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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 417869, 20 pages
http://dx.doi.org/10.1155/2010/417869
Research Article

On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
2Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 66237 Brno, Czech Republic

Received 1 July 2010; Accepted 26 August 2010

Academic Editor: Allan C. Peterson

Copyright © 2010 Jaromír Baštinec et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

New nonoscillation and oscillation criteria are derived for scalar delay differential equations and and in the critical case including equations with several unbounded delays, without the usual assumption that the parameters and of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.

1. Introduction

It is well known that a scalar linear equation with delay has a nonoscillatory solution as . This means that there exists an eventually positive solution. The coefficient is called critical with the following meaning: for any , all solutions of the equation are oscillatory while, for , there exists an eventually positive solution.

In [1] the third author considered the equation where , (throughout this paper we assume that is sufficiently large), is a continuous function and the delay is a constant. For the critical case, he obtained the following result.

Theorem 1.1. Let an integer exists such that if where Then there exists an eventually positive solution of (1.3).
Let an integer and , exist such that if . Then all solutions of (1.3) oscillate.

In this theorem for , , , where , , and .

Further results on the critical case for (1.3) can be found in [26]. Theorem 1.1 was generalized in [7] for the following equation with a variable delay where , and , are continuous functions.

The main results of this paper include the following.

Theorem 1.2 (see [7]). Let if . Let an integer exists such that for where If moreover then there exists an eventually positive solution of (1.6) for .

Theorem 1.3 (see [7]). Let one assume that if and as . Then there exists an eventually positive solution of (1.6).

In this paper we obtain new nonoscillation and oscillation sufficient conditions for (1.6) in the critical case, independent of Theorems 1.11.3. We also obtain nonoscillation and oscillation conditions for equations with several delays, including equations with unbounded delays. To the best of our knowledge, we are the first to investigate the critical case of such equations.

2. Preliminaries

We consider a scalar delay differential equation subject to the following conditions:

, are Lebesgue measurable functions essentially bounded in each finite interval with .

are Lebesgue measurable functions, , , and , .

Along with (2.1) we consider an initial value problem We also assume that the following hypothesis holds:

is a Lebesgue measurable function essentially bounded in each finite interval with , and is a Borel measurable bounded function.

Definition 2.1. A function absolutely continuous on each interval with , is called a solution of problem (2.2), (2.3) if it satisfies (2.2) for almost all and equalities (2.3) for .

Lemma 2.2 (see [8]). Let hold. Then there exists exactly one solution of problem (2.2), (2.3).

Definition 2.3. One will say that (2.1) has a nonoscillatory solution if, for some problem (2.2), (2.3) with , , there exists an eventually positive solution. Otherwise, all solutions of (2.1) oscillate.

To formulate a comparison result, consider the following equation: Let holds with and holds with , .

Lemma 2.4 (see [9, 10]). Let (2.4) have a nonoscillatory solution. If then (2.1) has a nonoscillatory solution as well.
Suppose that all solutions of (2.4) are oscillatory. If then all solutions of (2.1) are oscillatory as well.

Lemma 2.5 (see [9, 10]). Let exist such that Then there exists a positive solution of (2.1) for .

Lemma 2.6 (see [9, 10]). A nonoscillatory solution of (2.1) exists if and only if, for some there exists a nonnegative locally integrable function , such that

3. Differential Equation with a Single Delay

Equation (1.6) is a special case of (2.1) for , , and .

Our first result is a simple consequence of Theorem 1.1 and Lemma 2.4. Theorem 1.1 was obtained under the assumption that and are continuous functions. But the proof of this theorem remains valid even for more general conditions

Theorem 3.1. Let , for , and let condition (a) of Theorem 1.1 holds. Then (1.6) has a nonoscillatory solution.
Let for , and let condition (b) of Theorem 1.1 holds. Then all solutions of (1.6) oscillate.

Proof. We set , . Obviously for . By Theorem 1.1, (1.3) has a nonoscillatory solution. By Lemma 2.4 (with , ), (1.6) also has a nonoscillatory solution.
The proof of this part is much the same (using Theorem 1.1 and Lemma 2.4) as the proof of part

Theorems 1.2 and 1.3 can be applied to equations with one unbounded delay. Here, we want to give some new nonoscillation and oscillation conditions for equations with one delay, also including equations with unbounded delays. We remove some conditions of Theorems 1.2 and 1.3, in particular conditions (1.8) and (1.9). Moreover, the delay function used in Theorems 1.2 and 1.3 as a coefficient appears in our conditions in both integral and nonintegral expressions.

For every integer , and we define where

Theorem 3.2. Let for sufficiently large and : a.e. be a locally integrable function, and let there exists such that , .
If there exists a such that and, for a fixed integer, then there exists an eventually positive solution of (1.6).
If there exists a such that and, for a fixed integer and , , if then all solutions of (1.6) oscillate.

Proof. (a) For the proof we will use a transformation applied to delay equations for the first time in [11]. Consider (1.6) for . Denote Since, by (3.2), , is a strictly increasing function and, hence, there exists an inverse function . Denote Since , we have (by (3.9) and (3.2)). From (3.9) we also have Substituting in (1.6), we have and (using (3.9)) Hence, (1.6) takes the form where . Equality implies that the oscillation properties of (1.6) and (3.12) are equivalent. We have (by (3.2), (3.10), (3.8), and (3.4)) Hence Consider an equation where is defined similar to by (1.4), where is replaced by and by , that is, By Theorem 1.1, (3.16) has a positive solution. Equation (3.12) is of type (2.1) with Now we use comparison of Lemma 2.4 where (2.4) is replaced by (3.16), that is, Since, by (3.5) and (3.1), and, by (3.15), equation (3.12) and, due to (3.13), equation (1.6) also has a positive (i.e., nonoscillatory) solution. Part (b) can be proved in much the same way.

Now we want to compare Theorem 3.2 and Theorems 1.2 and 1.3 for equations with unbounded delays. Note that Theorem 1.1 is not valid for such equations and Theorem 1.2 contains additional restrictions (1.8), (1.9). Theorem 1.3 is not explicitly valid for the critical case.

Example 3.3. Let (1.6) be of the form where . Here We set In accordance with Theorem 3.2 (case (a) where ), (3.22) has a nonoscillatory solution if Since (by Theorem 3.2, case () with ) all solutions of (3.22) oscillate if we conclude that the value is a critical value for the nonoscillation of (3.22).
The above statement is corroborated by Lemma 2.5 since, for , , , and , and for and (2.7) turns into an equality for .

To apply Theorem 1.2, we verify condition (1.8). But, unfortunately, for (3.22) we have Thus, this theorem is not applicable to (3.22).

To compare Theorem 1.3 with Theorem 3.2 we set (where is defined by (3.27)) in (3.22). By Theorem 3.2, (3.22) has a nonoscillatory solution. By Theorem 1.3, (3.22) has a nonoscillatory solution if (we set in (1.10)) But in our case and Theorem 1.3 fails for this equation.

4. Differential Equation with Several Delays

We start with the following question: for what functions and delay the equation can have a nonoscillatory solution? It is easy to see that should be vanishing.

Theorem 4.1. Let and . Then all solutions of (4.1) are oscillatory.

Proof. Consider first the equation Suppose that (4.2) has a nonoscillatory solution. We set Since , by Lemma 2.4, the equation has a nonoscillatory solution. After the substitution , (4.4) takes a form Since , all solutions of (4.5) are oscillatory by Theorem 1.1 (b). This is a contradiction. Hence, all solutions of (4.2) are oscillatory.
For sufficiently large , we have , . We set Now, Lemma 2.4 implies the statement of the theorem.

We consider general equation (2.1) with delays subject to restrictions (a1), (a2).

Theorem 4.2. (a) Let an integer and exist such that, for all sufficiently large , inequalities where is defined by (1.4), are valid. Then there exists an eventually positive solution of (2.1).
(b) Let an integer , and exist such that, for all sufficiently large , inequalities where is defined by (1.4), are valid. Then all solutions of (2.1) oscillate.

Proof. Let the assumptions of case (a) be valid. Then, by Theorem 1.1, the equation has a nonoscillatory solution. By Lemma 2.6 (with , , and ), there exist a and a locally integrable function such that We have Hence Now using Lemma 2.6 again, we conclude that there exists an eventually positive solution of (2.1).
Let the assumptions of case (b) be valid. Suppose, on the contrary, that (2.1) has a nonoscillatory solution. Using calculations similar to those of the previous part of the proof, one can deduce that (by Lemma 2.6) there exist a and a locally integrable function such that Hence (using Lemma 2.6 again), an equation should have a nonoscillatory solution. Due to being arbitrary, we easily get a contradiction to statement (b) of Theorem 1.1.

Example 4.3. We show that equation of type (2.1) has a nonoscillatory solution for any positive . Indeed, set , , , , , and . Then (4.8), where , holds and Part (a) of Theorem 4.2 is valid.

Now we consider (2.1) with unbounded delays.

Theorem 4.4. Let be sufficiently large, for , a.e., let locally integrable function, and let there exists such that if .
(a) If there exists a such that and, for a fixed integer , where is defined by (3.1), (3.2), then there exists an eventually positive solution of (2.1).
(b) If there exists a such that and, for a fixed integer and , , where is defined by (3.1), (3.2), then all solutions of (2.1) oscillate.

Proof. Let the assumptions of case (a) be valid. By Theorem 3.2, the equation has a nonoscillatory solution. Equation (4.24) is of a form of (2.4) with , , . This means that we see (4.24) as an equation with delayed terms.
Compare (4.24) with (2.1). We have and, due to (4.19), , . By Lemma 2.4, (2.1) has a nonoscillatory solution.
The proof of part (b) can be carried out in a way similar to that of the proof of part (a) and, therefore, it is omitted.

Example 4.5. Consider the equation of the type of (2.1): First let . We set , , and Moreover, we put By (3.1), All conditions of Theorem 4.4 part (a) hold, hence (4.25) has a nonoscillatory solution.

Similarly, one can show (by Theorem 4.4 part (b)) that, for , all solutions of (4.25) are oscillatory.

5. Differential Equation with Two Delays

In [12] the authors consider a differential equation with two delays where , , and , are positive constants. Let In accordance with [12] we say that (5.1) is in a critical state if there exists such that and, for any , , we have

Theorem 5.1 (see [12]). Let (5.1) be in a critical case, , and where . If , then all solutions of (5.1) oscillate.

Theorem 5.2 (see [12]). Let where , . If , then all solutions of (5.1) oscillate.

The aim of the following theorems is to obtain nonoscillation conditions for (5.1) in the above-mentioned critical case. This will complete the oscillation results given by Theorems 5.1 and 5.2. Note that, in Theorems 5.3 and 5.4 below, delays and being not defined by (5.2) are arbitrary and subject only to the restrictions indicated.

Theorem 5.3. Let and for , where , and let there exists a such that Then (5.1) has a nonoscillatory solution.

Proof. There exist and such that, owing to (5.8) and (5.10), if . By Lemma 2.4, with , in (2.4), the existence of a nonoscillatory solution of the equation implies the existence of a nonoscillatory solution of (5.1).
By Lemma 2.6, for the existence of a positive solution of (5.13), it is sufficient to find a nonnegative solution of the inequality where . Substituting for in inequality (5.14), we have Due to (5.9) and (5.11), we conclude that the last inequality holds, and, consequently, is a solution of inequality (5.14). By Lemma 2.6, (5.13) has a nonoscillatory solution. Hence, (5.1) has a nonoscillatory solution, too.

Theorem 5.4. Let , for , where If then (5.1) has a nonoscillatory solution.

Proof. There exist and such that, owing to (5.16) and (5.17), if . By Lemma 2.4, with , in (2.4), the existence of a nonoscillatory solution of the equation implies the existence of a nonoscillatory solution of (5.1). By Lemma 2.6, it is sufficient to find a nonnegative solution of the inequality where . Put , in inequality (5.21). We have Due to (5.18), we conclude that the last inequality holds, and, consequently, is a solution of inequality (5.21). Let for . By Lemma 2.6, (5.20) has a nonoscillatory solution. Hence, (5.1) also has a nonoscillatory solution.

Example 5.5. Consider (5.1) with and with , defined by (5.2), that is, where . Since then, by Theorem 5.2 (with and ), all solutions of (5.24) oscillate if Since by Theorem 5.4 (with and ), (5.24) has a nonoscillatory solution if

5.1. Generalization for Equations with Several Delays

It is easy to generalize Theorem 5.3 for a general equation (2.1) with several delays. Denote where . We omit the proof of this generalization since it is similar to that of Theorem 5.3.

Theorem 5.6. Let , , be positive constants such that for , where and let there exists a such that Then (2.1) has a nonoscillatory solution.

The following statement generalizes Theorem 5.4. We will formulate this result for (2.4).

Theorem 5.7. Let be a set of indices such that where and . Let If, for functions , , , all assumptions of Theorem 5.4 are true, then (2.4) has a nonoscillatory solution.

The proof of Theorem 5.7 is omitted as it can be done easily using Lemma 2.4 and Theorem 5.4.

6. Concluding Remarks

In conclusion we note that there exist numerous results on nonoscillation for various classes of delay differential equations in a noncritical case. We refer, for example, to monographs [6, 9, 13, 14], recent papers [1522], and references therein. Some of the books and papers mentioned discuss the critical case from various points of view different from our approach, and we mentioned them above. In the paper we investigate the critical case for delayed differential equations. It will be interesting as a motivation for further investigation along these lines to consider cases, critical is a sense to other classes of equations, in particular, for integrodifferential equations, differential equations with distributed delay, or differential equations of a neutral type. Finally, for nonoscillation results for difference equations we refer to [2328].

Acknowledgments

J. Baštinec was supported by the Grant 201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30529 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. L. Berezansky was partially supported by grant 25/5 “Systematic support of international academic staff at Faculty of Electrical Engineering and Communication, Brno University of Technology” (Ministry of Education, Youth and Sports of the Czech Republic) and by the Grant 201/10/1032 of the Czech Grant Agency (Prague). J. Diblík was supported by the Grant 201/08/0469 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. Z. Šmarda was supported by the Council of Czech Government Grant MSM 00216 30503 and MSM 00216 30529, and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology.

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