#### Abstract

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem for given on the rectangle are established, where the linear operators , , 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 , we consider the linear partial functional-differential equation where and (resp., ) denote the first-order (resp., the second-order mixed) partial derivatives. The operators , , and 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 absolutely continuous in the sense of Carathéodory possessing some additional properties (namely, inclusions (2.20)) which satisfies equality (1.1) almost everywhere on .

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 to (1.1) are prescribed on both characteristics and , that is, the initial conditions are Properties of the initial functions and will be specified in Section 3. It is worth to remember here that various initial and boundary value problems for the hyperbolic equation 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 and is considered.

#### 2. Notation and Definitions

The following notation is used throughout the paper.(i), , and are the sets of all natural, rational, and real numbers, respectively, .(ii), where and .(iii)The first-order partial derivatives of the function at the point are denoted by (or ) and (or ). The second-order mixed partial derivatives of the function at the point are denoted by and whereas we use if .(iv) is the Banach space of continuous functions equipped with the norm (v), where , is the linear space of continuous functions .(vi), where , is the linear space of absolutely continuous functions .(vii) is the Banach space of Lebesgue measurable and essentially bounded functions equipped with the norm (viii).(ix)For any , we put (x), where , is the linear space of Lebesgue measurable and essentially bounded functions .(xi)meas denotes the Lebesgue measure of the set , .(xii)If , are Banach spaces and is a linear bounded operator then denotes the norm of the operator , that is,

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

##### 2.1. Spaces , , and Set

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 be such that
**
Then the function is measurable. *

*Notation 1. * denotes the linear space of all functions satisfying conditions (2.5), and
If one identifies functions , from such that for a.e. then
defines a norm in the space .

Analogously, we introduce the space of functions which are “measurable in the first variable and continuous in the second one” and define the norm 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. * and are Banach spaces.*

* Proof. *We only prove the assertion for the space , the assertion of the lemma concerning the space can be proven analogously by exchanging the roles of the variables and .

Let be an arbitrary Cauchy sequence in . For a decreasing sequence of positive numbers with there exists an increasing sequence such that
for every , . Let . Then, for any , there is a set , meas , such that
Put . Then, clearly, we have meas and
Consequently, for any fixed , the sequence converges uniformly on , say to . Hence, converges point-wise on to for every fixed . Therefore, the function satisfies conditions (2.5). Since
holds for , all and a.e. , in view of (2.8) and (2.9), we obtain
Hence, and in , that is, the space 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. * stands for the set of functions admitting the integral representation
where , , , and .

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 be a Lebesgue integrable function and
**
Then *(1)*there exists a set such that and *(2)*there exists a set such that and *(3)*there exists a set such that and *

*Remark 2.4. *If , that is, the function admits integral representation (2.13), then by using Lemma 2.3 we get
Consequently, for any , we have

*Remark 2.5. *It follows from Remark 2.4 and [22, Satz 1, page 654] that if and only if is absolutely continuous in the sense of Carathéodory with the properties

##### 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 .

*Definition 2.6. *A linear operator is said to be *positive* if the relation
holds whenever the function is such that

*Example 2.7. *For any , we put
where and , are measurable functions. Then the operator is linear and bounded. Moreover, is positive if and only if for a.e. .

*Definition 2.8. *A linear operator is called *-Volterra operator* if, for any and such that
we have

*Remark 2.9. *It can be shown by using Lemma 5.8 stated below that the operator given by formula (2.23) is an -Volterra one if and only if

Now we introduce analogous notions for linear operators defined on the spaces and .

*Definition 2.10. *We say that a linear operator (resp., ) is *positive* if relation (2.21) is satisfied for every function (resp., ) such that

*Example 2.11. *For any (resp., ), we put
respectively,
where and , are measurable functions. Then the operators and are linear and bounded. Moreover, (resp., ) is positive if and only if (resp., ) for a.e. .

*Definition 2.12. *We say that a linear operator (resp., ) is an *-Volterra operator* (resp., a *-Volterra operator*) if, for any (resp., ) and (resp., ) such that
we have

*Remark 2.13. *One can show by using Lemma 5.9 (resp., Lemma 5.10) stated below that the operator (resp., ) given by formula (2.28) (resp., (2.29)) is a -Volterra one (resp., an -Volterra one) if and only if
respectively,

#### 3. Statement of Problem

On the rectangle , we consider the linear nonhomogeneous Darboux problem (1.1), (1.2) in which , , and are linear bounded operators, , and , are such that , and . By a solution to problem (1.1), (1.2), we mean a function possessing property (1.2) and satisfying equality (1.1) almost everywhere on . 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 equation where , 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 , , and . 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 deviations in which coefficients and argument deviations , are measurable functions. We obtain this equation from (1.1) if the operators , , and are defined by formulas (2.23), (2.28), and (2.29), respectively. Let us also mention that in the case, where equation (3.2) takes form (3.1).

#### 4. Main Results

At first, we put where

Clearly, , , and thus the operators , , mapping the space into itself are linear and bounded.

Theorem 4.1. *Let , where the operators , , are defined by relations (4.1), (4.2). If the spectral radius of the operator is less than one then problem (1.1), (1.2) is uniquely solvable for arbitrary and , such that , and .*

Theorem 4.1 implies the following.

Corollary 4.2. *If the inequality
**
holds then problem (1.1), (1.2) is uniquely solvable for arbitrary and , such that , and .*

*Remark 4.3. *On the rectangle , we consider the equation
subjected to the initial conditions
where
Clearly (4.4) is a particular case of (1.1). If , then problem (4.4), (4.5) has the trivial solution and the nontrivial solution . 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 , , and on the right-hand side of (1.1) are positive then we can estimate the spectral radius of the operator by using the well-known results due to Krasnosel'skij and we thus obtain the following.

Theorem 4.4. *Let the operators , , be positive and , where the operators , , are defined by relations (4.1), (4.2). Then the following four assertions are equivalent. *(1)*There exists a function such that .*(2)*The spectral radius of the operator is less than one.*(3)*Problem (1.1), (1.2) is uniquely solvable for arbitrary and such that , , and .**If, in addition, the initial functions and the forcing term are such that
then the solution to problem (1.1), (1.2) satisfies
*(4)*There exists a function such that*

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

Corollary 4.5. *Let , , and be positive -Volterra, -Volterra, and -Volterra operators, respectively, such that the inequalities
**(here, means in which for and all ) and
** we mean , where and a.e. hold for every and . **Then problem (1.1), (1.2) is uniquely solvable for arbitrary and , such that , and . If, in addition, the initial functions , and the forcing term are such that relations (4.7) hold, then the solution to problem (1.1), (1.2) satisfies inequalities (4.8).*

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

*Definition 4.6. *Let , , and . We say that the triplet belongs to the set if the implication
holds.

*Remark 4.7. *If , 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 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 .*

*Remark 4.9. *The inclusion ensures that every solution 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 , , and . 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 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 and argument deviations , are measurable functions.

As a consequence of Corollary 4.2 we obtain the following.

Corollary 4.10. *If the inequality
**
holds, then problem (3.2), (1.2) is uniquely solvable for arbitrary and , such that , , and .*

If the coefficients , , in the previous corollary are non-negative then the assertion of the corollary follows also from implication of Theorem 4.4. More precisely, the following statement holds.

Corollary 4.11. *Let and
**
Then problem (3.2), (1.2) is uniquely solvable for arbitrary and , such that , , and . If, in addition, the initial functions , , and the forcing term are such that relations (4.7) hold, then the solution to problem (3.2), (1.2) satisfies inequalities (4.8).*

Finally, Corollary 4.5 implies the following.

Corollary 4.12. *Let and argument deviations , , , and satisfy inequalities (2.26), (2.32), and (2.33). Then problem (3.2), (1.2) is uniquely solvable for arbitrary and , such that , , and . If, in addition, the initial functions , and the forcing term are such that relations (4.7) hold, then the solution 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 , , , and are not necessarily delays but the differences are small enough, that is, if (3.2) with deviating arguments is “close” to (3.1).

Corollary 4.13. *Let , and
**where
**
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 , , and , the operators , , and are defined by relations (4.1), (4.2). Moreover, having and , such that , , and , we put the authors understand in which = for . Similarly, means , where for a.e. and all for all . Clearly, .

Lemma 5.1. *If is a solution to problem (1.1), (1.2) then is a solution to the equation
**
in the space , where the operators , , and are defined by relations (4.1), (4.2) and the function is given by formula (5.1). **Conversely, if is a solution to (5.2) in the space with the operators , , and defined by relations (4.1), (4.2) and the function given by formula (5.1), then **
is a solution to problem (1.1), (1.2).*

* Proof. *If is a solution to problem (1.1), (1.2) then, by virtue of Remark 2.4, we get ,
Consequently, (1.1) yields that
where the operators , , and are defined by relations (4.1), (4.2) and the function is given by formula (5.1).

Conversely, let be a solution to (5.2) in the space with the operators , , and defined by relations (4.1), (4.2) and the function given by formula (5.1). Moreover, let the function be defined by relation (5.3), that is,
Then the function belongs to the set and verifies initial conditions (1.2). Furthermore, by using Lemma 2.3, we get
Consequently, (5.2) implies that 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 in a Banach space is called a *cone* if the following conditions are satisfied: (i) for all ,(ii) for all and an arbitrary ,(iii)if and then .

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

*Definition 5.4. *We say that a cone is *solid* if its interior is nonempty.

*Remark 5.5. *The presence of a cone in a Banach space allows one to introduce a natural partial ordering there. More precisely, two elements are said to be in the relation if and only if they satisfy the inclusion . If, moreover, is a solid cone then we write if and only if .

*Definition 5.6. *A cone is said to be *normal* if there is a constant such that, for every with the property , the relation 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 be a normal and solid cone in a Banach space and the operator leave invariant the cone , that is, . If there exists a constant and an element such that , then the spectral radius of the operator is less than .*

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

Lemma 5.8. *Let be a positive -Volterra operator. Then, for any function satisfying
**
one has
*

* Proof. *We first show that, for any , we have
Indeed, let be arbitrary but fixed. Put
Then, clearly ,
Since the operator is positive, we obtain
On the other hand, the operator is supposed to be an -Volterra one which guarantees the equality
and thus the desired relation (5.10) holds.

Now we put
It follows from Lemma 2.3 that there exists a set , meas , such that
and, moreover, there is a set , meas , with the properties

Let be arbitrary but fixed. Then, relation (5.10) yields
for and , whence we get
For any fixed, we pass to the limit in the latter inequality and thus, in view of equalities (5.16), we get
for . Now, letting in the previous relation and using equalities (5.17) give
That is, the desired inequality (5.9) holds because was arbitrary.

Lemma 5.9. *Let be a positive -Volterra operator such that inequality (4.14) holds for every . Then, for any function with the property
**
one has
*

*Proof. *Let , meas , be a set such that, for any , we have and
We first show that the relation
holds for every . Indeed, let be arbitrary but fixed. Put
Then, clearly, ,
Since the operator is positive and satisfies condition (4.14), we obtain
(by the authors mean , where for all and ). On the other hand, the operator is supposed to be an -Volterra one which guarantees the equality
and thus desired relation (5.25) holds for every . It means that, for any , there exists a set with meas such that
where, for each , we have with meas .

Put , where . Clearly, meas because the set is countable. Moreover, relation (5.30) yields that
Let , , and be arbitrary but fixed. Then there exists a sequence such that as . It follows from relation (5.31) that