‘ +22โ€–โ€–๐ฟโˆž๎€ท๐œ0๎€ธโ€–โ€–๐‘(๐‘ก,๐‘ฅ)โˆ’๐‘ก+21โ€–โ€–๐ฟโˆž๎€ท๐œ‡0๎€ธโˆถ๎‚ผโ‰ค๐œ”(๐‘ก,๐‘ฅ)โˆ’๐‘ฅ(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿe๎‚€11+ln๐œ”๎‚,๎‚ป๎€œesssup๐‘ก๐‘Ž๎€œ๐œ‡1๐‘ฅ(๐‘ก,๐‘ฅ)๐‘0โ€–โ€–๐‘(๐‘ ,๐œ‚)d๐œ‚d๐‘ +21โ€–โ€–๐ฟโˆž๎€ท๐œ‡1๎€ธ๎‚ผ<๐œ”(๐‘ก,๐‘ฅ)โˆ’๐‘ฅโˆถ(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿe,๎‚ป๎€œesssup๐œ2๐‘ก(๐‘ก,๐‘ฅ)๎€œ๐‘ฅ๐‘๐‘0โ€–โ€–๐‘(๐‘ ,๐œ‚)d๐œ‚d๐‘ +22โ€–โ€–๐ฟโˆž๎€ท๐œ2๎€ธ๎‚ผ<๐œ”(๐‘ก,๐‘ฅ)โˆ’๐‘กโˆถ(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿe,(4.19)where ๎€ฝโ€–โ€–๐‘๐œ”=2min1โ€–โ€–๐ฟโˆž,โ€–โ€–๐‘2โ€–โ€–๐ฟโˆž๎€พโ€–โ€–๐‘0โ€–โ€–๐ฟโˆž๎€ฝโ€–โ€–๐‘max{๐‘โˆ’๐‘Ž,๐‘‘โˆ’๐‘}+2min1โ€–โ€–๐ฟโˆž,โ€–โ€–๐‘2โ€–โ€–๐ฟโˆž๎€พ.(4.20) Then the assertion of Corollary 4.12 holds.

5. Auxiliary Statements and Proofs of Main Results

The proofs use several auxiliary statements given in the next subsection.

5.1. Auxiliary Statements

Remember that, for given operators โ„“0, โ„“1, and โ„“2, the operators ๐ด0, ๐ด1, and ๐ด2 are defined by relations (4.1), (4.2). Moreover, having ๐‘žโˆˆ๐ฟโˆž(๐’Ÿ;โ„) and ๐›ผโˆˆ๐ด๐ถ([๐‘Ž,๐‘];โ„), ๐›ฝโˆˆ๐ด๐ถ([๐‘,๐‘‘];โ„) such that ๐›ผ๎…žโˆˆ๐ฟโˆž([๐‘Ž,๐‘];โ„), ๐›ฝ๎…žโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„), and ๐›ผ(๐‘Ž)=๐›ฝ(๐‘), we put ๐‘ฆ=โ„“0(โˆ’๐›ผ(๐‘Ž)+๐›ผ+๐›ฝ)+โ„“1๎€ท๐›ผ๎…ž๎€ธ+โ„“2๎€ท๐›ฝ๎…ž๎€ธ+๐‘ž(5.1)(byโ„“0(โˆ’๐›ผ(๐‘Ž)+๐›ผ+๐›ฝ) the authors understand โ„“0(๐œŽ) in which ๐œŽ(๐‘ก,๐‘ฅ)=โˆ’๐›ผ(๐‘Ž)+๐›ผ(๐‘ก)+๐›ฝ(๐‘ฅ) for (๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ. Similarly, โ„“1(๐›ผ๎…ž)(resp.,โ„“2(๐›ฝ๎…ž)) means โ„“1(๐›ผ0)(resp.,โ„“2(๐›ฝ0)), where ๐›ผ0(๐‘ก,๐‘ฅ)=๐›ผ๎…ž(๐‘ก) for a.e.๐‘กโˆˆ[๐‘Ž,๐‘] and all ๐‘ฅโˆˆ[๐‘,๐‘‘](resp.,๐›ฝ0(๐‘ก,๐‘ฅ)=๐›ฝ๎…ž(๐‘ฅ) for all ๐‘กโˆˆ[๐‘Ž,๐‘]anda.e.๐‘ฅโˆˆ[๐‘,๐‘‘])). Clearly, ๐‘ฆโˆˆ๐ฟโˆž(๐’Ÿ;โ„).

Lemma 5.1. If ๐‘ข is a solution to problem (1.1), (1.2) then ๐‘ข(1,1) is a solution to the equation ๎€ท๐ด๐‘ง=0+๐ด1+๐ด2๎€ธ(๐‘ง)+๐‘ฆ(5.2) in the space ๐ฟโˆž(๐’Ÿ;โ„), where the operators ๐ด0, ๐ด1, and ๐ด2 are defined by relations (4.1), (4.2) and the function ๐‘ฆ is given by formula (5.1).
Conversely, if ๐‘ง is a solution to (5.2) in the space ๐ฟโˆž(๐’Ÿ;โ„) with the operators ๐ด0, ๐ด1, and ๐ด2 defined by relations (4.1), (4.2) and the function ๐‘ฆ given by formula (5.1), then ๎€œ๐‘ข(๐‘ก,๐‘ฅ)=โˆ’๐›ผ(๐‘Ž)+๐›ผ(๐‘ก)+๐›ฝ(๐‘ฅ)+๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘๐‘ง(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ(5.3) is a solution to problem (1.1), (1.2).

Proof. If ๐‘ข is a solution to problem (1.1), (1.2) then, by virtue of Remark 2.4, we get ๐‘ข(1,1)โˆˆ๐ฟโˆž(๐’Ÿ;โ„), ๎€œ๐‘ข(๐‘ก,๐‘ฅ)=โˆ’๐›ผ(๐‘Ž)+๐›ผ(๐‘ก)+๐›ฝ(๐‘ฅ)+๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘๐‘ข(1,1)(๐‘ข๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ,(1,0)(๐‘ก,๐‘ฅ)=๐›ผ๎…ž๎€œ(๐‘ก)+๐‘ฅ๐‘๐‘ข(1,1)[][],๐‘ข(๐‘ก,๐œ‚)d๐œ‚fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘(0,1)(๐‘ก,๐‘ฅ)=๐›ฝ๎…ž๎€œ(๐‘ฅ)+๐‘ก๐‘Ž๐‘ข(1,1)[][].(๐‘ ,๐‘ฅ)d๐‘ forall๐‘กโˆˆ๐‘Ž,๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘(5.4) Consequently, (1.1) yields that ๐‘ข(1,1)=๎€ท๐ด0+๐ด1+๐ด2๐‘ข๎€ธ๎€ท(1,1)๎€ธ+๐‘ฆ,(5.5) where the operators ๐ด0, ๐ด1, and ๐ด2 are defined by relations (4.1), (4.2) and the function ๐‘ฆ is given by formula (5.1).
Conversely, let ๐‘ง be a solution to (5.2) in the space ๐ฟโˆž(๐’Ÿ;โ„) with the operators ๐ด0, ๐ด1, and ๐ด2 defined by relations (4.1), (4.2) and the function ๐‘ฆ given by formula (5.1). Moreover, let the function ๐‘ข be defined by relation (5.3), that is,๎€œ๐‘ข(๐‘ก,๐‘ฅ)=๐›ผ(๐‘Ž)+๐‘ก๐‘Ž๐›ผ๎…ž(๎€œ๐‘ )d๐‘ +๐‘ฅ๐‘๐›ฝ๎…ž(๎€œ๐œ‚)d๐œ‚+๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘๐‘ง(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(5.6) Then the function ๐‘ข belongs to the set ๐ถโˆ—(๐’Ÿ;โ„) and verifies initial conditions (1.2). Furthermore, by using Lemma 2.3, we get ๐‘ข(1,0)(๐‘ก,๐‘ฅ)=๐›ผ๎…ž๎€œ(๐‘ก)+๐‘ฅ๐‘[][],๐‘ข๐‘ง(๐‘ก,๐œ‚)d๐œ‚fora.e.๐‘กโˆˆ๐‘Ž,๐‘andall๐‘ฅโˆˆ๐‘,๐‘‘(0,1)(๐‘ก,๐‘ฅ)=๐›ฝ๎…ž๎€œ(๐‘ฅ)+๐‘ก๐‘Ž[][],๐‘ข๐‘ง(๐‘ ,๐‘ฅ)d๐‘ forall๐‘กโˆˆ๐‘Ž,๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘(1,1)(๐‘ก,๐‘ฅ)=๐‘ง(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(5.7) Consequently, (5.2) implies that ๐‘ข is also a solution to (1.1).

Now we recall some definitions from the theory of linear operators leaving invariant a cone in a Banach space (see, e.g., [24, 25] and references therein).

Definition 5.2. A nonempty closed set ๐พ in a Banach space ๐‘‹ is called a cone if the following conditions are satisfied: (i)๐‘ฅ+๐‘ฆโˆˆ๐พ for all ๐‘ฅ,๐‘ฆโˆˆ๐พ,(ii)๐œ†๐‘ฅโˆˆ๐พ for all ๐‘ฅโˆˆ๐พ and an arbitrary ๐œ†โ‰ฅ0,(iii)if ๐‘ฅโˆˆ๐พ and โˆ’๐‘ฅโˆˆ๐พ then ๐‘ฅ=0.

Remark 5.3. In the original terminology introduced by Kreฤญn and Rutman [25], a set ๐พ satisfying conditions (i) and (ii) of Definition 5.2 is called a linear semigroup.

Definition 5.4. We say that a cone ๐พโŠ†๐‘‹ is solid if its interior Int๐พ is nonempty.

Remark 5.5. The presence of a cone ๐พ in a Banach space ๐‘‹ allows one to introduce a natural partial ordering there. More precisely, two elements ๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹ are said to be in the relation ๐‘ฅ2โ‰ฅ๐พ๐‘ฅ1 if and only if they satisfy the inclusion ๐‘ฅ2โˆ’๐‘ฅ1โˆˆ๐พ. If, moreover, ๐พ is a solid cone then we write ๐‘ฅ2โ‰ซ๐พ๐‘ฅ1 if and only if ๐‘ฅ2โˆ’๐‘ฅ1โˆˆInt๐พ.

Definition 5.6. A cone ๐พโŠ†๐‘‹ is said to be normal if there is a constant ๐‘โ‰ฅ0 such that, for every ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ with the property 0โ‰ค๐พ๐‘ฅโ‰ค๐พ๐‘ฆ, the relation โ€–๐‘ฅโ€–๐‘‹โ‰ค๐‘โ€–๐‘ฆโ€–๐‘‹ holds.

The proof of the main part of Theorem 4.4 is based on the following result.

Lemma 5.7 (see [24, Theorem 5.6]). Let ๐พ be a normal and solid cone in a Banach space ๐‘‹ and the operator ๐ดโˆถ๐‘‹โ†’๐‘‹ leave invariant the cone ๐พ, that is, ๐ด(๐พ)โŠ†๐พ. If there exists a constant ๐›ฟ>0 and an element ๐‘ฅ0โˆˆInt๐พ such that ๐›ฟ๐‘ฅ0โˆ’๐ด(๐‘ฅ0)โˆˆInt๐พ, then the spectral radius of the operator ๐ด is less than ๐›ฟ.

Finally, we establish three lemmas dealing with Volterra type operators which we need to prove Corollary 4.5.

Lemma 5.8. Let โ„“0โˆถ๐ถ(๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) be a positive (๐‘Ž,๐‘)-Volterra operator. Then, for any function ๐›พโˆˆ๐ถ(๐’Ÿ;โ„) satisfying ๐›พ๎€ท๐‘ก1,๐‘ฅ1๎€ธ๎€ท๐‘กโ‰ค๐›พ2,๐‘ฅ2๎€ธfor๐‘Žโ‰ค๐‘ก1โ‰ค๐‘ก2โ‰ค๐‘,๐‘โ‰ค๐‘ฅ1โ‰ค๐‘ฅ2โ‰ค๐‘‘,(5.8) one has โ„“0(๐›พ)(๐‘ก,๐‘ฅ)โ‰คโ„“0(1)(๐‘ก,๐‘ฅ)๐›พ(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(5.9)

Proof. We first show that, for any (๐‘ก,๐‘ฅ)โˆˆ]๐‘Ž,๐‘]ร—]๐‘,๐‘‘], we have โ„“0(๐›พ)(๐‘ ,๐œ‚)โ‰คโ„“0[]ร—[](1)(๐‘ ,๐œ‚)๐›พ(๐‘ก,๐‘ฅ)fora.e.(๐‘ ,๐œ‚)โˆˆ๐‘Ž,๐‘ก๐‘,๐‘ฅ.(5.10) Indeed, let (๐‘ก,๐‘ฅ)โˆˆ]๐‘Ž,๐‘]ร—]๐‘,๐‘‘] be arbitrary but fixed. Put ๐›พ0(๐‘ ,๐œ‚)=๐›พ(min{๐‘ ,๐‘ก},min{๐œ‚,๐‘ฅ})for(๐‘ ,๐œ‚)โˆˆ๐’Ÿ.(5.11) Then, clearly ๐›พ0โˆˆ๐ถ(๐’Ÿ;โ„), ๐›พ0๐›พ(๐‘ ,๐œ‚)โ‰ค๐›พ(๐‘ก,๐‘ฅ)for(๐‘ ,๐œ‚)โˆˆ๐’Ÿ,0[]ร—[].(๐‘ ,๐œ‚)=๐›พ(๐‘ ,๐œ‚)for(๐‘ ,๐œ‚)โˆˆ๐‘Ž,๐‘ก๐‘,๐‘ฅ(5.12) Since the operator โ„“0 is positive, we obtain โ„“0๎€ท๐›พ0๎€ธ(๐‘ ,๐œ‚)โ‰คโ„“0(๐›พ(๐‘ก,๐‘ฅ))(๐‘ ,๐œ‚)=๐›พ(๐‘ก,๐‘ฅ)โ„“0(1)(๐‘ ,๐œ‚)fora.e.(๐‘ ,๐œ‚)โˆˆ๐’Ÿ.(5.13) On the other hand, the operator โ„“0 is supposed to be an (๐‘Ž,๐‘)-Volterra one which guarantees the equality โ„“0๎€ท๐›พ0๎€ธ(๐‘ ,๐œ‚)=โ„“0[]ร—[](๐›พ)(๐‘ ,๐œ‚)fora.e.(๐‘ ,๐œ‚)โˆˆ๐‘Ž,๐‘ก๐‘,๐‘ฅ,(5.14) and thus the desired relation (5.10) holds.
Now we put ๎€œ๐‘ข(๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘โ„“0(๎€œ๐›พ)(๐‘ ,๐œ‚)d๐œ‚d๐‘ ,๐‘ฃ(๐‘ก,๐‘ฅ)=๐‘ก๐‘Ž๎€œ๐‘ฅ๐‘โ„“0(1)(๐‘ ,๐œ‚)d๐œ‚d๐‘ for(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(5.15) It follows from Lemma 2.3 that there exists a set ๐ธ1โŠ†]๐‘Ž,๐‘], measโ€‰๐ธ1=๐‘โˆ’๐‘Ž, such that ๐‘ข(1,0)๎€œ(๐‘ก,๐‘ฅ)=๐‘ฅ๐‘โ„“0(๐›พ)(๐‘ก,๐œ‚)d๐œ‚for๐‘กโˆˆ๐ธ1[],๐‘ฃ,๐‘ฅโˆˆ๐‘,๐‘‘(1,0)๎€œ(๐‘ก,๐‘ฅ)=๐‘ฅ๐‘โ„“0(1)(๐‘ก,๐œ‚)d๐œ‚for๐‘กโˆˆ๐ธ1[],,๐‘ฅโˆˆ๐‘,๐‘‘(5.16) and, moreover, there is a set ๐ธโŠ†๐ธ1ร—]๐‘,๐‘‘], measโ€‰๐ธ=(๐‘โˆ’๐‘Ž)(๐‘‘โˆ’๐‘), with the properties ๐‘ข(1,1)(๐‘ก,๐‘ฅ)=โ„“0(๐›พ)(๐‘ก,๐‘ฅ),๐‘ฃ(1,1)(๐‘ก,๐‘ฅ)=โ„“0(1)(๐‘ก,๐‘ฅ)for(๐‘ก,๐‘ฅ)โˆˆ๐ธ.(5.17)
Let (๐‘ก,๐‘ฅ)โˆˆ๐ธ be arbitrary but fixed. Then, relation (5.10) yields 1๎€œโ„Ž๐‘˜๐‘ก๐‘กโˆ’โ„Ž๎€œ๐‘ฅ๐‘ฅโˆ’๐‘˜โ„“0(๐›พ)(๐‘ ,๐œ‚)d๐œ‚d๐‘ โ‰ค๐›พ(๐‘ก,๐‘ฅ)๎€œโ„Ž๐‘˜๐‘ก๐‘กโˆ’โ„Ž๎€œ๐‘ฅ๐‘ฅโˆ’๐‘˜โ„“0(1)(๐‘ ,๐œ‚)d๐œ‚d๐‘ (5.18) for โ„Žโˆˆ]0,๐‘กโˆ’๐‘Ž] and ๐‘˜โˆˆ]0,๐‘ฅโˆ’๐‘], whence we get 1๐‘˜๎‚ธ๐‘ข(๐‘ก,๐‘ฅ)โˆ’๐‘ข(๐‘กโˆ’โ„Ž,๐‘ฅ)โ„Žโˆ’๐‘ข(๐‘ก,๐‘ฅโˆ’๐‘˜)โˆ’๐‘ข(๐‘กโˆ’โ„Ž,๐‘ฅโˆ’๐‘˜)โ„Ž๎‚นโ‰ค๐›พ(๐‘ก,๐‘ฅ)๐‘˜๎‚ธ๐‘ฃ(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ(๐‘กโˆ’โ„Ž,๐‘ฅ)โ„Žโˆ’๐‘ฃ(๐‘ก,๐‘ฅโˆ’๐‘˜)โˆ’๐‘ฃ(๐‘กโˆ’โ„Ž,๐‘ฅโˆ’๐‘˜)โ„Ž๎‚น,]]]].forโ„Žโˆˆ0,๐‘กโˆ’๐‘Ž,๐‘˜โˆˆ0,๐‘ฅโˆ’๐‘(5.19) For any ๐‘˜โˆˆ]0,๐‘ฅโˆ’๐‘] fixed, we pass to the limit โ„Žโ†’0+ in the latter inequality and thus, in view of equalities (5.16), we get 1๐‘˜๎€บ๐‘ข(1,0)(๐‘ก,๐‘ฅ)โˆ’๐‘ข(1,0)(๎€ปโ‰ค๐‘ก,๐‘ฅโˆ’๐‘˜)๐›พ(๐‘ก,๐‘ฅ)๐‘˜๎€บ๐‘ฃ(1,0)(๐‘ก,๐‘ฅ)โˆ’๐‘ฃ(1,0)(๎€ป๐‘ก,๐‘ฅโˆ’๐‘˜),(5.20) for ๐‘˜โˆˆ]0,๐‘ฅโˆ’๐‘]. Now, letting ๐‘˜โ†’0+ in the previous relation and using equalities (5.17) give โ„“0(๐›พ)(๐‘ก,๐‘ฅ)=๐‘ข(1,1)(๐‘ก,๐‘ฅ)โ‰ค๐›พ(๐‘ก,๐‘ฅ)๐‘ฃ(1,1)(๐‘ก,๐‘ฅ)=๐›พ(๐‘ก,๐‘ฅ)โ„“0(1)(๐‘ก,๐‘ฅ).(5.21) That is, the desired inequality (5.9) holds because (๐‘ก,๐‘ฅ)โˆˆ๐ธ was arbitrary.

Lemma 5.9. Let โ„“2โˆถ๐‘[1](๐’Ÿ;โ„)โ†’๐ฟโˆž(๐’Ÿ;โ„) be a positive ๐‘Ž-Volterra operator such that inequality (4.14) holds for every ๐‘งโˆˆ๐ฟโˆž([๐‘,๐‘‘];โ„). Then, for any function ๐›พโˆˆ๐‘[1](๐’Ÿ;โ„) with the property ๐›พ๎€ท๐‘ก1๎€ธ๎€ท๐‘ก,๐‘ฅโ‰ค๐›พ2๎€ธ,๐‘ฅfor๐‘Žโ‰ค๐‘ก1โ‰ค๐‘ก2[]โ‰ค๐‘anda.e.๐‘ฅโˆˆ๐‘,๐‘‘,(5.22) one has โ„“2(๐›พ)(๐‘ก,๐‘ฅ)โ‰คโ„“2(1)(๐‘ก,๐‘ฅ)๐›พ(๐‘ก,๐‘ฅ)fora.e.(๐‘ก,๐‘ฅ)โˆˆ๐’Ÿ.(5.23)

Proof. Let ๐ธ1โŠ†[๐‘,๐‘‘], measโ€‰๐ธ1=๐‘‘โˆ’๐‘, be a set such that, for any ๐‘ฅโˆˆ๐ธ1, we have ๐›พ(โ‹…,๐‘ฅ)โˆˆ๐ถ([๐‘Ž,๐‘];โ„) and ๐›พ๎€ท๐‘ก1๎€ธ๎€ท๐‘ก,๐‘ฅโ‰ค๐›พ2๎€ธ,๐‘ฅfor๐‘Žโ‰ค๐‘ก1โ‰ค๐‘ก2โ‰ค๐‘.(5.24) We first show that the relation โ„“2(๐›พ)(๐‘ ,๐‘ฅ)โ‰คโ„“2[]ร—[](1)(๐‘ ,๐‘ฅ)๐›พ(๐‘ก,๐‘ฅ)fora.e.(๐‘ ,๐‘ฅ)โˆˆ