- Open Access
Carathéodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments
© Lomtatidze and Šremr; licensee Springer. 2014
- Received: 28 January 2014
- Accepted: 20 February 2014
- Published: 12 March 2014
New effective conditions are found for the validity of a theorem on differential inequalities corresponding to the Darboux problem for linear hyperbolic differential equations with argument deviations.
- Carathéodory solution
- hyperbolic differential equation with arguments deviations
- Darboux problem
- differential inequality
where are Lebesgue integrable functions, and are measurable functions, and , are absolutely continuous functions such that . By a solution to problem (1.1), (1.2) we mean a function absolutely continuous on in the sense of Carathéodorya which satisfies equality (1.1) almost everywhere in and verifies the initial conditions (1.2).
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 [1–8]. 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 [4, 8–11].
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 linear operator on the right-hand side is investigated and (1.1) is considered as a particular case of it.
We have introduced the following definition in .
From those results it follows that, in the case (1.4), the hyperbolic equation (1.1) 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 1.1 is in compliance with the formulation of a theorem on differential inequalities given in [, Theorem 1], where the case (1.4) is also considered.
Unfortunately, the Riemann functions can be explicitly written only in some simple cases. In particular, for (1.9) with a constant non-positive coefficient p the following proposition holds (see, e.g., [, Section 3.4] or [, Example 8.1]).
where denotes the first positive zero of the Bessel function .
In Section 2, we consider the case (1.5) and we present new effective conditions for the validity of the inclusion that are proved later in Section 3 by comparing (1.1) with a linear hyperbolic equation without argument deviations.
where Γ is the standard gamma function.
Then the theorem on differential inequalities holds for (1.1), i.e., .
Remark 2.1 As we have mentioned above, assumption (1.7) in Theorem 2.1 is necessary for the validity of the inclusion in the case where inequality (1.5) holds (see ).
no matter how small is. Indeed, if p, τ, and μ are defined by (1.6) and (1.13) with , then assumptions (2.3) and (2.4) of Theorem 2.1 hold with , , and inequality (2.2) takes the form (1.14), which is, in this case, necessary for the validity of the inclusion as is stated in Proposition 1.1.
Remark 2.3 Observe that if for a.e. then the left-hand side of inequality (2.3) is equal to zero. Therefore, assumption (2.3) of Theorem 2.1 says how ‘close’ must be to t, and this ‘closeness’ is understood through the composition of the functions and z. Similarly, ‘closeness’ of to x is required in assumption (2.4).
The meaning of assumptions (2.3) and (2.4) of Theorem 2.1 is more transparent in the following two corollaries.
respectively, where the function z is defined by (2.5).
and thus inequality (2.6) with yields the validity of inequality (2.10). Analogously, inequality (2.11) follows from inequality (2.7) with .
The following notation is used throughout this section.
The first-order partial derivatives of a function at a point are denoted by
The second-order mixed partial derivative of a function at a point is denoted by
To prove the main results stated in the previous section we need the next three lemmas.
and is the first positive zero of the function .
The function is continuous for .
Proof (i), (ii): It follows from (2.1), the definition of the function (see, e.g., [, Chapter III, Section 3.12]), and the fact that for every .
(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).
(iv): See [, Chapter XV, Section 15.21].
(v): It follows from assertions (i) and (iii).
(see, e.g., [, Chapter III, Section 3.12]). Consequently, by direct calculation we can check that the function satisfies the desired equality for every . It remains to mention that for , the validity of the desired equality is obvious. □
Lemma 3.2 ([, Theorem 2.1])
- (2), i.e., the function admits the representation
The function satisfies the conditions:
the function is absolutely continuous for every and the function is absolutely continuous;
the function is absolutely continuous for almost all ;
the function is Lebesgue integrable.
for a.e. and all continuous functions . □
respectively. Finally, by virtue of inequalities (2.2), (3.3) and (3.8), (3.18), (3.19), it follows from (3.7) and (3.17) that the function γ satisfies also assumption (3.1) of Lemma 3.3 and thus . □
i.e., inequality (2.3) holds for a.e. . We can show in a similar manner that inequality (2.4) holds for a.e. , where the function z is defined by (2.5). Therefore, the assertion of the corollary follows from Theorem 2.1. □
i.e., inequality (2.3) with holds for a.e. . We can show in a similar manner that inequality (2.4) with holds for a.e. , as well. Consequently, all assumptions of Theorem 2.1 with are satisfied and thus . □
The research was supported by RVO: 67985840.
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