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

Abstract

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem 𝑢^(1,1)(𝑡,𝑥)=ℓ₀(𝑢)(𝑡,𝑥)+ℓ₁(𝑢^(1,0))(𝑡,𝑥)+ℓ₂(𝑢^(0,1))(𝑡,𝑥)+𝑞(𝑡,𝑥),𝑢(𝑡,𝑐)=𝛼(𝑡) for 𝑡∈[𝑎,𝑏],𝑢(𝑎,𝑥)=𝛽(𝑥)for𝑥∈[𝑐,𝑑] given on the rectangle [𝑎,𝑏]×[𝑐,𝑑] are established, where the linear operators ℓ₀, ℓ₁, ℓ₂ 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 𝒟=[𝑎,𝑏]×[𝑐,𝑑], we consider the linear partial functional-differential equation𝑢^(1,1)(𝑡,𝑥)=ℓ₀(𝑢)(𝑡,𝑥)+ℓ₁𝑢^(1,0)(𝑡,𝑥)+ℓ₂𝑢^(0,1)(𝑡,𝑥)+𝑞(𝑡,𝑥),(1.1) where 𝑢^(1,0) and 𝑢^(0,1) (resp., 𝑢^(1,1)) denote the first-order (resp., the second-order mixed) partial derivatives. The operators ℓ₀, ℓ₁, and ℓ₂ 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 𝑢∶𝒟→ℝ absolutely continuous in the sense of Carathéodory possessing some additional properties (namely, inclusions (2.20)) which satisfies equality (1.1) almost everywhere on 𝒟.

Three main initial value problems for the hyperbolic equations are studied in the literature—Darboux, Cauchy, and Goursat problems. In this paper, we consider the Darboux problem in which case the values of a solution 𝑢 to (1.1) are prescribed on both characteristics 𝑡=𝑎 and 𝑥=𝑐, that is, the initial conditions are𝑢(𝑡,𝑐)=𝛼(𝑡)for𝑡∈[𝑎,𝑏],𝑢(𝑎,𝑥)=𝛽(𝑥)for𝑥∈[𝑐,𝑑].(1.2) Properties of the initial functions 𝛼 and 𝛽 will be specified in Section 3. It is worth to remember here that various initial and boundary value problems for the hyperbolic equation 𝑢_𝑡𝑥=𝑓𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥(1.3) with continuous as well as discontinuous right-hand sides but without argument deviations have been studied in detail (see, e.g., [1–13] and references therein). As for the hyperbolic functional-differential equations, we can mention for example the works [14–16] (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 ℓ₁=0 and ℓ₂=0 is considered.

2. Notation and Definitions

The following notation is used throughout the paper.(i)ℕ, ℚ, and ℝ are the sets of all natural, rational, and real numbers, respectively, ℝ₊=[0,+∞[.(ii)𝒟=[𝑎,𝑏]×[𝑐,𝑑], where −∞<𝑎<𝑏<+∞ and −∞<𝑐<𝑑<+∞.(iii)The first-order partial derivatives of the function 𝑢∶𝒟→ℝ at the point (𝑡,𝑥)∈𝒟 are denoted by 𝑢^(1,0)(𝑡,𝑥) (or 𝑢_𝑡(𝑡,𝑥)) and 𝑢^(0,1)(𝑡,𝑥) (or 𝑢_𝑥(𝑡,𝑥)). The second-order mixed partial derivatives of the function 𝑢∶𝒟→ℝ at the point (𝑡,𝑥)∈𝒟 are denoted by 𝑢_𝑡𝑥(𝑡,𝑥) and 𝑢_𝑥𝑡(𝑡,𝑥) whereas we use 𝑢^(1,1)(𝑡,𝑥) if 𝑢_𝑡𝑥(𝑡,𝑥)=𝑢_𝑥𝑡(𝑡,𝑥).(iv)𝐶(𝒟;ℝ) is the Banach space of continuous functions 𝑢∶𝒟→ℝ equipped with the norm ‖𝑢‖_𝐶=max||𝑢(𝑡,𝑥)||∶(𝑡,𝑥)∈𝒟.(2.1)(v)𝐶([𝛼,𝛽];ℝ), where −∞<𝛼<𝛽<+∞, is the linear space of continuous functions 𝑣∶[𝛼,𝛽]→ℝ.(vi)𝐴𝐶([𝛼,𝛽];ℝ), where −∞<𝛼<𝛽<+∞, is the linear space of absolutely continuous functions 𝑣∶[𝛼,𝛽]→ℝ.(vii)𝐿^∞(𝒟;ℝ) is the Banach space of Lebesgue measurable and essentially bounded functions 𝑝∶𝒟→ℝ equipped with the norm ‖𝑝‖_𝐿^∞=esssup||𝑝(𝑡,𝑥)||∶(𝑡,𝑥)∈𝒟.(2.2)(viii)𝐿^∞(𝒟;ℝ₊)={𝑝∈𝐿^∞(𝒟;ℝ)∶𝑝(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟}.(ix)For any 𝑧₁,𝑧₂∈𝐿^∞(𝒟;ℝ), we put 𝑧𝑧₂≥𝑧₁⟺𝑧₂(𝑡,𝑥)−𝑧₁(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟,₂≫𝑧₁⟺𝑧₂(𝑡,𝑥)−𝑧₁(𝑡,𝑥)≥𝜀fora.e.(𝑡,𝑥)∈𝒟withsome𝜀>0.(2.3)(x)𝐿^∞([𝛼,𝛽];ℝ), where −∞<𝛼<𝛽<+∞, is the linear space of Lebesgue measurable and essentially bounded functions 𝑓∶[𝛼,𝛽]→ℝ.(xi)meas 𝐴 denotes the Lebesgue measure of the set 𝐴⊂𝑅^𝑚, 𝑚=1,2.(xii)If 𝑋, 𝑌 are Banach spaces and 𝑇∶𝑋→𝑌 is a linear bounded operator then ‖𝑇‖ denotes the norm of the operator 𝑇, that is, ‖𝑇‖=sup‖𝑇(𝑧)‖_𝑌∶𝑧∈𝑋,‖𝑧‖_𝑋≤1.(2.4)

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

2.1. Spaces 𝑍_[1](𝒟;ℝ), 𝑍_[2](𝒟;ℝ), and Set 𝐶^∗(𝒟;ℝ)

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 𝑢∶𝒟→ℝ be such that 𝑢𝑢((⋅𝑡,,𝑥⋅))∶∶[[𝑎𝑐,,𝑑𝑏]]⟶⟶ℝℝiisscmoenatisnuuraobulseffoorraa.lel.𝑡𝑥∈∈[𝑎[𝑐,,𝑏𝑑]].,(2.5) Then the function max{|𝑢(𝑡,⋅)|∶𝑡∈[𝑎,𝑏]}∶[𝑐,𝑑]→ℝ is measurable.

Notation 1. 𝑍_[1](𝒟;ℝ) denotes the linear space of all functions 𝑢∶𝒟→ℝ satisfying conditions (2.5), and esssupmax||𝑢(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏]∶𝑥∈[𝑐,𝑑]<+∞.(2.6) If one identifies functions 𝑢₁, 𝑢₂ from 𝑍_[1](𝒟;ℝ) such that 𝑢₁(⋅,𝑥)≡𝑢₂(⋅,𝑥) for a.e. 𝑥∈[𝑐,𝑑] then ‖𝑢‖_𝑍[1]=esssupmax||𝑢(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏]∶𝑥∈[𝑐,𝑑](2.7) defines a norm in the space 𝑍_[1](𝒟;ℝ).
Analogously, we introduce the space 𝑍_[2](𝒟;ℝ) of functions which are “measurable in the first variable and continuous in the second one” and define the norm ‖⋅‖_𝑍[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. 𝑍_[1](𝒟;ℝ) and 𝑍_[2](𝒟;ℝ) are Banach spaces.

Proof. We only prove the assertion for the space 𝑍_[1](𝒟;ℝ), the assertion of the lemma concerning the space 𝑍_[2](𝒟;ℝ) can be proven analogously by exchanging the roles of the variables 𝑡 and 𝑥.
Let {𝑢_𝑘}^+∞𝑘=1 be an arbitrary Cauchy sequence in 𝑍_[1](𝒟;ℝ). For a decreasing sequence of positive numbers {𝜀_𝑖}^+∞𝑖=1 with ∑^+∞𝑖=1𝜀_𝑖<+∞ there exists an increasing sequence {𝑘_𝑖}^+∞𝑖=1 such thatesssupmax||𝑢_𝑛(𝑡,𝑥)−𝑢_𝑘(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏]∶𝑥∈[𝑐,𝑑]<𝜀_𝑖,(2.8) for every 𝑛,𝑘≥𝑘_𝑖, 𝑖∈ℕ. Let 𝑣_𝑖=𝑢_𝑘𝑖(𝑖=1,2,…). Then, for any 𝑖∈ℕ, there is a set 𝐸_𝑖⊆[𝑐,𝑑], meas 𝐸_𝑖=𝑑−𝑐, such that max||𝑣_𝑖+1(𝑡,𝑥)−𝑣_𝑖(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏]<𝜀_𝑖for𝑥∈𝐸_𝑖,𝑖∈ℕ.(2.9) Put 𝐸=∩^+∞𝑖=1𝐸_𝑖. Then, clearly, we have meas 𝐸=𝑑−𝑐 and max≤||𝑣_𝑛(𝑡,𝑥)−𝑣_𝑘(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏_𝑛−1]_𝑚_=𝑘≤max||𝑣_𝑚+1(𝑡,𝑥)−𝑣_𝑚(𝑡,𝑥)||∶𝑡∈[𝑎,𝑏_+∞]_𝑚_=𝑘𝜀_𝑚for𝑥∈𝐸,𝑛>𝑘.(2.10) Consequently, for any fixed 𝑥∈𝐸, the sequence {𝑣_𝑖(⋅,𝑥)}^+∞𝑖=1 converges uniformly on [𝑎,𝑏], say to 𝑢(⋅,𝑥). Hence, {𝑣_𝑖(𝑡,⋅)}^+∞𝑖=1 converges point-wise on 𝐸 to 𝑢(𝑡,⋅) for every fixed 𝑡∈[𝑎,𝑏]. Therefore, the function 𝑢 satisfies conditions (2.5). Since 𝑢_𝑘(𝑡,𝑥)−𝑢(𝑡,𝑥)==𝑢𝑢_𝑘𝑘(𝑡,𝑥)−𝑢_𝑘𝑖(𝑡,𝑥)+lim_𝑛→+∞𝑣_𝑖(𝑡,𝑥)−𝑣_𝑛(𝑡,𝑥)(𝑡,𝑥)−𝑢_𝑘𝑖(𝑡,𝑥)+lim_{𝑛→+∞𝑛−1}_𝑚=𝑖𝑣_𝑚(𝑡,𝑥)−𝑣_𝑚+1(𝑡,𝑥)(2.11) holds for 𝑖,𝑘∈ℕ, all 𝑡∈[𝑎,𝑏] and a.e. 𝑥∈[𝑐,𝑑], in view of (2.8) and (2.9), we obtain ‖‖𝑢_𝑘−𝑢‖‖_𝑍[1]≤𝜀_𝑖+^+∞_𝑚=𝑖𝜀_𝑚for𝑘≥𝑘_𝑖,𝑖∈ℕ.(2.12) Hence, 𝑢∈𝑍_[1](𝒟;ℝ) and 𝑢_𝑛→𝑢 in 𝑍_[1](𝒟;ℝ), that is, the space 𝑍_[1](𝒟;ℝ) is complete.

For the investigation of hyperbolic differential equations with discontinuous right-hand side, the concept of a Carathéodory 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 Carathéodory 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. 𝐶^∗(𝒟;ℝ) stands for the set of functions 𝑢∶𝒟→ℝ admitting the integral representation 𝑢(𝑡,𝑥)=𝑧+^𝑡𝑎𝑓(𝑠)d𝑠+^𝑥𝑐𝑔(𝜂)d𝜂+^𝑡𝑎^𝑥𝑐ℎ(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟,(2.13) where 𝑧∈ℝ, 𝑓∈𝐿^∞([𝑎,𝑏];ℝ), 𝑔∈𝐿^∞([𝑐,𝑑];ℝ), and ℎ∈𝐿^∞(𝒟;ℝ).

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 ℎ∶𝒟→ℝ be a Lebesgue integrable function and 𝑣(𝑡,𝑥)=^𝑡𝑎^𝑥𝑐ℎ(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟.(2.14) Then (1)there exists a set 𝐸⊆[𝑎,𝑏] such that meas𝐸=𝑏−𝑎 and 𝑣^(1,0)(𝑡,𝑥)=^𝑥𝑐ℎ(𝑡,𝜂)d𝜂for𝑡∈𝐸,𝑥∈[𝑐,𝑑],(2.15)(2)there exists a set 𝐹⊆[𝑐,𝑑] such that meas𝐹=𝑑−𝑐 and 𝑣^(0,1)(𝑡,𝑥)=^𝑡𝑎ℎ(𝑠,𝑥)d𝑠for𝑡∈[𝑎,𝑏]and𝑥∈𝐹,(2.16)(3)there exists a set 𝐺⊆𝒟 such that meas𝐺=(𝑏−𝑎)(𝑑−𝑐) and 𝑣^(1,1)(𝑡,𝑥)=ℎ(𝑡,𝑥)for(𝑡,𝑥)∈𝐺.(2.17)

Remark 2.4. If 𝑢∈𝐶^∗(𝒟;ℝ), that is, the function 𝑢 admits integral representation (2.13), then by using Lemma 2.3 we get 𝑢𝑢^(1,0)(𝑡,𝑥)=𝑓(𝑡)+^𝑥𝑐ℎ(𝑡,𝜂)d𝜂fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],𝑢(𝑡,𝑥)=𝑔(𝑥)+^𝑡𝑎ℎ(𝑠,𝑥)d𝑠forall𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑_(1,1)],(𝑡,𝑥)=ℎ(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟.(2.18) Consequently, for any 𝑢∈𝐶^∗(𝒟;ℝ), we have 𝑢^(1,0)∈𝑍_[2](𝒟;ℝ),𝑢^(0,1)∈𝑍_[1](𝒟;ℝ),𝑢^(1,1)∈𝐿^∞(𝒟;ℝ).(2.19)

Remark 2.5. It follows from Remark 2.4 and [22, Satz 1, page 654] that 𝑢∈𝐶^∗(𝒟;ℝ) if and only if 𝑢∶𝒟→ℝ is absolutely continuous in the sense of Carathéodory with the properties 𝑢^(1,0)(⋅,𝑐)∈𝐿^∞([𝑎,𝑏];ℝ),𝑢^(0,1)(𝑎,⋅)∈𝐿^∞([𝑐,𝑑];ℝ),𝑢^(1,1)∈𝐿^∞(𝒟;ℝ).(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 𝐶(𝒟;ℝ).

Definition 2.6. A linear operator ℓ∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) is said to be positive if the relation ℓ(𝑢)(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟(2.21) holds whenever the function 𝑢∈𝐶(𝒟;ℝ) is such that 𝑢(𝑡,𝑥)≥0for(𝑡,𝑥)∈𝒟.(2.22)

Example 2.7. For any 𝑣∈𝐶(𝒟;ℝ), we put ℓ₀(𝑣)(𝑡,𝑥)=𝑝₀(𝑡,𝑥)𝑣𝜏₀(𝑡,𝑥),𝜇₀(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟,(2.23) where 𝑝₀∈𝐿^∞(𝒟;ℝ) and 𝜏₀∶𝒟→[𝑎,𝑏], 𝜇₀∶𝒟→[𝑐,𝑑] are measurable functions. Then the operator ℓ₀∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) is linear and bounded. Moreover, ℓ₀ is positive if and only if 𝑝₀(𝑡,𝑥)≥0 for a.e. (𝑡,𝑥)∈𝒟.

Definition 2.8. A linear operator ℓ∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) is called (𝑎,𝑐)-Volterra operator if, for any (𝑡₀,𝑥₀)∈𝒟 and 𝑢∈𝐶(𝒟;ℝ) such that 𝑢(𝑡,𝑥)=0for(𝑡,𝑥)∈𝑎,𝑡₀×𝑐,𝑥₀,(2.24) we have ℓ(𝑢)(𝑡,𝑥)=0fora.e.(𝑡,𝑥)∈𝑎,𝑡₀×𝑐,𝑥₀.(2.25)

Remark 2.9. It can be shown by using Lemma 5.8 stated below that the operator ℓ₀ given by formula (2.23) is an (𝑎,𝑐)-Volterra one if and only if ||||𝑝𝑝₀(𝑡,𝑥)||𝜏₀(𝑡,𝑥)−𝑡≤0fora.e.(𝑡,𝑥)∈𝒟,₀(𝑡,𝑥)||𝜇₀(𝑡,𝑥)−𝑥≤0fora.e.(𝑡,𝑥)∈𝒟.(2.26)

Now we introduce analogous notions for linear operators defined on the spaces 𝑍_[1](𝒟;ℝ) and 𝑍_[2](𝒟;ℝ).

Definition 2.10. We say that a linear operator ℓ∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) (resp., ℓ∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ)) is positive if relation (2.21) is satisfied for every function 𝑢∈𝑍_[1](𝒟;ℝ) (resp., 𝑢∈𝑍_[2](𝒟;ℝ)) such that 𝑢(r(e𝑡s,p𝑥.),≥𝑢(0𝑡,𝑥f)or≥𝑡0∈fo[r𝑎,a𝑏.e].a𝑡n∈da[𝑎.e,.𝑏𝑥]∈and[𝑐a,𝑑ll]𝑥∈[𝑐,𝑑]).(2.27)

Example 2.11. For any 𝑣∈𝑍_[2](𝒟;ℝ) (resp., 𝑣∈𝑍_[1](𝒟;ℝ)), we put ℓ₁(𝑣)(𝑡,𝑥)=𝑝₁(𝑡,𝑥)𝑣𝑡,𝜇₁(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟,(2.28) respectively, ℓ₂(𝑣)(𝑡,𝑥)=𝑝₂(𝑡,𝑥)𝑣𝜏₂(𝑡,𝑥),𝑥fora.e.(𝑡,𝑥)∈𝒟,(2.29) where 𝑝₁,𝑝₂∈𝐿^∞(𝒟;ℝ) and 𝜇₁∶𝒟→[𝑐,𝑑], 𝜏₂∶𝒟→[𝑎,𝑏] are measurable functions. Then the operators ℓ₁∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) and ℓ₂∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) are linear and bounded. Moreover, ℓ₁ (resp., ℓ₂) is positive if and only if 𝑝₁(𝑡,𝑥)≥0 (resp., 𝑝₂(𝑡,𝑥)≥0) for a.e. (𝑡,𝑥)∈𝒟.

Definition 2.12. We say that a linear operator ℓ∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) (resp., ℓ∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ)) is an 𝑎-Volterra operator (resp., a 𝑐-Volterra operator) if, for any 𝑡₀∈[𝑎,𝑏] (resp., 𝑥₀∈[𝑐,𝑑]) and 𝑢∈𝑍_[1](𝒟;ℝ) (resp., 𝑢∈𝑍_[2](𝒟;ℝ)) such that 𝑢r(e𝑡,s𝑥p).,=𝑢0(𝑡,𝑥fo)r=𝑡0∈fo𝑎r,a𝑡.₀e.𝑡an∈d[a𝑎.e,.𝑏]𝑥a∈nd[𝑐a,l𝑑l]𝑥∈𝑐,𝑥₀,(2.30) we have ℓr(e𝑢s)p(.𝑡,,𝑥ℓ)(=𝑢)0(𝑡,𝑥fo)r=a.0e.fo(r𝑡,a𝑥.)e.∈(𝑡,𝑎𝑥,)𝑡₀∈×[𝑎[,𝑐𝑏,]𝑑×]𝑐,𝑥₀.(2.31)

Remark 2.13. One can show by using Lemma 5.9 (resp., Lemma 5.10) stated below that the operator ℓ₁ (resp., ℓ₂) given by formula (2.28) (resp., (2.29)) is a 𝑐-Volterra one (resp., an 𝑎-Volterra one) if and only if ||𝑝₁(𝑡,𝑥)||𝜇₁(𝑡,𝑥)−𝑥≤0fora.e.(𝑡,𝑥)∈𝒟,(2.32) respectively, ||𝑝₂(𝑡,𝑥)||𝜏₂(𝑡,𝑥)−𝑡≤0fora.e.(𝑡,𝑥)∈𝒟.(2.33)

3. Statement of Problem

On the rectangle 𝒟, we consider the linear nonhomogeneous Darboux problem (1.1), (1.2) in which ℓ₀∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ), ℓ₁∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ), and ℓ₂∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) are linear bounded operators, 𝑞∈𝐿^∞(𝒟;ℝ), and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) are such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ),𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐). By a solution to problem (1.1), (1.2), we mean a function 𝑢∈𝐶^∗(𝒟;ℝ) possessing property (1.2) and satisfying equality (1.1) almost everywhere on 𝒟. 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 equation𝑢_𝑡𝑥=𝑝₀(𝑡,𝑥)𝑢+𝑝₁(𝑡,𝑥)𝑢_𝑡+𝑝₂(𝑡,𝑥)𝑢_𝑥+𝑞(𝑡,𝑥),(3.1) where 𝑝₀,𝑝₁,𝑝₂,𝑞∈𝐿^∞(𝒟;ℝ), 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 𝑝₀, 𝑝₁, and 𝑝₂. 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 deviations𝑢^(1,1)(𝑡,𝑥)=+𝑝₀𝑝(₂𝑡,𝑥)𝑢𝜏₀(𝑡,𝑥),𝜇₀(𝑡,𝑥)+𝑝₁(𝑡,𝑥)𝑢^(1,0)𝑡,𝜇₁(𝑡,𝑥)(𝑡,𝑥)𝑢^(0,1)𝜏₂(𝑡,𝑥),𝑥+𝑞(𝑡,𝑥),(3.2) in which coefficients 𝑝₀,𝑝₁,𝑝₂,𝑞∈𝐿^∞(𝒟;ℝ) and argument deviations 𝜏₀,𝜏₂∶𝒟→[𝑎,𝑏], 𝜇₀,𝜇₁∶𝒟→[𝑐,𝑑] are measurable functions. We obtain this equation from (1.1) if the operators ℓ₀, ℓ₁, and ℓ₂ are defined by formulas (2.23), (2.28), and (2.29), respectively. Let us also mention that in the case, where𝜏_𝑘(𝑡,𝑥)=𝑡,𝜇_𝑗(𝑡,𝑥)=𝑥fora.e.(𝑡,𝑥)∈𝒟(𝑘=0,2,𝑗=0,1),(3.3) equation (3.2) takes form (3.1).

4. Main Results

At first, we put𝐴_𝑘(𝑧)=ℓ_𝑘𝜑_𝑘(𝑧)for𝑧∈𝐿^∞(𝒟;ℝ),𝑘=0,1,2,(4.1) where 𝜑₀(𝑧)(𝑡,𝑥)=^𝑡𝑎^𝑥𝑐𝜑𝑧(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟,₁(𝜑𝑧)(𝑡,𝑥)=^𝑥𝑐𝑧(𝑡,𝜂)d𝜂fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑₂],(𝑧)(𝑡,𝑥)=^𝑡𝑎𝑧(𝑠,𝑥)d𝑠for𝑡∈[𝑎,𝑏],a.e.𝑥∈[𝑐,𝑑].(4.2)

Clearly, 𝜑₀∶𝐿^∞(𝒟;ℝ)→𝐶(𝒟;ℝ), 𝜑₁∶𝐿^∞(𝒟;ℝ)→𝑍_[2](𝒟;ℝ), 𝜑₂∶𝐿^∞(𝒟;ℝ)→𝑍_[1](𝒟;ℝ) and thus the operators 𝐴₀, 𝐴₁, 𝐴₂ mapping the space 𝐿^∞(𝒟;ℝ) into itself are linear and bounded.

Theorem 4.1. Let 𝐴=𝐴₀+𝐴₁+𝐴₂, where the operators 𝐴₀, 𝐴₁, 𝐴₂ are defined by relations (4.1), (4.2). If the spectral radius of the operator 𝐴 is less than one then problem (1.1), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ),𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).

Theorem 4.1 implies the following.

Corollary 4.2. If the inequality (𝑏−𝑎)(𝑑−𝑐)‖‖ℓ₀‖‖+(𝑑−𝑐)‖‖ℓ₁‖‖+(𝑏−𝑎)‖‖ℓ₂‖‖<1(4.3) holds then problem (1.1), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ),𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).

Remark 4.3. On the rectangle [𝑎,𝑏]×[𝑐,𝑑], we consider the equation 𝑢^(1,1)(𝑡,𝑥)=𝑝₀𝑢(𝑏,𝑑)+𝑝₁𝑢^(1,0)(𝑡,𝑑)+𝑝₂𝑢^(0,1)(𝑏,𝑥)(4.4) subjected to the initial conditions 𝑢(𝑡,𝑐)=0for𝑡∈[𝑎,𝑏],𝑢(𝑎,𝑥)=0for𝑥∈[𝑐,𝑑],(4.5) where 𝑝₀=𝑚⁰(𝑏−𝑎)(𝑑−𝑐),𝑝¹=𝑚¹𝑑−𝑐,𝑝²=𝑚²𝑏−𝑎.(4.6) Clearly (4.4) is a particular case of (1.1). If 𝑚₀+𝑚₁+𝑚₂=1, then problem (4.4), (4.5) has the trivial solution 𝑢(𝑡,𝑥)≡0 and the nontrivial solution 𝑢(𝑡,𝑥)≡(𝑡−𝑎)(𝑥−𝑐). 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 ℓ₀, ℓ₁, and ℓ₂ on the right-hand side of (1.1) are positive then we can estimate the spectral radius of the operator 𝐴 by using the well-known results due to Krasnosel'skij and we thus obtain the following.

Theorem 4.4. Let the operators ℓ₀, ℓ₁, ℓ₂ be positive and 𝐴=𝐴₀+𝐴₁+𝐴₂, where the operators 𝐴₀, 𝐴₁, 𝐴₂ are defined by relations (4.1), (4.2). Then the following four assertions are equivalent. (1)There exists a function 𝑧₀∈𝐿^∞(𝒟;ℝ₊) such that 𝑧₀≫𝐴(𝑧₀).(2)The spectral radius of the operator 𝐴 is less than one.(3)Problem (1.1), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ),𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).If, in addition, the initial functions 𝛼,𝛽and the forcing term 𝑞are such that 𝛼(𝑎)≥0,𝛼^(𝑡)≥0,𝛽^(𝑥)≥0,𝑞(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟,(4.7) then the solution 𝑢to problem (1.1), (1.2) satisfies 𝑢𝑢(𝑡,𝑥)≥0for(𝑡,𝑥)∈𝒟,_(1,0)𝑢(𝑡,𝑥)≥0fora.e.𝑡∈[a,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)≥0for𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑].(4.8)(4)There exists a function 𝛾∈𝐶^∗(𝒟;ℝ) such that𝛾𝛾^(1,1)≫ℓ₀(𝛾)𝛾+(𝑎ℓ,₁𝑐)𝛾≥0^(1,0),+ℓ₂𝛾^(0,1),(4(.41.09)_(1,0))(𝑡,𝑐)≥0fora𝛾.e.𝑡∈[𝑎,𝑏],𝛾^(0,1)(𝑎,𝑥)≥0fora.e.𝑥∈[𝑐,𝑑],(4.11)_(1,1)(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟.(4.12)

For Volterra-type operators ℓ₀, ℓ₁, and ℓ₂, we derive from the previous theorem the following.

Corollary 4.5. Let ℓ₀, ℓ₁, and ℓ₂ be positive (𝑎,𝑐)-Volterra, 𝑐-Volterra, and 𝑎-Volterra operators, respectively, such that the inequalities ℓ₁(𝑦)(𝑡,𝑥)≤𝑦(𝑡)ℓ₁(1)(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟(4.13)(here,ℓ₁(𝑦) means ℓ₁(𝑦) in which 𝑦(𝑡,𝑥)=𝑦(𝑡) for 𝑎.𝑒.𝑡∈[𝑎,𝑏] and all 𝑥∈[𝑐,𝑑]) and ℓ₂(𝑧)(𝑡,𝑥)≤𝑧(𝑥)ℓ₂(1)(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟(4.14)(byℓ₂(𝑧) we mean ℓ₂(𝑧), where 𝑧(𝑡,𝑥)=𝑧(𝑥)forall𝑡∈[𝑎,𝑏] and a.e. 𝑥∈[𝑐,𝑑]) hold for every 𝑦∈𝐿^∞([𝑎,𝑏];ℝ) and 𝑧∈𝐿^∞([𝑐,𝑑];ℝ).
Then problem (1.1), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ),𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐). If, in addition, the initial functions 𝛼, 𝛽 and the forcing term 𝑞 are such that relations (4.7) hold, then the solution 𝑢 to problem (1.1), (1.2) satisfies inequalities (4.8).

Following our previous results concerning the case, where ℓ₁=0 and ℓ₂=0 (see [18]), we can introduce the following.

Definition 4.6. Let ℓ₀∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ), ℓ₁∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ), and ℓ₂∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ). We say that the triplet (ℓ₀,ℓ₁,ℓ₂) belongs to the set 𝒮^𝑎𝑐 if the implication 𝑢𝑢∈𝐶^∗(𝒟;ℝ),_(1,1)(𝑡,𝑥)≥+ℓ₀ℓ(₂𝑢)(𝑡,𝑥)+ℓ₁𝑢^(1,0)(𝑡,𝑥)𝑢𝑢(𝑎,𝑐)≥0,_(1,0)𝑢^(0,1)(𝑡,𝑥),fora.e.(𝑡,𝑥)∈𝒟,𝑢(𝑡,𝑐)≥0fora.e.𝑡∈[𝑎,𝑏_(0,1)],(𝑎,𝑥)≥⟹0f𝑢orsaat.ies.fi𝑥es∈(4[.𝑐8,)𝑑],(4.15) holds.

Remark 4.7. If (ℓ₀,ℓ₁,ℓ₂)∈𝒮^𝑎𝑐, 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 (ℓ₀,ℓ₁,ℓ₂)∈𝒮^𝑎𝑐 is fulfilled.

Theorem 4.4 immediately yields the following.

Corollary 4.8. If one of assertions (1)–(4) stated in Theorem 4.4 holds then (ℓ₀,ℓ₁,ℓ₂)∈𝒮^_ac.

Remark 4.9. The inclusion (ℓ₀,ℓ₁,ℓ₂)∈𝒮^𝑎𝑐 ensures that every solution 𝑢 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 𝑞, 𝛼, and 𝛽. 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 𝐴 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 𝑝₀,𝑝₁,𝑝₂,𝑞∈𝐿^∞(𝒟;ℝ) and argument deviations 𝜏₀,𝜏₂∶𝒟→[𝑎,𝑏], 𝜇₀,𝜇₁∶𝒟→[𝑐,𝑑] are measurable functions.

As a consequence of Corollary 4.2 we obtain the following.

Corollary 4.10. If the inequality (b−𝑎)(𝑑−𝑐)‖‖𝑝₀‖‖_𝐿∞+(𝑑−𝑐)‖‖𝑝₁‖‖_𝐿∞+(𝑏−𝑎)‖‖𝑝₂‖‖_𝐿∞<1(4.16) holds, then problem (3.2), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).

If the coefficients 𝑝₀, 𝑝₁, 𝑝₂ in the previous corollary are non-negative then the assertion of the corollary follows also from implication (4)⇒(3) of Theorem 4.4. More precisely, the following statement holds.

Corollary 4.11. Let 𝑝₀,𝑝₁,𝑝₂∈𝐿^∞(𝒟;ℝ₊) and esssup+𝑝𝑝₀₂(𝑡,𝑥)𝜏₀(𝑡,𝑥)−𝑎𝜇₀(𝑡,𝑥)−𝑐+𝑝₁(𝑡,𝑥)𝜇₁(𝑡,𝑥)−𝑐(𝑡,𝑥)𝜏₂(𝑡,𝑥)−𝑎∶(𝑡,𝑥)∈𝒟<1.(4.17) Then problem (3.2), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐). If, in addition, the initial functions 𝛼, 𝛽, and the forcing term 𝑞 are such that relations (4.7) hold, then the solution 𝑢 to problem (3.2), (1.2) satisfies inequalities (4.8).

Finally, Corollary 4.5 implies the following.

Corollary 4.12. Let 𝑝₀,𝑝₁,𝑝₂∈𝐿^∞(𝒟;ℝ₊) and argument deviations 𝜏₀, 𝜇₀, 𝜇₁, and 𝜏₂ satisfy inequalities (2.26), (2.32), and (2.33). Then problem (3.2), (1.2) is uniquely solvable for arbitrary 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐). If, in addition, the initial functions 𝛼, 𝛽 and the forcing term 𝑞 are such that relations (4.7) hold, then the solution 𝑢 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 𝜏₀, 𝜇₀, 𝜇₁, and 𝜏₂ are not necessarily delays but the differences 𝜏_𝑘(𝑡,𝑥)−𝑡,𝜇_𝑗(𝑡,𝑥)−𝑥(𝑘=0,2,𝑗=0,1)(4.18) are small enough, that is, if (3.2) with deviating arguments is “close” to (3.1).

Corollary 4.13. Let 𝑝₀,𝑝₁,𝑝₂∈𝐿^∞(𝒟;ℝ₊),𝑝_𝑘≢0(𝑘=1,2), and esssup^{𝜏0𝑡(𝑡,𝑥)}^{𝜇0𝑐(𝑡,𝑥)}𝑝₀(𝑠,𝜂)d𝜂d𝑠+^𝑡𝑎^{𝜇0𝑥(𝑡,𝑥)}+2‖‖𝑝₂𝑝₀(𝑠,𝜂)d𝜂d𝑠‖‖_𝐿∞𝜏₀(𝑡,𝑥)−𝑡+2‖‖𝑝₁‖‖_𝐿∞𝜇₀(𝑡,𝑥)−𝑥∶(𝑡,𝑥)∈𝒟≤𝜔esssup_𝑡𝑎e1+ln𝜔1,^{𝜇1𝑥(𝑡,𝑥)}𝑝₀(𝑠,𝜂)d𝜂d𝑠+2‖‖𝑝₁‖‖_𝐿∞𝜇₁(𝑡,𝑥)−𝑥∶(𝑡,𝑥)∈𝒟<𝜔esssup_{𝜏2𝑡(𝑡,𝑥)}e,^𝑥𝑐𝑝₀(𝑠,𝜂)d𝜂d𝑠+2‖‖𝑝₂‖‖_𝐿∞𝜏₂(𝑡,𝑥)−𝑡∶(𝑡,𝑥)∈𝒟<𝜔e,(4.19)where 𝜔=2min‖‖𝑝¹‖‖_𝐿∞,‖‖𝑝₂‖‖_𝐿∞‖‖𝑝₀‖‖_𝐿∞max{𝑏−𝑎,𝑑−𝑐}+2min‖‖𝑝₁‖‖_𝐿∞,‖‖𝑝₂‖‖_𝐿∞.(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 ℓ₀, ℓ₁, and ℓ₂, the operators 𝐴₀, 𝐴₁, and 𝐴₂ are defined by relations (4.1), (4.2). Moreover, having 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ), 𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐), we put 𝑦=ℓ₀(−𝛼(𝑎)+𝛼+𝛽)+ℓ₁𝛼^+ℓ₂𝛽^+𝑞(5.1)(byℓ₀(−𝛼(𝑎)+𝛼+𝛽) the authors understand ℓ₀(𝜎) in which 𝜎(𝑡,𝑥)=−𝛼(𝑎)+𝛼(𝑡)+𝛽(𝑥) for (𝑡,𝑥)∈𝒟. Similarly, ℓ₁(𝛼^)(resp.,ℓ₂(𝛽^)) means ℓ₁(𝛼₀)(resp.,ℓ₂(𝛽₀)), where 𝛼₀(𝑡,𝑥)=𝛼^(𝑡) for a.e.𝑡∈[𝑎,𝑏] and all 𝑥∈[𝑐,𝑑](resp.,𝛽₀(𝑡,𝑥)=𝛽^(𝑥) for all 𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑])). Clearly, 𝑦∈𝐿^∞(𝒟;ℝ).

Lemma 5.1. If 𝑢 is a solution to problem (1.1), (1.2) then 𝑢^(1,1) is a solution to the equation 𝑧=𝐴₀+𝐴₁+𝐴₂(𝑧)+𝑦(5.2) in the space 𝐿^∞(𝒟;ℝ), where the operators 𝐴₀, 𝐴₁, and 𝐴₂ are defined by relations (4.1), (4.2) and the function 𝑦 is given by formula (5.1).
Conversely, if 𝑧 is a solution to (5.2) in the space 𝐿^∞(𝒟;ℝ) with the operators 𝐴₀, 𝐴₁, and 𝐴₂ defined by relations (4.1), (4.2) and the function 𝑦 given by formula (5.1), then 𝑢(𝑡,𝑥)=−𝛼(𝑎)+𝛼(𝑡)+𝛽(𝑥)+^𝑡𝑎^𝑥𝑐𝑧(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟(5.3) is a solution to problem (1.1), (1.2).

Proof. If 𝑢 is a solution to problem (1.1), (1.2) then, by virtue of Remark 2.4, we get 𝑢^(1,1)∈𝐿^∞(𝒟;ℝ), 𝑢(𝑡,𝑥)=−𝛼(𝑎)+𝛼(𝑡)+𝛽(𝑥)+^𝑡𝑎^𝑥𝑐𝑢𝑢^(1,1)(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟,_(1,0)𝑢(𝑡,𝑥)=𝛼^(𝑡)+^𝑥𝑐𝑢^(1,1)(𝑡,𝜂)d𝜂fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)=𝛽^(𝑥)+^𝑡𝑎𝑢^(1,1)(𝑠,𝑥)d𝑠forall𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑].(5.4) Consequently, (1.1) yields that 𝑢^(1,1)=𝐴₀+𝐴₁+𝐴₂𝑢^(1,1)+𝑦,(5.5) where the operators 𝐴₀, 𝐴₁, and 𝐴₂ are defined by relations (4.1), (4.2) and the function 𝑦 is given by formula (5.1).
Conversely, let 𝑧 be a solution to (5.2) in the space 𝐿^∞(𝒟;ℝ) with the operators 𝐴₀, 𝐴₁, and 𝐴₂ defined by relations (4.1), (4.2) and the function 𝑦 given by formula (5.1). Moreover, let the function 𝑢 be defined by relation (5.3), that is,𝑢(𝑡,𝑥)=𝛼(𝑎)+^𝑡𝑎𝛼^(𝑠)d𝑠+^𝑥𝑐𝛽^(𝜂)d𝜂+^𝑡𝑎^𝑥𝑐𝑧(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟.(5.6) Then the function 𝑢 belongs to the set 𝐶^∗(𝒟;ℝ) and verifies initial conditions (1.2). Furthermore, by using Lemma 2.3, we get 𝑢𝑢^(1,0)(𝑡,𝑥)=𝛼^(𝑡)+^𝑥𝑐𝑧(𝑡,𝜂)d𝜂fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],𝑢(𝑡,𝑥)=𝛽^(𝑥)+^𝑡𝑎𝑧(𝑠,𝑥)d𝑠forall𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑_(1,1)],(𝑡,𝑥)=𝑧(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟.(5.7) Consequently, (5.2) implies that 𝑢 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 𝐾 in a Banach space 𝑋 is called a cone if the following conditions are satisfied: (i)𝑥+𝑦∈𝐾 for all 𝑥,𝑦∈𝐾,(ii)𝜆𝑥∈𝐾 for all 𝑥∈𝐾 and an arbitrary 𝜆≥0,(iii)if 𝑥∈𝐾 and −𝑥∈𝐾 then 𝑥=0.

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

Definition 5.4. We say that a cone 𝐾⊆𝑋 is solid if its interior Int𝐾 is nonempty.

Remark 5.5. The presence of a cone 𝐾 in a Banach space 𝑋 allows one to introduce a natural partial ordering there. More precisely, two elements 𝑥₁,𝑥₂∈𝑋 are said to be in the relation 𝑥₂≥_𝐾𝑥₁ if and only if they satisfy the inclusion 𝑥₂−𝑥₁∈𝐾. If, moreover, 𝐾 is a solid cone then we write 𝑥₂≫_𝐾𝑥₁ if and only if 𝑥₂−𝑥₁∈Int𝐾.

Definition 5.6. A cone 𝐾⊆𝑋 is said to be normal if there is a constant 𝑁≥0 such that, for every 𝑥,𝑦∈𝑋 with the property 0≤_𝐾𝑥≤_𝐾𝑦, the relation ‖𝑥‖_𝑋≤𝑁‖𝑦‖_𝑋 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 𝐾 be a normal and solid cone in a Banach space 𝑋 and the operator 𝐴∶𝑋→𝑋 leave invariant the cone 𝐾, that is, 𝐴(𝐾)⊆𝐾. If there exists a constant 𝛿>0 and an element 𝑥₀∈Int𝐾 such that 𝛿𝑥₀−𝐴(𝑥₀)∈Int𝐾, then the spectral radius of the operator 𝐴 is less than 𝛿.

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

Lemma 5.8. Let ℓ₀∶𝐶(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) be a positive (𝑎,𝑐)-Volterra operator. Then, for any function 𝛾∈𝐶(𝒟;ℝ) satisfying 𝛾𝑡₁,𝑥₁≤𝛾𝑡₂,𝑥₂for𝑎≤𝑡₁≤𝑡₂≤𝑏,𝑐≤𝑥₁≤𝑥₂≤𝑑,(5.8) one has ℓ₀(𝛾)(𝑡,𝑥)≤ℓ₀(1)(𝑡,𝑥)𝛾(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟.(5.9)

Proof. We first show that, for any (𝑡,𝑥)∈]𝑎,𝑏]×]𝑐,𝑑], we have ℓ₀(𝛾)(𝑠,𝜂)≤ℓ₀(1)(𝑠,𝜂)𝛾(𝑡,𝑥)fora.e.(𝑠,𝜂)∈[𝑎,𝑡]×[𝑐,𝑥].(5.10) Indeed, let (𝑡,𝑥)∈]𝑎,𝑏]×]𝑐,𝑑] be arbitrary but fixed. Put 𝛾₀(𝑠,𝜂)=𝛾(min{𝑠,𝑡},min{𝜂,𝑥})for(𝑠,𝜂)∈𝒟.(5.11) Then, clearly 𝛾₀∈𝐶(𝒟;ℝ), 𝛾𝛾₀(𝑠,𝜂)≤𝛾(𝑡,𝑥)for(𝑠,𝜂)∈𝒟,₀(𝑠,𝜂)=𝛾(𝑠,𝜂)for(𝑠,𝜂)∈[𝑎,𝑡]×[𝑐,𝑥].(5.12) Since the operator ℓ₀ is positive, we obtain ℓ₀𝛾₀(𝑠,𝜂)≤ℓ₀(𝛾(𝑡,𝑥))(𝑠,𝜂)=𝛾(𝑡,𝑥)ℓ₀(1)(𝑠,𝜂)fora.e.(𝑠,𝜂)∈𝒟.(5.13) On the other hand, the operator ℓ₀ is supposed to be an (𝑎,𝑐)-Volterra one which guarantees the equality ℓ₀𝛾₀(𝑠,𝜂)=ℓ₀(𝛾)(𝑠,𝜂)fora.e.(𝑠,𝜂)∈[𝑎,𝑡]×[𝑐,𝑥],(5.14) and thus the desired relation (5.10) holds.
Now we put 𝑢(𝑡,𝑥)=^𝑡𝑎^𝑥𝑐𝑣(𝑡,𝑥)=_𝑡𝑎ℓ₀(𝛾)(𝑠,𝜂)d𝜂d𝑠,^𝑥𝑐ℓ₀(1)(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟.(5.15) It follows from Lemma 2.3 that there exists a set 𝐸₁⊆]𝑎,𝑏], meas 𝐸₁=𝑏−𝑎, such that 𝑣𝑢^(1,0)(𝑡,𝑥)=^𝑥𝑐ℓ₀(𝛾)(𝑡,𝜂)d𝜂for𝑡∈𝐸₁,𝑥∈[𝑐,𝑑_(1,0)],(𝑡,𝑥)=^𝑥𝑐ℓ₀(1)(𝑡,𝜂)d𝜂for𝑡∈𝐸₁,𝑥∈[𝑐,𝑑],(5.16) and, moreover, there is a set 𝐸⊆𝐸₁×]𝑐,𝑑], meas 𝐸=(𝑏−𝑎)(𝑑−𝑐), with the properties 𝑢^(1,1)(𝑡,𝑥)=ℓ₀(𝛾)(𝑡,𝑥),𝑣^(1,1)(𝑡,𝑥)=ℓ₀(1)(𝑡,𝑥)for(𝑡,𝑥)∈𝐸.(5.17)
Let (𝑡,𝑥)∈𝐸 be arbitrary but fixed. Then, relation (5.10) yields 1ℎ𝑘^{𝑡𝑡−ℎ}^{𝑥𝑥−𝑘}ℓ₀(𝛾)(𝑠,𝜂)d𝜂d𝑠≤𝛾(𝑡,𝑥)ℎ𝑘^{𝑡𝑡−ℎ}^{𝑥𝑥−𝑘}ℓ₀(1)(𝑠,𝜂)d𝜂d𝑠(5.18) for ℎ∈]0,𝑡−𝑎] and 𝑘∈]0,𝑥−𝑐], whence we get 1𝑘𝑢≤(𝑡,𝛾𝑥()𝑡,−𝑥)𝑢ℎ(𝑡−ℎ,𝑥)−𝑢(𝑡,𝑥−𝑘)−𝑢ℎ(𝑡−ℎ,𝑥−𝑘)𝑘𝑣(𝑡,𝑥)−𝑣ℎ(𝑡−foℎr,ℎ𝑥)∈−]0𝑣,(𝑡𝑡,−𝑥𝑎−],𝑘)𝑘−∈𝑣ℎ](0𝑡,−𝑥ℎ−,𝑥𝑐]−.𝑘),(5.19) For any 𝑘∈]0,𝑥−𝑐] fixed, we pass to the limit ℎ→0+ in the latter inequality and thus, in view of equalities (5.16), we get 1𝑘𝑢^(1,0)(𝑡,𝑥)−𝑢^(1,0)(𝑡,𝑥−𝑘)≤𝛾(𝑡,𝑥)𝑘𝑣^(1,0)(𝑡,𝑥)−𝑣^(1,0)(𝑡,𝑥−𝑘),(5.20) for 𝑘∈]0,𝑥−𝑐]. Now, letting 𝑘→0+ in the previous relation and using equalities (5.17) give ℓ₀(𝛾)(𝑡,𝑥)=𝑢^(1,1)(𝑡,𝑥)≤𝛾(𝑡,𝑥)𝑣^(1,1)(𝑡,𝑥)=𝛾(𝑡,𝑥)ℓ₀(1)(𝑡,𝑥).(5.21) That is, the desired inequality (5.9) holds because (𝑡,𝑥)∈𝐸 was arbitrary.

Lemma 5.9. Let ℓ₂∶𝑍_[1](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) be a positive 𝑎-Volterra operator such that inequality (4.14) holds for every 𝑧∈𝐿^∞([𝑐,𝑑];ℝ). Then, for any function 𝛾∈𝑍_[1](𝒟;ℝ) with the property 𝛾𝑡₁,𝑥≤𝛾𝑡₂,𝑥for𝑎≤𝑡₁≤𝑡₂≤𝑏anda.e.𝑥∈[𝑐,𝑑],(5.22) one has ℓ₂(𝛾)(𝑡,𝑥)≤ℓ₂(1)(𝑡,𝑥)𝛾(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟.(5.23)

Proof. Let 𝐸₁⊆[𝑐,𝑑], meas 𝐸₁=𝑑−𝑐, be a set such that, for any 𝑥∈𝐸₁, we have 𝛾(⋅,𝑥)∈𝐶([𝑎,𝑏];ℝ) and 𝛾𝑡₁,𝑥≤𝛾𝑡₂,𝑥for𝑎≤𝑡₁≤𝑡₂≤𝑏.(5.24) We first show that the relation ℓ₂(𝛾)(𝑠,𝑥)≤ℓ₂(1)(𝑠,𝑥)𝛾(𝑡,𝑥)fora.e.(𝑠,𝑥)∈[𝑎,𝑡]×[𝑐,𝑑](5.25) holds for every 𝑡∈]𝑎,𝑏]. Indeed, let 𝑡∈]𝑎,𝑏] be arbitrary but fixed. Put 𝛾₀(𝑠,𝑥)=𝛾𝛾(₁min{𝑠,𝑡},𝑥)for𝑠∈[𝑎,𝑏],𝑥∈𝐸₁,(𝑥)=𝛾(𝑡,𝑥)for𝑥∈𝐸₁.(5.26) Then, clearly, 𝛾₀∈𝑍_[1](𝒟;ℝ),𝛾₁∈𝐿^∞([𝑐,𝑑];ℝ), 𝛾𝛾₀₀(𝑠,𝑥)≤𝛾₁(𝑥)for𝑠∈[𝑎,𝑏],𝑥∈𝐸₁,(𝑠,𝑥)=𝛾(𝑠,𝑥)for𝑠∈[𝑎,𝑡],𝑥∈𝐸₁.(5.27) Since the operator ℓ₂ is positive and satisfies condition (4.14), we obtain ℓ₂𝛾₀(𝑠,𝑥)≤ℓ₂𝛾₁(𝑠,𝑥)≤𝛾₁(𝑥)ℓ₂(1)(𝑠,𝑥)fora.e.(𝑠,𝑥)∈𝒟(5.28)(by ℓ₂(𝛾₁) the authors mean ℓ₂(𝛾₁), where 𝛾₁(𝑠,𝑥)=𝛾₁(𝑥) for all 𝑠∈[𝑎,𝑏] and 𝑥∈𝐸₁). On the other hand, the operator ℓ₂ is supposed to be an 𝑎-Volterra one which guarantees the equality ℓ₂𝛾₀(𝑠,𝑥)=ℓ₂(𝛾)(𝑠,𝑥)fora.e.(𝑠,𝑥)∈[𝑎,𝑡]×[𝑐,𝑑],(5.29) and thus desired relation (5.25) holds for every 𝑡∈]𝑎,𝑏]. It means that, for any 𝑡∈]𝑎,𝑏], there exists a set 𝐴_𝑡⊆[𝑐,𝑑] with meas 𝐴_𝑡=𝑑−𝑐 such that ℓ₂(𝛾)(𝑠,𝑥)≤ℓ₂(1)(𝑠,𝑥)𝛾(𝑡,𝑥)for𝑥∈𝐴_𝑡,𝑠∈𝐵_𝑡(𝑥),(5.30) where, for each 𝑥∈𝐴_𝑡, we have 𝐵_𝑡(𝑥)⊆[𝑎,𝑏] with meas 𝐵_𝑡(𝑥)=𝑡−𝑎.
Put 𝐸₂=∩_𝑡∈𝐶𝐴_𝑡, where 𝐶=]𝑎,𝑏]∩ℚ. Clearly, meas 𝐸₂=𝑑−𝑐 because the set 𝐶 is countable. Moreover, relation (5.30) yields that^𝑡𝑡₀ℓ₂(𝛾)(𝑠,𝑥)d𝑠≤𝛾(𝑡,𝑥)^𝑡𝑡₀ℓ₂(1)(𝑠,𝑥)d𝑠for𝑥∈𝐸₂,𝑡∈𝐶,𝑡₀∈[𝑎,𝑡].(5.31) Let 𝑡∈]𝑎,𝑏], 𝑡₀∈[𝑎,𝑡[, and 𝑥∈𝐸₁∩𝐸₂ be arbitrary but fixed. Then there exists a sequence {𝑡_𝑛}^+∞𝑛=1⊂[𝑡₀,𝑏]∩ℚ such that 𝑡_𝑛→𝑡 as 𝑛→+∞. It follows from relation (5.31) that ^𝑡𝑛_𝑡0ℓ₂(𝛾)(𝑠,𝑥)d𝑠≤𝛾𝑡_𝑛,𝑥^𝑡𝑛_𝑡0ℓ₂(1)(𝑠,𝑥)d𝑠for𝑛∈ℕ,(5.32) whence we get ^𝑡𝑡₀ℓ₂(𝛾)(𝑠,𝑥)d𝑠≤𝛾(𝑡,𝑥)^𝑡𝑡₀ℓ₂(1)(𝑠,𝑥)d𝑠.(5.33) Consequently, we have proved that ^𝑡𝑡₀ℓ₂(𝛾)(𝑠,𝑥)d𝑠≤𝛾(𝑡,𝑥)^𝑡𝑡₀ℓ₂(1)(𝑠,𝑥)d𝑠,for𝑥∈𝐸₁∩𝐸₂,𝑎≤𝑡₀<𝑡≤𝑏.(5.34)
Now we put 𝑢(𝑡,𝑥)=^𝑡𝑎ℓ₂(𝛾)(𝑠,𝑥)d𝑠,𝑣(𝑡,𝑥)=^𝑡𝑎ℓ₂(1)(𝑠,𝑥)d𝑠for𝑡∈[𝑎,𝑏],𝑥∈𝐸₁∩𝐸₂.(5.35) Lemma 2.3 guarantees that there exists a set 𝐸⊆]𝑎,𝑏]×(𝐸₁∩𝐸₂) such that meas 𝐸=(𝑏−𝑎)(𝑑−𝑐) and 𝑢^(1,0)(𝑡,𝑥)=ℓ₂(𝛾)(𝑡,𝑥),𝑣^(1,0)(𝑡,𝑥)=ℓ₂(1)(𝑡,𝑥)for(𝑡,𝑥)∈𝐸.(5.36) Let (𝑡,𝑥)∈𝐸 be arbitrary but fixed. We choose a sequence {ℎ_𝑛}^+∞𝑛=1 of numbers from the interval ]0,𝑡−𝑎] such that lim_𝑛→+∞ℎ_𝑛=0.(5.37) Then relation (5.34) yields that 1ℎ_𝑛^{𝑡𝑡−ℎ}_𝑛ℓ₂(𝛾)(𝑠,𝑥)d𝑠≤𝛾(𝑡,𝑥)ℎ_𝑛^{𝑡𝑡−ℎ}_𝑛ℓ₂(1)(𝑠,𝑥)d𝑠for𝑛∈ℕ.(5.38) Letting 𝑛→+∞ in the previous relation and using equalities (5.36) give ℓ₂(𝛾)(𝑡,𝑥)=𝑢^(1,0)(𝑡,𝑥)≤𝛾(𝑡,𝑥)𝑣^(1,0)(𝑡,𝑥)=𝛾(𝑡,𝑥)ℓ₂(1)(𝑡,𝑥).(5.39) Consequently, desired inequality (5.23) holds because (𝑡,𝑥)∈𝐸 was arbitrary.

Lemma 5.10. Let ℓ₁∶𝑍_[2](𝒟;ℝ)→𝐿^∞(𝒟;ℝ) be a positive 𝑐-Volterra operator such that inequality (4.13) holds for every 𝑦∈𝐿^∞([𝑎,𝑏];ℝ). Then, for any function 𝛾∈𝑍_[2](𝒟;ℝ) with the property 𝛾𝑡,𝑥₁≤𝛾𝑡,𝑥₂fora.e.𝑡∈[𝑎,𝑏]andall𝑐≤𝑥₁≤𝑥₂≤𝑑,(5.40) one has ℓ₁(𝛾)(𝑡,𝑥)≤ℓ₁(1)(𝑡,𝑥)𝛾(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟.(5.41)

Proof. Lemma can be proven analogously as Lemma 5.9 by exchanging the roles of the variables 𝑡 and 𝑥.

5.2. Proofs of Main Results

Now we are in a position to prove the main results stated in Section 4.

Proof of Theorem 4.1. Since the spectral radius of the linear bounded operator 𝐴∶𝐿^∞(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) is less than one, (5.2) has a unique solution for an arbitrary 𝑦∈𝐿^∞(𝒟;ℝ) and thus, in view of Lemma 5.1, problem (1.1), (1.2) has a unique solution for every 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ),𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ),𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).

Proof of Corollary 4.2. It is easy to show that the norms of the linear bounded operators 𝐴₀,𝐴₁,𝐴₂∶𝐿^∞(𝒟;ℝ)→𝐿^∞(𝒟;ℝ) defined by relations (4.1), (4.2) satisfy the estimates ‖‖𝐴₀‖‖≤(𝑏−𝑎)(𝑑−𝑐)‖‖ℓ₀‖‖,‖‖𝐴₁‖‖≤(𝑑−𝑐)‖‖ℓ₁‖‖,‖‖𝐴₂‖‖≤(𝑏−𝑎)‖‖ℓ₂‖‖.(5.42) Consequently, the assumption (4.3) guarantees that the spectral radius of the operator 𝐴₀+𝐴₁+𝐴₂ is less than one. Hence, the assertion of the corollary follows from Theorem 4.1.

Proof of Theorem 4.4. To prove the theorem, it is sufficient to show the following four implications.
(1)⇒(2): assume that the assertion (1) of the theorem holds. We put 𝐾=𝐿^∞(𝒟;ℝ₊). It is not difficult to verify that 𝐾 forms a normal and solid cone in the Banach space 𝐿^∞(𝒟;ℝ). Moreover, a function 𝑧∈𝐿^∞(𝒟;ℝ) satisfies the relation 𝑧≫0 if and only if the inclusion 𝑧∈Int𝐾 holds.
On the other hand, by virtue of (4.1) and (4.2), the operator 𝐴 leaves the cone 𝐾 invariant, that is, 𝐴(𝐾)⊆𝐾 because the operators ℓ₀, ℓ₁, and ℓ₂ are supposed to be positive. Therefore, the assumptions 𝑧₀∈𝐾 and 𝑧₀≫𝐴(𝑧₀) yield that 𝑧₀≫0 as well.
Applying Lemma 5.7 with 𝑋=𝐿^∞(𝒟;ℝ) and 𝛿=1, we obtain the desired assertion (2) of our theorem.
(2)⇒(3): assume that the spectral radius of the operator 𝐴 is less than one. Then, for an arbitrary 𝑦∈𝐿^∞(𝒟;ℝ), (5.2) has a unique solution 𝑧 and, moreover, this solution admits the series representation 𝑧=𝑦+𝐴(𝑦)+𝐴²(𝑦)+⋯.(5.43) Consequently, in view of Lemma 5.1, problem (1.1), (1.2) has a unique solution 𝑢 for every 𝑞∈𝐿^∞(𝒟;ℝ) and 𝛼∈𝐴𝐶([𝑎,𝑏];ℝ),𝛽∈𝐴𝐶([𝑐,𝑑];ℝ) such that 𝛼^∈𝐿^∞([𝑎,𝑏];ℝ), 𝛽^∈𝐿^∞([𝑐,𝑑];ℝ), and 𝛼(𝑎)=𝛽(𝑐).
Assume, in addition, that the initial functions 𝛼, 𝛽 and the forcing term 𝑞 are such that relations (4.7) hold. The above-used Lemma 5.1 guarantees that the solution 𝑢 to problem (1.1), (1.2) admits the integral representation 𝑢(𝑡,𝑥)=𝛼(𝑎)+^𝑡𝑎𝛼^(𝑠)d𝑠+^𝑥𝑐𝛽^(𝜂)d𝜂+^𝑡𝑎^𝑥𝑐𝑧(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟,(5.44) in which 𝑧 is given by formula (5.43) with 𝑦 defined by relation (5.1). Clearly, assumption (4.7) yields 𝑦≥0 and thus the series representation (5.43) ensures that 𝑧≥0 because the operator 𝐴 leaves invariant the cone 𝐿^∞(𝒟;ℝ₊). Hence, by virtue of Remark 2.4, desired property (4.8) of the solution 𝑢 follows from integral representation (5.44).
(3)⇒(4): assume that the assertion (3) of the theorem holds. Then, clearly the problem 𝛾^(1,1)(𝑡,𝑥)=ℓ₀(𝛾)(𝑡,𝑥)+ℓ₁𝛾^(1,0)(𝑡,𝑥)+ℓ₂𝛾^(0,1)(𝑡,𝑥)+1,(5.45)𝛾(𝑡,𝑐)=0for𝑡∈[𝑎,𝑏],𝛾(𝑎,𝑥)=0for𝑥∈[𝑐,𝑑](5.46) has a unique solution 𝛾 and the function 𝛾∈𝐶^∗(𝒟;ℝ) satisfies the inequalities 𝛾𝛾(𝑡,𝑥)≥0for(𝑡,𝑥)∈𝒟,_(1,0)𝛾(𝑡,𝑥)≥0fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)≥0for𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑].(5.47) Hence, conditions (4.9)–(4.11) are fulfilled. Moreover, inequality (4.12) follows from (5.45) because the operators ℓ₀, ℓ₁, and ℓ₂ are positive and 𝛾 satisfies relations (5.47).
(4)⇒(1): assume that there exists a function 𝛾∈𝐶^∗(𝒟;ℝ) satisfying inequalities (4.9)–(4.12). Then, by virtue of Remark 2.4, we get 𝛾^(1,1)∈𝐿^∞(𝒟;ℝ₊), 𝛾(𝑡,𝑥)≥^𝑡𝑎^𝑥𝑐𝛾𝛾^(1,1)(𝑠,𝜂)d𝜂d𝑠for(𝑡,𝑥)∈𝒟,_(1,0)𝛾(𝑡,𝑥)≥^𝑥𝑐𝛾^(1,1)(𝑡,𝜂)d𝜂fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)≥^𝑡𝑎𝛾^(1,1)(𝑠,𝑥)d𝑠,forall𝑡∈[𝑎,𝑏],a.e.𝑥∈[𝑐,𝑑].(5.48) Consequently, assertion (1) of the theorem holds with 𝑧₀=𝛾^(1,1) because the operators ℓ₀, ℓ₁, and ℓ₂ are positive.

Proof of Corollary 4.5. We put 𝛾(𝑡,𝑥)=e^{∫𝑡𝑎∫𝑥𝑐(1+ℓ0(1)(𝑠,𝜂))d𝜂d𝑠+2𝑔2(𝑡−𝑎)+2𝑔1(𝑥−𝑐)}for(𝑡,𝑥)∈𝒟,(5.49) where 𝑔₁=‖ℓ₁(1)‖_𝐿^∞ and 𝑔₂=‖ℓ₂(1)‖_𝐿^∞. It can be verified that 𝛾∈𝐶^∗(𝒟;ℝ) and satisfies the inequality 𝛾(𝑡,𝑥)≥1for(𝑡,𝑥)∈𝒟.(5.50) In view of Lemma 2.3, from (5.49) we get 𝛾^(1,0)(𝑡,𝑥)=^𝑥𝑐1+ℓ₀(1)(𝑡,𝜂)d𝜂+2𝑔₂𝛾𝛾(𝑡,𝑥),fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)=^𝑡𝑎1+ℓ₀(1)(𝑠,𝑥)d𝑠+2𝑔₁𝛾(𝑡,𝑥),for𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑].(5.51) Therefore, inequalities (4.10) and (4.11) are fulfilled and 𝛾^(1,0)(𝑡,𝑥)≥0,𝛾^(0,1)(𝑡,𝑥)≥0fora.e.(𝑡,𝑥)∈𝒟,(5.52) because the operator ℓ₀ is positive. Moreover, by using Lemma 2.3 and inequalities (5.50) and (5.52), it follows from equalities (5.51) that 𝛾^(1,1)(𝑡,𝑥)=1+ℓ₀(1)(𝑡,𝑥)𝛾(𝑡,𝑥)+^𝑥𝑐1+ℓ₀(1)(𝑡,𝜂)d𝜂+2𝑔₂𝛾^(0,1)=1+ℓ₀(𝑡,𝑥)+1(1)(𝑡,𝑥)𝛾(𝑡,𝑥)2^𝑡𝑎1+ℓ₀(1)(𝑠,𝑥)d𝑠+2𝑔₁𝛾^(1,0)+1(𝑡,𝑥)2^𝑥𝑐1+ℓ₀(1)(𝑡,𝜂)d𝜂+2𝑔₂𝛾^(0,1)≥1+ℓ₀(𝑡,𝑥)≥ℓ₀(1)(𝑡,𝑥)𝛾(𝑡,𝑥)+𝑔₁𝛾^(1,0)(𝑡,𝑥)+𝑔₂𝛾^(0,1)(𝑡,𝑥)(1)(𝑡,𝑥)𝛾(𝑡,𝑥)+ℓ₁(1)(𝑡,𝑥)𝛾^(1,0)(𝑡,𝑥)+ℓ₂(1)(𝑡,𝑥)𝛾^(0,1)(𝑡,𝑥)+1,(5.53) for a.e. (𝑡,𝑥)∈𝒟 and thus inequality (4.12) is satisfied.
On the other hand, equalities (5.49), (5.51) guarantee the validity of the inequalities𝛾𝛾𝑡₁,𝑥₁≤𝛾𝑡₂,𝑥₂for𝑎≤𝑡₁≤𝑡₂≤𝑏,𝑐≤𝑥₁≤𝑥₂≤𝑑,_(1,0)𝛾𝑡,𝑥₁≤𝛾^(1,0)𝑡,𝑥₂fora.e.𝑡∈[𝑎,𝑏]andall𝑐≤𝑥₁≤𝑥₂≤𝑑,_(0,1)𝑡₁,𝑥≤𝛾^(0,1)𝑡₂,𝑥for𝑎≤𝑡₁≤𝑡₂≤𝑏anda.e.𝑥∈[𝑐,𝑑].(5.54) Therefore, by using Lemmas 5.8–5.10, from inequality (5.53) we get 𝛾^(1,1)(𝑡,𝑥)≥ℓ₀(𝛾)(𝑡,𝑥)+ℓ₁𝛾^(1,0)(𝑡,𝑥)+ℓ₂𝛾^(0,1)(𝑡,𝑥)+1fora.e.(𝑡,𝑥)∈𝒟.(5.55) that is, relation (4.9) holds.
Consequently, we have found a function 𝛾 satisfying conditions (4.9)–(4.12) and thus the assertion of the corollary follows from Theorem 4.4.

Proof of Corollary 4.10. It is clear that (3.2) is a particular case of (1.1) in which the operators ℓ₀, ℓ₁, and ℓ₂ are defined by formulas (2.23), (2.28), and (2.29), respectively (see Examples 2.7 and 2.11). Since we have ‖‖ℓ_𝑘‖‖=‖‖𝑝_𝑘‖‖_𝐿∞for𝑘=0,1,2,(5.56) the validity of the corollary follows immediately from Corollary 4.2.

Proof of Corollary 4.11. It is clear that (3.2) is a particular case of (1.1) in which the operators ℓ₀,ℓ₁, and ℓ₂ are defined by formulas (2.23), (2.28), and (2.29), respectively, and that the operators indicated are positive (see Examples 2.7 and 2.11). Moreover, in view of inequality (4.17), there exists 𝜀>0 such that 1≥+𝑝₀𝑝(₂𝑡,𝑥)𝜏₀(𝑡,𝑥)−𝑎𝜇₀(𝑡,𝑥)−𝑐+𝑝₁(𝑡,𝑥)𝜇₁(𝑡,𝑥)−𝑐(𝑡,𝑥)𝜏₂(𝑡,𝑥)−𝑎+𝜀fora.e.(𝑡,𝑥)∈𝒟.(5.57) Therefore, the function 𝛾 defined by the relation 𝛾(𝑡,𝑥)=(𝑡−𝑎)(𝑥−𝑐)for(𝑡,𝑥)∈𝒟(5.58) satisfies inequalities (4.9)–(4.12) and thus the assertion of the corollary follows from Theorem 4.4.

Proof of Corollary 4.12. It is clear that (3.2) is a particular case of (1.1) in which the operators ℓ₀, ℓ₁, and ℓ₂ are defined by formulas (2.23), (2.28), and (2.29), respectively, and that conditions (4.13) and (4.14) are fulfilled (see Examples 2.7 and 2.11). Moreover, the operators ℓ₀, ℓ₁ and ℓ₂ are positive and, in view of Remarks 2.9 and 2.13, the operators indicated are, respectively, (𝑎,𝑐)-Volterra, 𝑐-Volterra, and 𝑎-Volterra ones.
Consequently, the validity of the corollary follows from Corollary 4.5.

Proof of Corollary 4.13. It is clear that (3.2) is a particular case of (1.1) in which the operators ℓ₀, ℓ₁, and ℓ₂ are defined by formulas (2.23), (2.28), and (2.29), respectively, and that the operators indicated are positive (see Examples 2.7 and 2.11).
Let 𝑝_𝑘̃=‖𝑝_𝑘‖_𝐿^∞(𝑘=0,1,2) and 𝑦=e/𝜔. According to (4.19) and (4.20), there exist numbers 𝜀>0 and 𝛿∈]0,1[ such that inequalities ^{𝜏0𝑡(𝑡,𝑥)}^{𝜇0𝑐(𝑡,𝑥)}𝜀+𝑝₀(𝑠,𝜂)d𝜂d𝑠+^𝑡𝑎^{𝜇0𝑥(𝑡,𝑥)}𝜀+𝑝₀+2𝑝₂̃(𝑠,𝜂)d𝜂d𝑠≤1𝜏₀(𝑡,𝑥)−𝑡+2𝑝₁̃𝜇₀(𝑡,𝑥)−𝑥𝑦ln𝑦+e𝑦_∫𝑦[^𝑏𝑎_∫^𝑑𝑐_{(𝜀+𝑝0(𝑠,𝜂))d𝜂d𝑠+2}𝛿𝑝̃_{2(𝑏−𝑎)+2}𝑝̃_{1(𝑑−𝑐)]}−𝛿_𝑡𝑎,^{𝜇1𝑥(𝑡,𝑥)}𝜀+𝑝₀≤1(𝑠,𝜂)d𝜂d𝑠+2p̃₁𝜇₁(𝑡,𝑥)−𝑥𝑦ln2𝑝²̃𝑦𝜀+𝑝₀̃(𝑑−𝑐)+2𝑝₂̃_{𝜏2𝑡(𝑡,𝑥)},^𝑥𝑐𝜀+𝑝₀≤1(𝑠,𝜂)d𝜂d𝑠+2𝑝₂̃𝜏₂(𝑡,𝑥)−𝑡𝑦ln2𝑝¹̃𝑦𝜀+𝑝₀̃(𝑏−𝑎)+2𝑝₁̃,(5.59) hold for a.e. (𝑡,𝑥)∈𝒟. Now we put 𝛾(𝑡,𝑥)=e^{𝑦𝑧(𝑡,𝑥)}−𝛿for(𝑡,𝑥)∈𝒟,(5.60) where 𝑧(𝑡,𝑥)=^𝑡𝑎^𝑥𝑐𝜀+𝑝₀(𝑠,𝜂)d𝜂d𝑠+2𝑝₂̃(𝑡−𝑎)+2𝑝₁̃(𝑥−𝑐)for(𝑡,𝑥)∈𝒟.(5.61) It can be verified that 𝛾∈𝐶^∗(𝒟;ℝ) and satisfies 𝛾(𝑡,𝑥)≥1−𝛿>0for(𝑡,𝑥)∈𝒟.(5.62) In view of Lemma 2.3, from (5.62) we get 𝛾^(1,0)(𝑡,𝑥)=𝑦^𝑥𝑐𝜀+𝑝₀(𝑡,𝜂)d𝜂+2𝑝₂̃𝛾(𝛾(𝑡,𝑥)+𝛿),fora.e.𝑡∈[𝑎,𝑏]andall𝑥∈[𝑐,𝑑_(0,1)],(𝑡,𝑥)=𝑦^𝑡𝑎𝜀+𝑝₀(𝑠,𝑥)d𝑠+2𝑝₁̃(𝛾(𝑡,𝑥)+𝛿),for𝑡∈[𝑎,𝑏]anda.e.𝑥∈[𝑐,𝑑].(5.63) Therefore, relations (4.10), (4.11), and (5.52) are fulfilled. Moreover, by using Lemma 2.3 and inequalities (5.52) and (5.62), it follows from equalities (5.63) that 𝛾^(1,1)(𝑡,𝑥)=𝑦𝜀+𝑝₀(𝑡,𝑥)(𝛾(𝑡,𝑥)+𝛿)+𝑦^𝑥𝑐𝜀+𝑝₀(𝑡,𝜂)d𝜂+2𝑝₂̃𝛾^(0,1)=𝑦𝜀(𝛾(𝑡,𝑥)+𝛿)+𝑦𝑝₀(𝑡,𝑥)+𝑦(𝑡,𝑥)(𝛾(𝑡,𝑥)+𝛿)2^𝑡𝑎𝜀+𝑝₀(𝑠,𝑥)d𝑠+2𝑝₁̃𝛾^(1,0)+𝑦(𝑡,𝑥)2^𝑥𝑐𝜀+𝑝₀(𝑡,𝜂)d𝜂+2𝑝₂̃𝛾^(0,1)≥𝑦𝜀+𝑝₀(𝑡,𝑥)≥𝑝₀(𝑡,𝑥)𝑦(𝛾(𝑡,𝑥)+𝛿)+𝑦𝑝₁̃𝛾^(1,0)(𝑡,𝑥)+𝑦𝑝₂̃𝛾^(0,1)(𝑡,𝑥)(𝑡,𝑥)𝑦(𝛾(𝑡,𝑥)+𝛿)+𝑝₁(𝑡,𝑥)𝑦𝛾^(1,0)(𝑡,𝑥)+𝑝₂(𝑡,𝑥)𝑦𝛾^(0,1)(𝑡,𝑥)+𝑦𝜀,(5.64) for a.e. (𝑡,𝑥)∈𝒟 and thus inequality (4.12) is satisfied. Now observe that inequalities (5.59) can be rewritten to the forms 𝑧𝜏₀(𝑡,𝑥),𝜇₀(𝑡,𝑥)−𝑧(𝑡,𝑥)≤1𝑧𝑡,𝜇₁𝑦ln𝑦+e𝑦𝛿^{𝑦𝑧(𝜏0(𝑡,𝑥),𝜇0(𝑡,𝑥))}−𝛿,(𝑡,𝑥)−𝑧(𝑡,𝑥)≤1𝑦ln2𝑝²̃𝑦∫^{𝜇1𝑐(𝑡,𝑥)}𝜀+𝑝₀(𝑡,𝜂)d𝜂+2𝑝₂̃𝑧𝜏₂,(𝑡,𝑥),𝑥−𝑧(𝑡,𝑥)≤1𝑦ln2𝑝¹̃𝑦∫^{𝜏2𝑎(𝑡,𝑥)}𝜀+𝑝₀(𝑠,𝑥)d𝑠+2𝑝₁̃,(5.65) for a.e. (𝑡,𝑥)∈𝒟 and thus, by using relations (5.60), (5.63), we get 𝑦𝛾𝑦_(1,0)(𝛾(𝑡,𝑥)+𝛿)≥𝛾𝜏₀(𝑡,𝑥),𝜇₀(𝑡,𝑥)fora.e.(𝑡,𝑥)∈𝒟,(𝑡,𝑥)≥𝛾^(1,0)𝑡,𝜇₁(𝑡,𝑥),𝑦𝛾^(0,1)(𝑡,𝑥)≥𝛾^(0,1)𝜏₂(𝑡,𝑥),𝑥(5.66) for a.e. (𝑡,𝑥)∈𝒟. Consequently, it follows from (5.64) that inequality (4.9) holds.
We have constructed a function 𝛾 satisfying conditions (4.9)–(4.12) and thus the assertion of the corollary follows from Theorem 4.4.

Acknowledgment

For A. Lomtatidze and J. Šremr, the research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503.