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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 638271, 15 pages
On Constants in Nonoscillation Criteria for Half-Linear Differential Equations
Department of Mathematics, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic
Received 15 June 2011; Accepted 22 August 2011
Academic Editor: Victor M. Perez Garcia
Copyright © 2011 Simona Fišnarová and Robert Mařík. 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.
We study the half-linear differential equation , where , . Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.
In this paper we consider the equation where , , , for some . Under a solution of this equation, we understand every continuously differentiable function such that is differentiable and (1.1) holds on . This equation is called half-linear, since a constant multiple of any solution is also a solution of (1.1).
If , then (1.1) reduces to the linear equation
For detailed discussion related to general theory as well as applications, we refer to . According to , the classical linear Sturmian comparison theory extends to (1.1) and hence, if a solution has infinitely many zeros in a neighborhood of infinity, then the same is true for every solution. In this case, we say that (1.1) is oscillatory. In the opposite case, we say that (1.1) is nonoscillatory, as the following definition shows. Note that due to homogeneity of the set of all solutions, we can restrict ourselves to solutions which are positive in a neighborhood of infinity.
Došlý and Řezníčková  viewed (1.1) as a perturbation of another nonoscillatory half-linear differential equation and proved the following result. Note that denotes the conjugate number to in Theorem A and in the whole paper, that is, holds.
Theorem A is sharp in the sense that a convenient choice of the function allows to prove explicit sharp nonoscillation criteria. As a particular example, choosing and , Došlý and Řezníčková derived the following result for the perturbed Euler differential equation
A variant of Theorem A without convergent integral of is the following.
Theorem C (see [3, Theorem 2]). Let be a positive function such that for large , say , Suppose that (1.5) and (1.6) with replaced by hold. If the integral is convergent, and then (1.1) is nonoscillatory.
If we take and , then Theorem C can be applied to the perturbed Riemann-Weber equation and we obtain the following statement.
The aim of this paper is to improve Theorems A, B, C, and D and show that the constants in the inequalities involving and can be replaced by a certain one-parametric expression. Roughly speaking, these theorems claim that the nonoscillation is preserved if the perturbation which is measured by the expressions is bounded in a strip between and for large . We show that there is a possibility to shift this strip down. In other words, we show that if the inequality involving limes inferior is not satisfied, it can be relaxed provided the condition involving limes superior is strengthened properly. Together with these results, we prove also similar results of a different type, where (1.1) is viewed as a standalone equation and not as a perturbation of another equation (Theorems 3.1 and 3.2).
2. Preliminary Results
The main tool used in the paper is the method based on the Riccati equation which can be obtained from (1.1) by substitution . Our results are obtained from the following necessary and sufficient condition for nonoscillation of (1.1) which can be found, for example, in [1, Theorem 2.2.1].
Lemma 2.1. Equation (1.1) is nonoscillatory if and only if there exists a differentiable function which satisfies the Riccati type inequality for large .
Our results heavily depend on the following relationship between the Riccati type differential operator and the so-called modified Riccati operator (the operator on the right-hand side of (2.3)).
Lemma 2.2 ([4, Lemma 2.2]). Let and be differentiable functions and , then one has the identity where .
3. Main Results
In contrast to Theorems A and C in the first pair of theorems, we do not consider (1.1) as a perturbation of a nonoscillatory equation, but we consider this equation as a standalone problem.
Theorem 3.1. Let be a function such that and , both for large . Suppose that the following conditions hold: If for some , then (1.1) is nonoscillatory.
Proof. Denote , and
Consider the function . This function satisfies and . Hence, by the Taylor formula, the function can be approximated by in a neighborhood of .
Conditions of the theorem imply that hence, for every , there exists such that holds for . At the same time can be taken so small (will be specified later how) and so large that for we have Consequently, Let . By Lemma 2.2, we have Consider the function in the numerator of the last fraction We have and by a direct computation and hence . This means that can be taken so small that Combining (3.9) and (3.12) we have for and (1.1) is nonoscillatory by Lemma 2.1.
Theorem 3.2. Let be a function such that and , both for large . Suppose that If for some , then (1.1) is nonoscillatory.
Proof. With we take and the proof is the same as the proof of Theorem 3.1.
Proof. Denote , as in Theorem 3.1. Further Similarly as in the proof of Theorem 3.1 we get (3.4), and for sufficiently small , there exists such that holds for . Using this estimate and Lemma 2.2, we see that the function satisfies for . From (3.17) it follows that for large . Using this and (3.12) we get for large . Hence (1.1) is nonoscillatory by Lemma 2.1.
Corollary 3.4. If there exists such that then (1.8) is nonoscillatory.
Theorem 3.5. Let be a function such that and , both for large . Suppose that the following conditions hold: If for some , then (1.1) is nonoscillatory.
Corollary 3.6. If converges and there exists such that then (1.13) is nonoscillatory.
In the following theorem we employ the technique used in previous results directly for Riccati operator from (2.1) rather than for modified Riccati operator from (2.3). This method yields a result which is (as far as we know) new even in the linear case (Theorem 3.7) and offers also a simple and alternative proof of known results (see Remark 6).
Theorem 3.7. Let the following conditions hold: and for some Then (1.1) is nonoscillatory.
Proof. It follows from the fact that is for explicit formula for the curve given parametrically by , for . This curve is increasing for and (3.33) means that the point is below this curve. The same is ensured by inequalities (3.37).
4. Concluding Remarks and Comments
Remark 1. If we put in Theorems 3.3 and 3.5, we get Theorems A and C. The constant from the condition with limes superior is maximal with this choice. As far as we know, Theorems 3.3 and 3.5 are new if and Theorems 3.1 and 3.2 are new for every . Similarly, if we put in Theorem 3.7, then we get [1, Theorem 3.1.5].
Remark 2. If , then both are decreasing functions of the variable and thus the bounds for and which guarantee nonoscillation in Theorems 3.3 and 3.5 also decrease. Closer investigation shows that the bound for limes inferior decreases faster and thus the maximal allowed difference between limes superior and inferior is allowed to be bigger if both are small. A similar remark applies also to the results from Theorems 3.1 and 3.2.
Remark 3. The nonoscillation criteria in the previous theorems are written in the form where the definition of varies for each particular theorem. Note that if inequalities (4.1) hold for some which satisfies , then they hold also for . In view of this fact it is reasonable to suppose .
Remark 4. Denote It is easy to show that the parametric curve , , , is an increasing function with nonparametric equation . Since (4.1) expresses the fact that the point is below this curve, inequalities (4.1) are satisfied if
Remark 5. Let us compare our results with known results on a particular example of perturbed Euler equation (1.8). Let us denote
and rewrite the conditions from Corollary 3.4 into
It is easy to see that every equation in the form (1.8) can be associated with some point in the plane and . On the other hand, for every point in this plane which satisfies , we can construct an equation in the form (1.8) which is associated with this point.
As mentioned before, Došlý and Řezníčková  proved that (1.8) is oscillatory if and nonoscillatory if and . This gives two regions in plane with resolved oscillation properties of (1.8): the unbounded region of oscillation has the form of the angle with vertex , rays and , open up and the bounded region of nonoscillation has the form of triangle with vertexes , , and . Corollary 3.4 allows us to extend the region of nonoscillation by the unbounded region which is between the line and the curve given for parametrically by , or, equivalently, given by for . All these regions are shown on Figure 1.
In particular, if , then and (1.8) is oscillatory if and nonoscillatory if . This observation has been made already by Elbert and Schneider in  and has a close connection with the so-called conditional oscillation, see  for more details related to conditional oscillation.
Remark 6. If the integral is convergent and if we use the method from Theorem 3.7 and Corollary 3.8 with , we get the second part of [1, Theorem 3.3.6], which is originally due for to Kandelaki et al. ([7, Theorem 1.6]).
Remark 8. If we use the additional condition then the conclusion related to that of Theorems 3.2 and 3.5 can be derived from known results. Really, denote and , where is a positive function such that and suppose that and is convergent. Under these conditions, an alternative version of Theorem 3.2 can be derived using the so-called linearization technique. This technique is based on comparison of the (non)oscillation of (1.1) with that of a certain linear equation. The relation between these equations is hidden in identity (2.3) which (after the quadratization of the last term on the right-hand side) relates the associated Riccati operators. More precisely, nonoscillation of (1.1) is implied by nonoscillation of the linear equation where is arbitrary. Applying the linear version () of the criteria discussed in Remark 6 to the above linear equation, we obtain the conditions for limes inferior and superior from Theorem 3.2. Similarly if , where is a positive solution of (1.3), then we get the conditions for limes inferior and superior from Theorem 3.5. Note that if , then and are the coefficients of the equation which results from (1.2) (the special case of (1.1) for ) upon the transformation . We refer to [8–10] for results concerning the linearization technique and for the half-linear Hille-Nehari type criteria derived using this technique from the classical criteria mentioned in Remark 7.
Research supported by the Grant P201/10/1032 of the Czech Science Foundation.
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