A note on oscillation of second-order delay differential equations

Applied Mathematics Letters 69 (2017) 126–132

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Applied Mathematics Letters www.elsevier.com/locate/aml

A note on oscillation of second-order delay differential equations Jozef Džurina, Irena Jadlovská * Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, B. Němcovej 32, 042 00 Košice, Slovakia

article

abstract

info

Article history: Received 5 December 2016 Received in revised form 6 February 2017 Accepted 6 February 2017 Available online 21 February 2017

The purpose of this paper is to study the second-order half-linear delay differential equation

(

(

)α ) ′

r(t) y ′ (t)

+ q(t)y α (τ (t)) = 0

∫∞

under the condition r−1/α (t)dt < ∞. Contrary to most existing results, oscillation of the studied equation is attained via only one condition. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Half-linear differential equation Delay Second-order Oscillation

1. Introduction In this paper, we are concerned with the oscillatory behavior of the second-order half-linear delay differential equation of the form (

r(t)(y ′ (t))

α )′

+ q(t)y α (τ (t)) = 0,

t ≥ t0 ,

(1.1)

where α > 0 is a quotient of odd positive integers, r, τ ∈ C 1 ([t0 , ∞), (0, ∞)) and q ∈ C([t0 , ∞), [0, ∞)). We also suppose that, for all t ≥ t0 , τ (t) ≤ t, τ ′ (t) ≥ 0, limt→∞ τ (t) = ∞, and q does not vanish identically on any half-line of the form [t∗ , ∞). Under the solution of Eq. (1.1) we mean a function y ∈ C([ta , ∞), R) with ta = τ (tb ), for some tb ≥ t0 , α which has the property r(y ′ ) ∈ C 1 ([ta , ∞), R) and satisfies (1.1) on [tb , ∞). We consider only those solutions of (1.1) which exist on some half-line [tb , ∞) and satisfy the condition sup{|x(t)| : tc ≤ t < ∞} > 0 for any tc ≥ tb . As is customary, a solution y(t) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate. * Corresponding author. E-mail addresses: [email protected] (J. Džurina), [email protected] (I. Jadlovská). http://dx.doi.org/10.1016/j.aml.2017.02.003 0893-9659/© 2017 Elsevier Ltd. All rights reserved.

J. Džurina, I. Jadlovská / Applied Mathematics Letters 69 (2017) 126–132

Following Trench [1], we shall say that Eq. (1.1) is in canonical form if ∫ t R(t) := r−1/α (s)ds → ∞ as t → ∞.

127

(1.2)

t0

Conversely, we say that (1.1) is in non-canonical form if ∫ ∞ π(t) := r−1/α (s)ds < ∞.

(1.3)

t

There is a significant difference in the structure of nonoscillatory (say positive) solutions between canonical and non-canonical equations. It is well known that the first derivative of any positive solution y of (1.1) is of one sign eventually, while (1.2) ensures that this solution is increasing eventually. Most often, Eq. (1.1) has been studied exactly in canonical form, see, e.g., [2–4] and the references therein. On the other hand, when investigating non-canonical equations, both sign possibilities of the first derivative of any positive solution y have to be treated. A common approach in the literature (see [5–12]) for investigation of such equations consists in extending known results for canonical ones. The objective of this paper is to study oscillatory and asymptotic properties of (1.1) in non-canonical form. Thus, in the sequel and without further mentioning, it will be always assumed that (1.3) holds. In what follows, we briefly review several important oscillation results established for second-order non-canonical equations which can be seen as a motivation for this paper. Theorem A ([11, Theorem 3.1]). Assume that ) )α+1 ( ∫ ∞( τ ′ (t) α α dt = ∞ R (τ (t))q(t) − α+1 R(τ (t))r1/α (τ (t)) and there exists a continuously differential function ρ(t) such that ρ(t) > 0, ρ′ (t) ≥ 0 for t ≥ t0 , ∫ ∞ ρ(t)q(t)dt = ∞

(1.4)

(1.5)

and ∫

∞

(

1 r(t)ρ(t)

∫

t

)1/α dt = ∞. ρ(s)q(s)ds

(1.6)

Then every solution y(t) of (1.1) oscillates or limt→∞ y(t) = 0. Note that (1.4) in Theorem A eliminates existence of positive increasing solutions of (1.1), while conditions (1.5)–(1.6) ensure that any positive decreasing solution converges to zero in the neighborhood of infinity. Recently, Maˇr´ık [9] revised Theorem A and provided its simplified version. Theorem B ([9, Theorem 2]). With no lack of generality we can put ρ ≡ 1 in Theorem A and the pair of conditions (1.5) and (1.6) can be safely and with no lack of generality replaced by one condition ∫

∞

(

1 r(t)

∫

t

)1/α q(s)ds dt = ∞.

(1.7)

Erbe et al. [6,7], Hassan [8], and Saker [10] independently obtained the analogue of Theorem B in such sense that (1.4) has been replaced by another condition resulting from the use of different techniques. In [6], authors also presented the following criterion which removes the possibility that any positive solution of (1.1) is decreasing.

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Theorem C ([6, Lemma 2.2]). Assume that )1/α ∫ t ∫ ∞( 1 q(s)π α (s)ds dt = ∞ r(t)

(1.8)

and (1.1) has a positive solution y(t) on [t0 , ∞). Then there exists t1 ≥ t0 such that y ′ (t) > 0 on [t1 , ∞). Ye and Xu [12] obtained a sufficient condition for nonexistence of decreasing solution of (1.1) which is different to that in Theorem C. Theorem D ([12, Theorem 2.4]). Assume that (1.4) and ∫ ∞ q(s)π α+1 (s)ds = ∞

(1.9)

hold. Then (1.1) is oscillatory. It is well-known that the second-order Euler differential equation ( 2 ′ )′ t y (t) + q0 y(t) = 0, t ≥ 1, q0 > 0,

(1.10)

is oscillatory if q0 > 1/4. However, (1.8) nor (1.9) cannot be applied to (1.10), since ∫ ∞ ln t dt < ∞. t2 Using the generalized Riccati substitution, Agarwal et al. [5] recently proved oscillation of (1.1) via two independent conditions of limsup type which are applicable to (1.10). Moreover, this result is sharp for (1.10). Theorem E ([5, Theorem 2.2]). Assume that α ≥ 1 and there exist functions ρ, δ ∈ C 1 ([t0 , ∞)(0, ∞)) such that ) )α+1 ( ′ ∫ t ( (ρ (s))+ r(τ (s)) lim sup ds = ∞ (1.11) ρ(s)q(s) − (α + 1)α+1 ρα (s)(τ ′ (s))α t→∞ t0 and ∫

t

lim sup t→∞

t0

( )α+1 ) δ(s)r(s) (φ(s))+ ψ(s) − ds = ∞, (α + 1)α+1

(

(1.12)

where (

1−α ψ(t) := δ(t) q(t) + 1/α r (t)π α+1 (t)

) ,

φ(t) :=

δ ′ (t) 1+α + 1/α . δ(t) r (t)π(t)

Then (1.1) is oscillatory. Contrarily to above-mentioned existing results, the main aim of this paper is to ensure oscillation of (1.1) via only one condition. The method is twofold: first, we show by careful observation that condition (1.4) (or any of its analogue) in Theorems A–D is redundant. Also, we provide a simple refinement of Theorem D to be applicable on equations such as (1.10). Second, we attempt to deduce oscillation of (1.1) in non-canonical form from that of the corresponding canonical linear differential equation (˜ r(t)y ′ (t)) + q˜(t)y(τ (t)) = 0.

(1.13)

This comparison result is easily verifiable and different to that presented in Theorem E. In order to illustrate the significance of our main results, a particular example of Euler type half-linear equation is provided. As is convenient and with no loss of generality, we may deal only with positive solutions of Eq. (1.1). All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.

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2. Main results Using elementary arguments, we start with a simplified version of Theorems A–B and Theorems C–D, respectively. Theorem 1. Assume that (1.7) holds. Then every solution y(t) of (1.1) oscillates or limt→∞ y(t) = 0. Proof . Assume to the contrary that y(t) is an eventually positive solution of (1.1). Then either y ′ (t) > 0 or y ′ (t) < 0 holds eventually, say for t ≥ t1 , t1 ∈ [t0 , ∞). If y ′ (t) < 0 then it follows that limt→∞ y(t) = c ≥ 0 exists. Similar as in the proof of [9, Theorem 2], one can show that (1.7) yields c = 0. Assume now that y ′ (t) > 0. Define w(t) := r(t)(y ′ (t))α /y α (τ (t)) > 0. Then we see that w(t) satisfies the equation w′ (t) = −q(t) + w(t)

y ′ (τ (t)) ′ τ (t) ≤ −q(t). y(τ (t))

(2.1)

Integrating (2.1) from t1 to t, we have ∫

t

w(t) ≤ w(t1 ) −

q(s)ds.

(2.2)

t1

∫t But it follows from (1.7) and (1.3) that t q(s)ds is unbounded, which in turn contradicts to the positivity 1 of w(t). Thus the case y ′ (t) > 0 is impossible and the proof is complete. □ Theorem 2. Assume that one of the conditions (1.8) or (1.9) holds. Then (1.1) is oscillatory. Proof . Assume to the contrary that y(t) is an eventually positive solution of (1.1). Then either y ′ (t) > 0 or y ′ (t) < 0 holds eventually, say for t ≥ t1 , t1 ∈ [t0 , ∞). If y ′ (t) < 0, then proceeding as in the proof of [6, Lemma 2.2] ([12, Theorem 2.4]), we get a contradiction with the condition (1.8) (condition (1.9)). If y ′ (t) > 0, the proof is similar to that of Theorem 1 and so is omitted. □ Remark 1. Theorems 1 and 2 essentially simplify a number of related criteria in the literature. In fact, ∫ ∞ q(s)ds = ∞ (2.3) removes the possibility that any positive solution y of (1.1) is increasing. This is well known for canonical equations as Fite–Wintner’s Theorem, but it has not been surprisingly used when studying non-canonical equations in works [6–12]. Note that the assumption (2.3) clearly gives no restriction to, e.g., Euler equation (1.10). The following result, which provides another condition ensuring oscillation of (1.1), can be seen as an alternative of Theorem C which is applicable to (1.10). Theorem 3. Assume that ) ( ∫ t lim sup π α (t) q(s)ds > 1. t→∞

Then (1.1) is oscillatory.

(2.4)

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Proof . Let y(t) be an eventually positive solution of (1.1). Then either y ′ (t) > 0 or y ′ (t) < 0 holds eventually, say for t ≥ t1 , t1 ∈ [t0 , ∞). At first we assume that y ′ (t) < 0. Since r1/α (t)y ′ (t) is decreasing, we get ∫ ∞ ( ) 1 1/α ′ −r (s)y (s) ds ≥ −π(t)r1/α (t)y ′ (t) ≥ 0. (2.5) y(τ (t)) ≥ y(t) ≥ r1/α (s) t Integrating (1.1) from t1 to t and using (2.5), we obtain ∫t α −r(t)(y ′ (t)) ≥ t q(s)y α (τ (s))ds 1 ∫t ≥ y α (τ (t)) t q(s)ds 1 ∫t α ≥ −r(t)(y ′ (t)) π α (t) t q(s)ds.

(2.6)

1

Taking the limsup on both sides of the above inequality, we obtain a contradiction to (2.4). Assume now that y ′ (t) > 0. It obviously follows from (1.3) and (2.4) that (2.3) holds. By virtue of Remark 1, this is a contradiction and the proof is complete. □ The following lemma plays a crucial rule in the proof of our next main result. Lemma 1. Let α ≥ 1 and assume that y(t) is an eventually positive solution of (1.1). Then )α−1 ( 1/α r (t)y ′ (t) ≤ π 1−α (t). y(τ (t))

(2.7)

Proof . Let y(t) be an eventually positive solution of (1.1). Then either y ′ (t) > 0 or y ′ (t) < 0 hold eventually, say for t ≥ t1 , t1 ∈ [t0 , ∞). Assume first that y ′ (t) < 0. Then (2.7) immediately follows from (2.5) and the fact that α ≥ 1 is a quotient of odd positive integers. Suppose now that y ′ (t) > 0. First note that R(τ (t)) + π(τ (t)) = π(t1 ) > 0 together with (1.3) implies R(τ (t)) ≥ π(t). Then ∫ τ (t) 1 y(τ (t)) ≥ r1/α (s)y ′ (s)ds ≥ R(τ (t))r1/α (t)y ′ (t) ≥ π(t)r1/α (t)y ′ (t), (2.8) 1/α r (s) t1 which is obviously equivalent to (2.7). The proof is complete. □ Our last oscillation result for (1.1) is based on comparison with the second-order canonical linear delay differential equation (1.13). So far, such an approach has not been applied to the half-linear equation (1.1). Theorem 4. Let α ≥ 1. Assume that Eq. (1.13) with r˜(t) = π 2 (t)r1/α (t),

q˜(t) =

1 α π (t)π(τ (t))q(t) α

is oscillatory. Then (1.1) is oscillatory. Proof . Assume to the contrary that y(t) is an eventually positive solution of (1.1). By applying the ( α )′ chain-rule on r(t)(y ′ (t)) , Eq. (1.1) becomes ( )′ 1 ( )1−α r1/α (t)y ′ (t) + r1/α (t)y ′ (t) q(t)y α (τ (t)) = 0. (2.9) α In view of Lemma 1, it is easy to see that y is a solution of the linear second-order inequality ( )′ 1 r1/α (t)y ′ (t) + π α−1 (t)q(t)y(τ (t)) ≤ 0, (2.10) α

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131

for t ≥ t1 , t1 ∈ [t0 , ∞). According to the general theory of W. Trench [1], the inequality (2.10) can be rewritten in equivalent canonical form as ( ( )′ )′ y(t) 1 1 2 1/α π (t)r (t) + π α−1 (t)q(t)y(τ (t)) ≤ 0. π(t) π(t) α

(2.11)

The oscillation-preserving transformation y(t) = π(t)x(t) reduces (2.11) to ′

(˜ r(t)x′ (t)) + q˜(t)x(τ (t)) ≤ 0.

(2.12)

By [13, Corollary 1], the corresponding differential equation (1.13) also has a positive solution which contradicts to the theorem assumption. The proof is complete. □ In view of Theorem 4, applying any criterion for oscillation of (1.13) immediately yields to an oscillation criterion for (1.1). The following one is due to Dˇzurina [14, Theorem 4]. Corollary 1. Let α ≥ 1. Assume that there exists a t1 ∈ [t0 , ∞) such that ∫

t

(∫

τ (u)

lim inf t→∞

τ (t)

t1

) 1 1 ds q˜(u)du > , r˜(s) e

(2.13)

where r˜, q˜ are as in Theorem 4. Then (1.1) is oscillatory. We would like to illustrate our main results on the following example. Example 1. Consider the non-canonical Euler type differential equation (

tα+1 (y ′ (t))

α )′

+ q0 y α (λt) = 0,

q0 > 0,

λ ∈ (0, 1],

α ≥ 1,

t ≥ 1.

(2.14)

It is easy to verify that (1.7) holds. By Theorem 1, every eventually positive solution y of (2.14) converges to zero as t → ∞. Theorem 3 implies that (2.14) is oscillatory if q0 > 1. Finally, Theorem 4 reduces oscillation problem of (2.14) to that of the canonical linear equation (

t1−1/α x′

)′

+

q0 αα−2 x(λt) = 0. λ1/α t1+1/α

By Corollary 1, we conclude that (2.14) is oscillatory if q0 ln

α1−α 1 > . λ e

Thus, the smaller is the delay parameter λ, the better oscillation constant is acquired. Note that Theorems A–D cannot be applied on (2.14), while the criterion (1.12) in Theorem E does not take the length of the delay into account and thus provides a weaker result. Acknowledgment The work on this research has been supported by the internal grant project no. FEI-2015-22.

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