We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

1. Introduction

In this paper we study oscillatory properties of the delay second-order half-linear differential equation

We suppose that , , are continuous functions defined on such that , for large , for all , and . By we denote the conjugate number to the number , that is, .

Under the solution of (1) we understand any differentiable function which does not identically equal zero eventually, such that is differentiable and (1) holds for large .

The solution of (1) is said to be oscillatory if it has infinitely many zeros tending to infinity. Equation (1) is said to be oscillatory if all its solutions are oscillatory. In the opposite case, that is, if there exists an eventually positive solution of (1), (1) is said to be nonoscillatory.

It is well known that the behavior of delay differential equations is very different from the behavior of ordinary differential equations. Among others, the Sturm theory fails and oscillatory solutions may coexist with nonoscillatory solutions.

In certain special cases, it is possible to compare asymptotics of (1) with some other simpler equation. One of the typical objects for this comparison is the first order delay differential equation; see, for example, [13] for results on comparing (1) or its extension in the form of neutral differential equation with the first order delay differential inequality. Another simpler object than (1) suitable for comparison with (1) is the half-linear second-order ordinary differential equation see, for example, [47]. Note that some of these papers deal with a slightly more general equation However, if this more general equation is considered, conditions imposed on the nonlinearity usually state that (3) is a kind of majorant of (1) (in the sense used in the Sturmian theory of ordinary differential equations) and allow to extend the results readily from (1) to (3). An example of such conditions is for some (positive) function and all positive numbers , . Note also that some of the above cited papers deal more generally with neutral differential equations and (or) dynamic equations on time scales.

In this paper we compare (1) with the ordinary half-linear equation of the form (2). To make our paper more readable we restrict our attention to differential equations rather than equations on time scales. An extension of our results to neutral differential equations is provided at the end of this paper.

Let us recall the Riccati technique, which is one of the methods frequently used in oscillation theory of both (1) and (2) (it is easy to see that if , then (1) reduces to (2)). Suppose that (1) is nonoscillatory and let be its eventually positive solution. Then the function satisfies the Riccati type equation

The following lemma plays a crucial role in the qualitative theory of half-linear second order ordinary differential equations.

Lemma 1 (see [8, Theorem 2.2.1]). Denote and . The following statements are equivalent: (i)(2) is nonoscillatory, (ii)there is and a continuously differentiable function such that (iii)there is and a continuously differentiable function such that (iv)there is and a positive function with continuously differentiable such that

As we show below, the assumptions used in the paper ensure that the positive solutions are eventually increasing and concave down. The main step when we compare the ordinary half-linear differential equation and its delay counterpart (1) is to reduce (5) to the Riccati inequality of the form (7). The usual approach on how to remove the term from (5) is the following lemma, originally proved in [9] and then used in many subsequent papers.

Lemma 2. Suppose that is a function defined for some such that , , , and for . Then, for each there exists such that

Note that the proof of Lemma 2 does not exploit the fact that is a solution of (1) and the lemma holds for any positive increasing concave down function. The proof of (9) can be based on the fact that if on and , then is decreasing with respect to on (see [10, Theorem 128]). Thus where . Removing the dependence on may be implemented by using of a constant . The presence of one of the constants or in the estimates (9) and (10) is an important attribute of these estimates. As a consequence, the resulting integral oscillation citeria have to be formulated either with the constant , or as interval-type or Kamenev-type criteria, where the dependence on is usually not disturbing. A typical result looks like the following Theorem A.

Theorem A (see [11, Theorem 2.6]). Equation (1) with is oscillatory if the differential equation is oscillatory for some .

As another particular example of a criterion which suffers from the presence of the constants see [12, Theorem 2.1].

The above mentioned disadvantage has been removed for the linear delay equation under the condition Opluštil and Šremr utilized in recent papers [13, 14] (12) to derive a sharper estimate than the estimate from Lemma 2. Note that imposing (13) on does not yield any restriction in oscillation criteria for (12) since (12) is already known to be nonoscillatory if (13) fails. The same approach has been used for linear dynamic equations on time scales by Erbe, Peterson and Saker in [15].

The aim of this paper is to derive a result analogical to the estimate from [13, 14] and make it available also for delay half-linear differential equation. The nonlinearity of the equation causes, that the method from [13, 14] does not extend to (1) directly and we have to use an indirect approach which originates in the fact that the half-linear extension does not yield (13) as its special case, but includes the term instead of . This estimate suggests a new tool which can be used to improve some oscillation criteria for (1).

2. Preliminaries

The proof of the following statement can be found in [16].

Lemma 3. Let be an eventually positive solution of (1). If , then for large . Moreover, if , then for large .

The following lemma shows that under certain additional conditions we can utilize (1) to derive a sharper version of the estimate from Lemma 2.

Lemma 4. Suppose that (1) is nonoscillatory, and let be a solution of (1). If the conditions hold, then there exists such that

Proof. Conditions (14) and Lemma 3 imply that there exists such that , , for .
We show that for large . Since , it is sufficient to show that (17) holds for some . Suppose, by contradiction, that for all . Solving this inequality we get for , where . Hence, there exists such that Since is a solution of (1), we have Integrating the last inequality from to we obtain and from the fact that is positive we get the following finite upper bound for the integral of : for . However the condition (15) ensures that the left hand side of this inequality is unbounded. This contradiction proves (17) for large .
Hence there exists such that (17) holds for . This inequality together with the computation shows that the function is decreasing on . This fact and the fact that reveal that there exists such that which is equivalent to (16).

3. Oscillation of Delay Differential Equation

Theorem 5. Suppose that conditions (14) and (15) hold. If the ordinary differential equation is oscillatory, then (1) is also oscillatory.

Proof. Suppose, by contradiction, that (1) is nonoscillatory and (24) is oscillatory. Let be an eventually positive solution of (1). Using Lemma 4 we see that satisfies the inequality and hence, using equivalence between parts (i) and (iv) of Lemma 1, we see that (24) is nonoscillatory which contradicts our assumptions.

Remark 6. The oscillation criterion from Theorem 5 is general in the sense that the oscillation is given in terms of oscillation of a certain half-linear differential equation rather than in terms of explicit conditions on the coefficients of the equation. Most of the related papers continue the proofs by utilizing techniques used in the theory of half-linear ordinary differential equations (often simply copy of the proofs of known oscillation citeria) to reach effective conditions for oscillation. However, we feel our approach as an advantage, since it allows to utilize arbitrary from large family of oscillation criteria for half-linear oscillation equations to detect oscillation of delay equation. See also [8] for a comprehensive survey on oscillation criteria known up to 2005.

Remark 7. Note that a similar result like Theorem 5 can be proved also without Lemma 4 and using Lemma 2 instead. This results in a comparison of (1) with the equation where is a real parameter which satisfies . (Note that for we get Theorem A.) Equation (24) can be viewed in a certain sense as a continuation of (26) with respect to to the border value . Note that the problems related to oscillation of equation of the type (26) and dependence of oscillatory properties on the parameter are referred to as conditional oscillation. In general, oscillation of (26) implies oscillation of (24), but the opposite implication need not be true in general, see the paper [17] which (based on the results from [18]) suggests a method on how to construct a pair of equations of the type (24) and (26) with (24) oscillatory and (26) nonoscillatory.

Remark 8. Theorem 5 extends Theorem A, where oscillation of (1) is deduced from oscillation of (26). The following example shows that this extension is nonempty.

Example 9. Consider the perturbed Euler type half-linear delay differential equation where is real constant. According to Theorem 5, (27) is oscillatory if is oscillatory. Following [8, Theorem 5.2.2] (see also [19]) we treat (28) as a perturbation of the nonoscillatory equation with principal solution . A simple computation shows hence (28) is oscillatory by [8, Theorem 5.2.2]. Consequently, (27) is oscillatory for every .
We claim that the oscillation of (27) cannot be proved with Theorem A. Really, in our example (11) becomes where . This equation is nonoscillatory for every by Kneser type nonoscillation criterion [8, Theorem 1.4.5], and thus Theorem A fails to apply.

4. Oscillation of Neutral Differential Equation

In this section we use a slight modification of the estimates from the first part of the paper to derive similar results for the second order neutral differential equation where , , , , , .

Similarly as for (1), if is a solution of (32) on such that is positive on , then the function satisfies the Riccati type equation on .

Similarly like for the delay equation, the positive solution is increasing and concave down. More precisely, the following lemma holds. For linear version of this lemma see [2, Lemma 1] and for see [1, Lemma 2.1].

Lemma 10. Let be an eventually nonoscillatory solution of (32). If , then the corresponding function satisfies eventually. Moreover, if , then for large .

Proof. The proof is essentially the same as the proof of [1, Lemma 2.1]. We just relax the restriction on .
Without loss of generality we can suppose that is eventually positive solution of (32). There exists such that , and are positive on and for . Hence, is decreasing and either for large .
Suppose that there exists such that for . There exists a positive constant such that for . Integrating this inequality over the interval we get Letting we have a negative upper bound for the function and large . However, the positivity of both and implies positivity of . This contradiction proves that and eventually.
If , then and hence .

Lemma 11. Suppose that is an eventually positive nonoscillatory solution of (32) and is the corresponding function defined by (33). If , then eventually.

Proof. According to Lemma 10 there exists such that holds for . From here and from the fact that is increasing and is delay we have From here we conclude and hence (41) holds for .

The following lemma is an alternative to Lemma 4 for neutral differential equations.

Lemma 12. Suppose that (32) is nonoscillatory and is an eventually positive solution of (32). If and (14) holds, then the function is decreasing eventually.

Proof. Similarly like in Lemma 4 we find the derivative It is sufficient to show that eventually. Lemma 10 implies that there exists such that on . This shows that is decreasing on . As a consequence, if for some , then on .
Suppose by contradiction that there exists such that on . Solving this inequality we get Now integrating (32) from to and using (41) we get Taking sufficiently large and using (45) we obtain a negative upper bound for a positive function . This contradiction proves the lemma.

Now we can formulate the comparison theorem which relates neutral differential equations to ordinary second-order half-linear differential equations.

Theorem 13. Suppose that (45) and (14) hold. If the ordinary half-linear differential equation is oscillatory, then (32) is also oscillatory.

Proof. Having proved important estimates in the preceding two lemmas, the proof of the theorem is a modification of the proof of Theorem 5. If is an eventually positive solution of (32), then the function defined by satisfies (34). Using Lemmas 11 and 12 we see that Hence (49) is nonoscillatory by Lemma 1.

Remark 14. A version of Theorem 13 has been used implicitly in the proof of [20, Theorem 2.2] for dynamic equations. A closer estimation of the proof shows that one of the important steps is an application of inequality which in the continuous case reads as (10). However, Lemma 12 allows the estimate which appears to be sharper, since and the annoying dependence of the left-hand side on usually necessitates to replace it by a constant which may appear in the resulting oscillation criterion.


This research is supported by the Grant P201/10/1032 of the Czech Science Foundation.