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Carathéodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments
Boundary Value Problems volume 2014, Article number: 52 (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.
On the rectangle we consider the Darboux problem
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 .
Definition 1.1 Let be a Lebesgue integrable function and , be measurable functions. We say that the principle on differential inequalities (maximum principle) holds for (1.1) and we write if for any function absolutely continuous on in the sense of Carathéodoryb satisfying the inequalities
It is also mentioned in  that under the assumption , problem (1.1), (1.2) has a unique (Carathéodory) solution and this solution satisfies (1.3) provided
Moreover, some efficient conditions are given in  for the validity of the inclusion in the case, where
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.
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 (1.1) with
oscillatory solutions may occur. In this case, the properties of hyperbolic equation (1.1) are ‘closer’ to properties of ordinary differential equations of the second order. In , we got a general sufficient condition for the validity of the inclusion in the case (1.5) under the assumption that (1.1) is delayed in both arguments, i.e., if the inequalities
hold (see Lemma 3.3 below). Using that general result, we have also proved in  that if p, μ, and τ satisfy conditions (1.5) and (1.7), then provided
Note that assumption (1.7) is not restrictive in the case (1.5) because it is necessary, as is shown in . Moreover, inequality (1.8) cannot, in general, be improved (see [, Example 6.2]). However, it does not mean that inequality (1.8) is necessary and cannot be weakened in particular cases. In this paper, we give efficient criteria for the validity of the inclusion in the case when (1.5) holds optimally for equations which are ‘close’ to the equation without argument deviations,
It is well known that, without any additional assumptions, the Darboux problem (1.9), (1.2) has a unique (Carathéodory) solution u (see [, Existensatz, ] and [, Remarks (b), (c)]) and this solution admits the integral representation
Recall that, for any , the Riemann function is defined as a solution to (1.11) satisfying the initial conditions
Therefore, it follows from equality (1.10) and Definition 1.1 that a theorem on differential inequalities holds for (1.9) if and only if
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]).
Proposition 1.1 Let , the functions τ, μ be defined by relations (1.6), and
Then if and only if
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.
2 Main results
For any , let denote the Bessel function of the first kind and order ν and let be the first positive zero of the function . Moreover, we put
where Γ is the standard gamma function.
Theorem 2.1 Let be a Lebesgue integrable function and , be measurable functions satisfying conditions (1.5) and (1.7). Moreover, let there exist numbers , , and such that the inequalities
are fulfilled a.e. on , where
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 ).
Remark 2.2 Theorem 2.1 cannot be improved in the sense that assumption (2.2) cannot be, in general, replaced by the assumption
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.
Corollary 2.1 Let be a Lebesgue integrable function and , be measurable functions satisfying conditions (1.5) and (1.7). Moreover, let there exist numbers , , and such that inequalities (2.2),
are fulfilled a.e. on , where
Remark 2.4 It follows from the proof of Corollary 2.1 that the number on the right-hand side of inequalities (2.6) and (2.7) can be replaced by
respectively, where the function z is defined by (2.5).
Corollary 2.2 Let be a Lebesgue integrable function and , be measurable functions satisfying conditions (1.5) and (1.7). Moreover, let there exist numbers and such that the inequalities
are fulfilled a.e. on , where the number is defined by formula (2.8). Then .
Remark 2.5 Corollary 2.2 improves Corollary 2.1 with . Indeed, for a.e. such that we have
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.
Lemma 3.1 Let . Then the function defined by (2.1) has the following properties:
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).
(vi): The function is a solution to the Bessel equation and thus we have
(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])
The following three statements are equivalent:
The function is absolutely continuous on in the sense of Carathéodory.c
, i.e., the function admits the representation
where and , , and are Lebesgue integrable functions.
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.
Lemma 3.3 Let be a Lebesgue integrable function and , be measurable functions satisfying conditions (1.5) and (1.7). Assume that there exists a functiond such that
Proof It follows from [, Theorem 3.5] with the operator ℓ defined by the relation
for a.e. and all continuous functions . □
Proof of Theorem 2.1 Let
where the functions and z are defined by (2.1) and (2.5), respectively. It is clear that
and thus, in view of Lemma 3.1(ii), the function γ satisfies inequalities (3.2) and (3.3). Since the functions and are absolutely continuous for every and , by virtue of Lemma 3.1(v), we conclude that the functions and are absolutely continuous for every and , respectively. Moreover, we have
Now, in view of Lemma 3.1(v), it follows from (3.5) that the function is absolutely continuous for every and
for every . Therefore, by using equalities (2.5), (3.4)-(3.6), and Lemma 3.1(vi), we get
which shows, in particular, that the function is Lebesgue integrable on . Consequently, Lemma 3.2 guarantees that . Moreover, by using Lemma 3.1(iii), (iv), we get
and thus (3.7) implies that
Now, by virtue of Lemma 3.3, to prove the theorem it remains to show that the function γ satisfies differential inequality (3.1). For this purpose we put
Observe that, in view of equalities (3.5), (3.10), and Lemma 3.1(iii), we have
and thus, by using (2.1), (2.5), and (3.12), we get
We can show in a similar manner that
On the other hand, in view of (2.1), (2.5) and (3.12), and Lemma 3.1(iii), it follows from equality (3.10) that
whence we get
for a.e. because is an increasing absolutely continuous function for every . Similarly, we can show that
for a.e. . We have proved that and therefore, by using Lemma 3.2, we get
Multiplying both sides of the latter equality by and using inequalities (1.5), (1.7), (3.9), and properties (3.13), (3.14), for a.e. we obtain
Now, combining (2.3), (3.10), (3.15) and (2.4), (3.11), (3.16), we get
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 . □
Proof of Corollary 2.1 According to Lemma 3.1(iii) and (v), for any , , there exists such that
and thus we get
because the function is decreasing on as follows from Lemma 3.1(iii), (iv). Moreover, in view of assumption (1.7), for a.e. such that we have
where the function z is defined by (2.5). Consequently, it follows from (2.6), (3.20), and (3.21) that
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. □
Proof of Corollary 2.2 Let and the function z be defined by (2.5). For any we put
Since the function is absolutely continuous for every , by using Lemma 3.1(v), we conclude that the function is absolutely continuous for every , as well. Moreover, by virtue of Lemma 3.1(iii), we get
It follows from Lemma 3.1(iii), (iv) that the function is decreasing on and thus we have
Now (3.22) and (3.23) yield
Consequently, for any and , , we get
On the other hand, observe that, in view of assumption (1.7), for a.e. such that we have
Therefore, by virtue of assumption (2.10) and condition (3.23), (3.24) shows that
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 . □
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The research was supported by RVO: 67985840.
The authors declare that they have no competing interests.
AL and JŠ obtained the results in a joint research. Both authors read and approved the final manuscript.