Carathéodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments

where p,q : D → R are Lebesgue integrable functions, τ : D → [a,b] and μ : D → [c,d] are measurable functions, and φ : [a,b] → R, ψ : [c,d] → R are absolutely continuous functions such that φ(a) = ψ(c). By a solution to problem (.), (.) we mean a function u : D →R absolutely continuous onD in the sense of Carathéodorya which satisfies equality (.) almost everywhere in D and verifies the initial conditions (.). It is well known that theorems on differential inequalities (maximum principles in other terminology) play an important role in the theory of both ordinary and partial differential equations. For example, theorems on hyperbolic differential inequalities dealing with classical as well as Carathéodory solutions are studied in [–]. By using these statements, in particular, the method of lower and upper functions and monotone iterative techniques can be developed to derive solvability results for hyperbolic equations subjected to various initial conditions (Darboux, Cauchy, Goursat, etc.) as is done, e.g., in [, –]. In this paper we continue the study of theorems on linear hyperbolic differential inequalities initiated in [], where a more general functional-differential equation with a


Introduction
It is well known that theorems on differential inequalities (maximum principles in other terminology) play an important role in the theory of both ordinary and partial differential equations. For example, theorems on hyperbolic differential inequalities dealing with classical as well as Carathéodory solutions are studied in [-]. By using these statements, in particular, the method of lower and upper functions and monotone iterative techniques can be developed to derive solvability results for hyperbolic equations subjected to various initial conditions (Darboux, Cauchy, Goursat, etc.) as is done, e.g., in [, -].
In this paper we continue the study of theorems on linear hyperbolic differential inequalities initiated in [], where a more general functional-differential equation with a http://www.boundaryvalueproblems.com/content/2014/1/52 linear operator : C(D; R) → L(D; R) on the right-hand side is investigated and (.) is considered as a particular case of it.
We have introduced the following definition in [].
We say that the principle on differential inequalities (maximum principle) holds for (.) and we write (p, τ , μ) ∈ S ac (D) if for any function u : D → R absolutely continuous on D in the sense of Carathéodory b satisfying the inequalities holds.
It is also mentioned in [] that under the assumption (p, τ , μ) ∈ S ac (D), problem (.), (.) has a unique (Carathéodory) solution and this solution satisfies (.) provided Moreover, some efficient conditions are given in [] for the validity of the inclusion (p, τ , μ) ∈ S ac (D) in the case, where From those results it follows that, in the case (.), the hyperbolic equation (.) is similar in a certain sense to first-order ordinary differential equations, which is already noted in the book of Walter []. It is worth mentioning here that Definition . is in compliance with the formulation of a theorem on differential inequalities given in [, Theorem ], where the case (.) is also considered.
On the other hand, if then the explanation of Walter that hyperbolic equations are 'similar' to first-order ordinary differential equations does not hold because even for (.) with It is well known that, without any additional assumptions Recall that, for any (t  , x  ) ∈ D, the Riemann function Z(t  , x  , ·, ·) is defined as a solution to (.) satisfying the initial conditions Therefore, it follows from equality (.) and Definition . that a theorem on differential inequalities holds for (.) if and only if Proposition . Let k ≤ , the functions τ , μ be defined by relations (.), and where j  denotes the first positive zero of the Bessel function J  .
In Section , we consider the case (.) and we present new effective conditions for the validity of the inclusion (p, τ , μ) ∈ S ac (D) that are proved later in Section  by comparing (.) with a linear hyperbolic equation without argument deviations.

Main results
For any ν > -, let J ν denote the Bessel function of the first kind and order ν and let j ν be the first positive zero of the function J ν . Moreover, we put where is the standard gamma function.
Remark . Theorem . cannot be improved in the sense that assumption (.) cannot be, in general, replaced by the assumption Remark . It follows from the proof of Corollary . that the number j * -α on the righthand side of inequalities (.) and (.) can be replaced by respectively, where the function z is defined by (.).

Proofs
The following notation is used throughout this section.
• The first-order partial derivatives of a function v : • The second-order mixed partial derivative of a function v : To prove the main results stated in the previous section we need the next three lemmas.
Lemma . Let ν > -. Then the function E ν defined by (.) has the following properties: for s ≥ .
(ii) E ν () >  and j ν is the first positive zero of the function E ν .
(iii): Since the series in assertion (i) converges uniformly on every closed subinterval of [, +∞[, we can take its derivative term-by-term and thus assertion (iii) follows immediately from (i). and Then (p, τ , μ) ∈ S ac (D). http://www.boundaryvalueproblems.com/content/2014/1/52 Proof It follows from [, Theorem .] with the operator defined by the relation for a.e. (t, x) ∈ D and all continuous functions v : D → R.