Oscillation of Half-Linear Differential Equations with Delay

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.

Under the solution of (1) we understand any differentiable function () which does not identically equal zero eventually, such that ()Φ(  ()) 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, [1][2][3] 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, [4][5][6][7].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 , V.Note also that some of the above cited papers deal more generally with neutral differential equations and (or) dynamic equations on time scales.

Abstract and Applied Analysis
In this paper we compare (1) with the ordinary halflinear 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 () = ()Φ(  ())/Φ(()) satisfies the Riccati type equation The following lemma plays a crucial role in the qualitative theory of half-linear second order ordinary differential equations.
The following statements are equivalent: (iii) there is  ∈ R and a continuously differentiable function  : [, ∞) → R such that (iv) there is  ∈ R and a positive function 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 Φ((())/()) 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  > 0 such that () ∈  2 [, ∞), () > 0,   () > 0, and   () ≤ 0 for  ≥ .Then, for each  ∈ (0, 1) 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   () ≤ 0 on [, ∞) and () ≥ 0, 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  ∈ (0, 1).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  ∈ (0, 1), 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.
As another particular example of a criterion which suffers from the presence of the constants   ∈ (0, 1) see [12,Theorem 2.1].
The above mentioned disadvantage has been removed for the linear delay equation   () +  ()  ( ()) = 0 (12) 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 halflinear 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).
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 () > 0 be a solution of (1).If the conditions hold, then there exists  ∈ R such that Proof.Conditions ( 14) and Lemma 3 imply that there exists  0 such that () > 0,   () > 0,   () ≤ 0 for  ≥  0 .We show that for large .Since (  () − ())  =   () ≤ 0, it is sufficient to show that (17) holds for some  1 ≥  0 .Suppose, by contradiction, that   () − () > 0 for all  ≥  0 .Solving this inequality we get () >  for  ≥  0 , where  = ( 0 )/ 0 > 0. Hence, there exists  2 ≥  0 such that Since  is a solution of (1), we have Integrating the last inequality from  2 to  we obtain and from the fact that ()Φ(  ()) is positive we get the following finite upper bound for the integral of ()(()) −1 : for  ≥  2 .However the condition (15) ensures that the left hand side of this inequality is unbounded.This contradiction proves (17) for large .
Hence there exists  1 ≥  0 such that (17) holds for  ≥  1 .This inequality together with the computation shows that the function ()/ is decreasing on ( 1 , ∞).This fact and the fact that () ≤  reveal that there exists  ≥  1 such that which is equivalent to (16).

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  ∈ (0, 1).(Note that for  ≡ 1 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  = 1.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  > 0 is real constant.According to Theorem where  ∈ (0, 1).This equation is nonoscillatory for every  > 0 by Kneser type nonoscillation criterion [8, Theorem 1.4.5], and thus Theorem A fails to apply.
and the annoying dependence of the left-hand side on  usually necessitates to replace it by a constant  < 1 which may appear in the resulting oscillation criterion.
(11)rem 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