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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 458275, 11 pages
http://dx.doi.org/10.1155/2011/458275
Research Article

Oscillation of Second-Order Sublinear Impulsive Differential Equations

Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

Received 25 January 2011; Accepted 27 February 2011

Academic Editor: JosefΒ DiblΓ­k

Copyright Β© 2011 A. Zafer. 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

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: π‘₯β€²β€²(𝑑)+𝑝(𝑑)|π‘₯(𝜏(𝑑))|π›Όβˆ’1π‘₯(𝜏(𝑑))=0, π‘‘β‰ πœƒπ‘˜; Ξ”π‘₯β€²(𝑑)|𝑑=πœƒπ‘˜+π‘žπ‘˜|π‘₯(𝜏(πœƒπ‘˜))|π›Όβˆ’1π‘₯(𝜏(πœƒπ‘˜))=0; Ξ”π‘₯(𝑑)|𝑑=πœƒπ‘˜=0, (0<𝛼<1) by considering the cases 𝜏(𝑑)≀𝑑 and 𝜏(𝑑)=𝑑, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.

1. Introduction

We deal with second-order sublinear impulsive differential equations of the formπ‘₯ξ…žξ…ž||||(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))π›Όβˆ’1π‘₯(𝜏(𝑑))=0,π‘‘β‰ πœƒπ‘˜,Ξ”π‘₯ξ…ž||(𝑑)𝑑=πœƒπ‘˜+π‘žπ‘˜||π‘₯ξ€·πœξ€·πœƒπ‘˜||ξ€Έξ€Έπ›Όβˆ’1π‘₯ξ€·πœξ€·πœƒπ‘˜||ξ€Έξ€Έ=0,Ξ”π‘₯(𝑑)𝑑=πœƒπ‘˜=0,(1.1) where 0<𝛼<1, 𝑑β‰₯𝑑0, and π‘˜β‰₯π‘˜0 for some 𝑑0βˆˆβ„+ and π‘˜0βˆˆβ„•, {πœƒπ‘˜} is a strictly increasing unbounded sequence of positive real numbers, ||Δ𝑧(𝑑)𝑑=πœƒξ€·πœƒβˆΆ=𝑧+ξ€Έβˆ’π‘§(πœƒβˆ’ξ€·πœƒ),π‘§βˆ“ξ€ΈβˆΆ=limπ‘‘β†’πœƒβˆ“π‘§(𝑑).(1.2)

Let PLC(𝐽,𝑅) denote the set of all real-valued functions 𝑒 defined on 𝐽 such that 𝑒 is continuous for all π‘‘βˆˆπ½ except possibly at 𝑑=πœƒπ‘˜ where 𝑒(πœƒΒ±π‘˜) exists and 𝑒(πœƒπ‘˜)∢=𝑒(πœƒβˆ’π‘˜).

We assume in the sequel that (a)π‘βˆˆPLC([𝑑0,∞),ℝ), (b){π‘žπ‘˜} is a sequence of real numbers, (c)𝜏∈𝐢([𝑑0,∞),ℝ+), 𝜏(𝑑)≀𝑑, limπ‘‘β†’βˆžπœ(𝑑)=∞.

By a solution of (1.1) on an interval π½βŠ‚[𝑑0,∞), we mean a function π‘₯(𝑑) which is defined on 𝐽 such that π‘₯,π‘₯ξ…ž,π‘₯ξ…žξ…žβˆˆPLC(𝐽) and which satisfies (1.1). Because of the requirement Ξ”π‘₯(𝑑)|𝑑=πœƒπ‘˜=0 every solution of (1.1) is necessarily continuous.

As usual we assume that (1.1) has solutions which are nontrivial for all large 𝑑. Such a solution of (1.1) is called oscillatory if it has no last zero and nonoscillatory otherwise.

In case there is no impulse, (1.1) reduces to Emden-Fowler equation with delayπ‘₯ξ…žξ…ž||||(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))π›Όβˆ’1π‘₯(𝜏(𝑑))=0,0<𝛼<1,(1.3) and without delayπ‘₯ξ…žξ…ž+𝑝(𝑑)|π‘₯|π›Όβˆ’1π‘₯=0,0<𝛼<1.(1.4)

The problem of oscillation of solutions of (1.3) and (1.4) has been considered by many authors. Kusano and Onose [1] see also [2, 3] proved the following necessary and sufficient condition for oscillation of (1.3).

Theorem 1.1. If 𝑝(𝑑)β‰₯0, then a necessary and sufficient condition for every solution of (1.3) to be oscillatory is that ξ€œβˆž[]𝜏(𝑑)𝛼𝑝(𝑑)𝑑𝑑=∞.(1.5)

The condition 𝑝(𝑑)β‰₯0 is required only for the sufficiency part, and no similar criteria is available for 𝑝(𝑑) changing sign, except in the case 𝜏(𝑑)=𝑑. Without imposing a sign condition on 𝑝(𝑑), Belohorec [4] obtained the following sufficient condition for oscillation of (1.4).

Theorem 1.2. If ξ€œβˆžπ‘‘π›½π‘(𝑑)𝑑𝑑=∞(1.6) for some π›½βˆˆ[0,𝛼], then every solution of (1.4) is oscillatory.

Compared to the large body of papers on oscillation of differential equations, there is only little known about the oscillation of impulsive differential equations; see [5–7] for equations with delay and [8–13] for equations without delay. For some applications of such equations, we may refer to [14–18]. The books [19, 20] are good sources for a general theory of impulsive differential equations.

The object of this paper is to extend Theorems 1.1 and 1.2 to impulsive differential equations of the form (1.1). The results show that the impulsive perturbations may greatly change the oscillatory behavior of the solutions. A nonoscillatory solution of (1.3) or (1.4) may become oscillatory under impulsive perturbations.

The following two lemmas are crucial in the proof of our main theorems. The first lemma is contained in [21] and the second one is extracted from [22].

Lemma 1.3. If each 𝐴𝑖 is continuous on [π‘Ž,𝑏], then ξ€œπ‘π‘Žξ“π‘ β‰€πœƒπ‘–<𝑏𝐴𝑖(𝑠)𝑑𝑠=π‘Žβ‰€πœƒπ‘–<π‘ξ€œπœƒπ‘–π‘Žπ΄π‘–(𝑠)𝑑𝑠.(1.7)

Lemma 1.4. Fix𝐽=[π‘Ž,𝑏], let 𝑒,πœ†βˆˆπΆ(𝐽,ℝ+), β„ŽβˆˆπΆ(ℝ+,ℝ+), and π‘βˆˆβ„+, and let {πœ†π‘˜} a sequence of positive real numbers. If 𝑒(𝐽)βŠ‚πΌβŠ‚β„+ and ξ€œπ‘’(𝑑)≀𝑐+π‘‘π‘Žξ“πœ†(𝑠)β„Ž(𝑒(𝑠))𝑑𝑠+π‘Ž<πœƒπ‘˜<π‘‘πœ†π‘˜β„Žξ€·π‘’ξ€·πœƒπ‘˜ξ€Έξ€Έ,π‘‘βˆˆπ½,(1.8) then 𝑒(𝑑)β‰€πΊβˆ’1⎧βŽͺ⎨βŽͺβŽ©ξ€œπΊ(𝑐)+π‘‘π‘Žξ“πœ†(𝑠)𝑑𝑠+π‘Ž<πœƒπ‘˜<π‘‘πœ†π‘˜βŽ«βŽͺ⎬βŽͺ⎭[,π‘‘βˆˆπ‘Ž,𝛽),(1.9) where ξ€œπΊ(𝑒)=𝑒𝑒0𝑑π‘₯β„Ž(π‘₯),𝑒,𝑒0⎧βŽͺ⎨βŽͺβŽ©ξ€œβˆˆπΌ,𝛽=sup𝜈∈𝐽∢𝐺(𝑐)+π‘‘π‘Žξ“πœ†(𝑠)𝑑𝑠+π‘Ž<πœƒπ‘˜<π‘‘πœ†π‘˜βŽ«βŽͺ⎬βŽͺ⎭.∈𝐺(𝐼),π‘Žβ‰€π‘‘β‰€πœˆ(1.10)

2. The Main Results

We first establish a necessary and sufficient condition for oscillation of solutions of (1.1) when 𝜏(𝑑)≀𝑑.

Theorem 2.1. If ξ€œβˆž[]𝜏(𝑑)𝛼||||𝑝(𝑑)𝑑𝑑+βˆžξ“ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Ό||π‘žπ‘˜||<∞,(2.1) then (1.1) has a solution π‘₯(𝑑) satisfying limπ‘‘β†’βˆžπ‘₯(𝑑)𝑑=π‘Žβ‰ 0.(2.2)

Proof. Choose 𝑑1β‰₯max{1,𝑑0}. In view of Lemma 1.3 by integrating (1.1) twice from 𝑑0 to 𝑑, we obtain 𝑑π‘₯(𝑑)=π‘₯1ξ€Έβˆ’π‘₯ξ…žξ€·π‘‘1ξ€Έξ€·π‘‘βˆ’π‘‘1ξ€Έβˆ’ξ“π‘‘1β‰€πœƒπ‘˜<π‘‘π‘žπ‘˜||π‘₯ξ€·πœξ€·πœƒπ‘˜||ξ€Έξ€Έπ›Όβˆ’1π‘₯πœξ€·πœƒξ€·ξ€·π‘˜ξ€Έξ€Έξ€Έξ€·π‘‘βˆ’πœƒπ‘˜ξ€Έβˆ’ξ€œπ‘‘π‘‘1||||(π‘‘βˆ’π‘ )𝑝(𝑠)π‘₯(𝜏(𝑠))π›Όβˆ’1π‘₯((𝜏(𝑠)))𝑑𝑠,𝑑β‰₯𝑑1.(2.3) Set 𝑒(𝑑)=𝑐+𝑑1β‰€πœƒπ‘˜<𝑑||π‘žπ‘˜||||π‘₯ξ€·πœξ€·πœƒπ‘˜||𝛼+ξ€œπ‘‘π‘‘1||||||||𝑝(𝑠)π‘₯(𝜏(𝑠))𝛼𝑑𝑠,𝑑β‰₯𝑑1,(2.4) where 𝑐=|π‘₯(𝑑1)|+|π‘₯ξ…ž(𝑑1)|. Then ||||π‘₯(𝑑)≀𝑑𝑒(𝑑),𝑑β‰₯𝑑1.(2.5) Let 𝑑2β‰₯𝑑1 be such that 𝜏(𝑑)β‰₯𝑑1 for all 𝑑β‰₯𝑑2. Replacing 𝑑 by 𝜏(𝑑) in (2.5) and using the increasing character of 𝑒(𝑑), we see that ||||π‘₯(𝜏(𝑑))β‰€πœ(𝑑)𝑒(𝑑),𝑑β‰₯𝑑2.(2.6) From (2.4), we also see that π‘’ξ…ž||||||||(𝑑)=𝑝(𝑑)π‘₯(𝜏(𝑑))𝛼,π‘‘β‰ πœƒπ‘˜,||(2.7)Δ𝑒(𝑑)𝑑=πœƒπ‘˜=||π‘žπ‘˜||||π‘₯ξ€·πœξ€·πœƒπ‘˜||𝛼(2.8) for 𝑑β‰₯𝑑2 and πœƒπ‘˜β‰₯𝑑2. Now, in view of (2.6) and (2.8), an integration of (2.7) from 𝑑2 to 𝑑 leads to ξ€œπ‘’(𝑑)≀𝑐+𝑑𝑑2||||[]𝑝(𝑠)𝜏(𝑠)𝛼[]𝑒(𝑠)𝛼𝑑𝑠+𝑑2β‰€πœƒπ‘˜<𝑑||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Όξ€Ίπ‘’ξ€·πœƒπ‘˜ξ€Έξ€»π›Ό.(2.9) Applying Lemma 1.4 with β„Ž(π‘₯)=π‘₯𝛼||||[],πœ†(𝑠)=𝑝(𝑠)𝜏(𝑠)𝛼,πœ†π‘˜=||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Ό,(2.10) we easily see that 𝑒(𝑑)β‰€πΊβˆ’1⎧βŽͺ⎨βŽͺβŽ©ξ€œπΊ(𝑐)+𝑑𝑑2||||[]𝑝(𝑠)𝜏(𝑠)𝛼𝑑𝑠+𝑑2β‰€πœƒπ‘˜<𝑑||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›ΌβŽ«βŽͺ⎬βŽͺ⎭.(2.11) Since 𝑒𝐺(𝑒)=1βˆ’π›Όβˆ’π‘’1βˆ’π›Ό01βˆ’π›Ό1βˆ’π›Ό,πΊβˆ’1ξ€Ί(𝑒)=(1βˆ’π›Ό)𝑒+𝑒01βˆ’π›Όξ€»1/(1βˆ’π›Ό),(2.12) the inequality (2.11) becomes βŽ‘βŽ’βŽ’βŽ£π‘π‘’(𝑑)≀1βˆ’π›Όξ€œ+(1βˆ’π›Ό)𝑑𝑑1||||[]𝑝(𝑠)𝜏(𝑠)𝛼𝑑𝑠+(1βˆ’π›Ό)𝑑1β‰€πœƒπ‘˜<𝑑||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›ΌβŽ€βŽ₯βŽ₯⎦1/(1βˆ’π›Ό),(2.13) from which, on using (2.1), we have 𝑒(𝑑)≀𝑐1,𝑑β‰₯𝑑2,(2.14) where 𝑐1=βŽ‘βŽ’βŽ’βŽ£π‘1βˆ’π›Όξ€œ+(1βˆ’π›Ό)βˆžπ‘‘1||||[]𝑝(𝑠)𝜏(𝑠)𝛼𝑑𝑠+(1βˆ’π›Ό)𝑑1β‰€πœƒπ‘˜<∞||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›ΌβŽ€βŽ₯βŽ₯⎦1/(1βˆ’π›Ό).(2.15) In view of (2.5), (2.6), and (2.14) we see that ||||π‘₯(𝑑)≀𝑐1||||𝑑,π‘₯(𝜏(𝑑))≀𝑐1𝜏(𝑑),𝑑β‰₯𝑑2.(2.16)
To complete the proof it suffices to show that π‘₯ξ…ž(𝑑) approaches a nonzero limit as 𝑑 tends to ∞. To see this we integrate (1.1) from 𝑑2 to 𝑑 to getπ‘₯ξ…ž(𝑑)=π‘₯ξ…žξ€·π‘‘1ξ€Έβˆ’ξ€œπ‘‘π‘‘2||||𝑝(𝑠)π‘₯(𝜏(𝑠))π›Όβˆ’1π‘₯(𝜏(𝑠))π‘‘π‘ βˆ’π‘‘2β‰€πœƒπ‘˜<π‘‘π‘žπ‘˜||π‘₯ξ€·πœξ€·πœƒπ‘˜||ξ€Έξ€Έπ›Όβˆ’1π‘₯ξ€·πœξ€·πœƒπ‘˜.ξ€Έξ€Έ(2.17) Employing (2.16) we have ξ€œβˆžπ‘‘2||||𝑝(𝑠)π‘₯(𝜏(𝑠))𝛼𝑑𝑠≀𝑐𝛼1ξ€œβˆžπ‘‘2||||[]𝑝(𝑠)𝜏(𝑠)𝛼𝑑𝑠<∞,𝑑2β‰€πœƒπ‘˜<∞||π‘žπ‘˜π‘₯ξ€·πœξ€·πœƒπ‘˜||𝛼≀𝑐𝛼1𝑑2β‰€πœƒπ‘˜<∞||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Ό<∞.(2.18) Therefore, limπ‘‘β†’βˆžπ‘₯ξ…ž(𝑑)=𝐿 exists. Clearly, we can make 𝐿≠0 by requiring that π‘₯ξ…žξ€·π‘‘2ξ€Έ>𝑐𝛼1βŽ‘βŽ’βŽ’βŽ£ξ€œβˆžπ‘‘2||||[]𝑝(𝑠)𝜏(𝑠)𝛼𝑑𝑠+𝑑2β‰€πœƒπ‘˜<∞||π‘žπ‘˜||ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›ΌβŽ€βŽ₯βŽ₯⎦,(2.19) which is always possible by arranging 𝑑2.

Theorem 2.2. Suppose that 𝑝 and {π‘žπ‘˜} are nonnegative. Then every solution of (1.1) is oscillatory if and only if ξ€œβˆž[]𝜏(𝑑)𝛼𝑝(𝑑)𝑑𝑑+βˆžξ“ξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Όπ‘žπ‘˜=∞.(2.20)

Proof. Let (2.20) fail to hold. Then, by Theorem 2.1 we see that there is a solution π‘₯(𝑑) which satisfies (2.2). Clearly, such a solution is nonoscillatory. This proves the necessity.
To show the sufficiency, suppose that (2.20) is valid but there is a nonoscillatory solution π‘₯(𝑑) of (1.1). We may assume that π‘₯(𝑑) is eventually positive; the case π‘₯(𝑑) being eventually negative is similar. Clearly, there exists 𝑑1β‰₯𝑑0 such that π‘₯(𝜏(𝑑))>0 for all 𝑑β‰₯𝑑1. From (1.1), we have thatπ‘₯ξ…žξ…ž(𝑑)≀0for𝑑β‰₯𝑑1,π‘‘β‰ πœƒπ‘˜.(2.21) Thus, π‘₯ξ…ž(𝑑) is decreasing on every interval not containing 𝑑=πœƒπ‘˜. From the impulse conditions in (1.1), we also have Ξ”π‘₯ξ…ž(πœƒπ‘˜)≀0. Therefore, we deduce that π‘₯ξ…ž(𝑑) is nondecreasing on [𝑑1,∞).
We may claim that π‘₯ξ…ž(𝑑) is eventually positive. Because if π‘₯ξ…ž(𝑑)<0 eventually, then π‘₯(𝑑) becomes negative for large values of 𝑑. This is a contradiction.
It is now easy to show thatπ‘₯ξ€·(𝑑)β‰₯π‘‘βˆ’π‘‘1ξ€Έπ‘₯ξ…ž(𝑑),𝑑β‰₯𝑑1.(2.22) Therefore, 𝑑π‘₯(𝑑)β‰₯2π‘₯ξ…ž(𝑑),𝑑β‰₯𝑑2=2𝑑1.(2.23) Let 𝑑3β‰₯𝑑2 be such that 𝜏(𝑑)β‰₯𝑑2 for 𝑑β‰₯𝑑3. Using (2.23) and the nonincreasing character of π‘₯ξ…ž(𝑑), we have π‘₯(𝜏(𝑑))β‰₯𝜏(𝑑)2π‘₯ξ…ž(𝑑),𝑑β‰₯𝑑3,(2.24) and so, by (1.1), π‘₯ξ…žξ…ž(𝑑)+2βˆ’π›Ό[]𝑝(𝑑)𝜏(𝑑)𝛼π‘₯ξ…žξ€»(𝑑)𝛼≀0,π‘‘β‰ πœƒπ‘˜.(2.25) Dividing (2.25) by [π‘₯ξ…ž(𝑑)]𝛼 and integrating from 𝑑3 to 𝑑, we obtain 𝑑3β‰€πœƒπ‘˜<𝑑π‘₯ξ…žξ€·πœƒπ‘˜ξ€Έξ€»1βˆ’π›Όβˆ’ξ€Ίπ‘₯ξ…žξ€·πœƒπ‘˜ξ€Έβˆ’π‘žπ‘˜ξ€Ίπ‘₯ξ€·πœξ€·πœƒπ‘˜ξ€Έξ€Έξ€»π›Όξ€»1βˆ’π›Όξ‚‡+ξ€Ίπ‘₯ξ…žξ€»(𝑑)1βˆ’π›Όβˆ’ξ€Ίπ‘₯ξ…žξ€·π‘‘3ξ€Έξ€»1βˆ’π›Ό+(1βˆ’π›Ό)2βˆ’π›Όξ€œπ‘‘π‘‘3[]𝜏(𝑑)𝛼𝑝(𝑠)𝑑𝑠≀0(2.26) which clearly implies that 𝑑3β‰€πœƒπ‘˜<π‘‘π‘Žπ‘˜+(1βˆ’π›Ό)2βˆ’π›Όξ€œπ‘‘π‘‘3[]𝜏(𝑑)𝛼π‘₯𝑝(𝑠)π‘‘π‘ β‰€ξ…žξ€·π‘‘3ξ€Έξ€»1βˆ’π›Ό,(2.27) where π‘Žπ‘˜=ξ€Ίπ‘₯ξ…žξ€·πœƒπ‘˜ξ€Έξ€»1βˆ’π›Όξƒ¬ξƒ©π‘ž1βˆ’1βˆ’π‘˜ξ€Ίπ‘₯ξ€·πœξ€·πœƒπ‘˜ξ€Έξ€Έξ€»π›Όπ‘₯ξ…žξ€·πœƒπ‘˜ξ€Έξƒͺξƒ­1βˆ’π›Ό.(2.28) Since 1βˆ’(1βˆ’π‘’)1βˆ’π›Όβ‰₯(1βˆ’π›Ό)𝑒 for π‘’βˆˆ(0,∞) and 0<𝛼<1, by taking π‘žπ‘’=π‘˜ξ€Ίπ‘₯ξ€·πœξ€·πœƒπ‘˜ξ€Έξ€Έξ€»π›Όπ‘₯ξ…žξ€·πœƒπ‘˜ξ€Έ,(2.29) we see from (2.28) that π‘Žπ‘˜π‘žβ‰₯(1βˆ’π›Ό)π‘˜ξ€Ίπ‘₯ξ€·πœξ€·πœƒπ‘˜ξ€Έξ€Έξ€»π›Όξ€Ίπ‘₯ξ…žξ€·πœƒπ‘˜ξ€Έξ€»π›Ό.(2.30) But, (2.24) gives π‘₯ξ€·πœξ€·πœƒπ‘˜β‰₯πœξ€·πœƒξ€Έξ€Έπ‘˜ξ€Έ2π‘₯ξ…žξ€·πœξ€·πœƒπ‘˜β‰₯πœξ€·πœƒξ€Έξ€Έπ‘˜ξ€Έ2π‘₯ξ…žξ€·πœƒπ‘˜ξ€Έ,(2.31) and hence π‘Žπ‘˜β‰₯(1βˆ’π›Ό)2βˆ’π›Όξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Όπ‘žπ‘˜.(2.32)
Finally, (2.27) and (2.32) result inξ€œβˆžπ‘‘3[]𝜏(𝑑)𝛼𝑝(𝑑)𝑑𝑑+𝑑3<πœƒπ‘˜<βˆžξ€Ίπœξ€·πœƒπ‘˜ξ€Έξ€»π›Όπ‘žπ‘˜<∞,(2.33) which contradicts (2.20). The proof is complete.

Example 2.3. Consider the impulsive delay differential equation π‘₯ξ…žξ…ž(𝑑)+(π‘‘βˆ’1)βˆ’2||||π‘₯(π‘‘βˆ’1)βˆ’1/2π‘₯(π‘‘βˆ’1)=0,π‘‘β‰ π‘˜,Ξ”π‘₯ξ…ž||(𝑑)𝑑=π‘˜+(π‘˜βˆ’1)βˆ’1||||π‘₯(π‘˜βˆ’1)βˆ’1/2||π‘₯(π‘˜βˆ’1)=0,Ξ”π‘₯(𝑑)𝑑=π‘˜=0,(2.34) where 𝑑β‰₯2 and 𝑖β‰₯2.
We see that 𝜏(𝑑)=π‘‘βˆ’1, 𝛼=1/2, 𝑝(𝑑)=(π‘‘βˆ’1)βˆ’2, and π‘žπ‘˜=(π‘˜βˆ’1)βˆ’1, πœƒπ‘˜=π‘˜. Sinceξ€œβˆž(π‘‘βˆ’1)βˆ’3/2𝑑𝑑+βˆžξ“(π‘˜βˆ’1)βˆ’1/2=∞,(2.35) applying Theorem 2.2 we conclude that every solution of (2.34) is oscillatory.
We note that if the equation is not subject to any impulse condition, then, sinceξ€œβˆž(π‘‘βˆ’1)βˆ’5/2𝑑𝑑<∞,(2.36) the equation π‘₯ξ…žξ…ž(𝑑)+(π‘‘βˆ’1)βˆ’2||||π‘₯(π‘‘βˆ’1)βˆ’1/2π‘₯(π‘‘βˆ’1)=0(2.37) has a nonoscillatory solution by Theorem 1.1.

Let us now consider (1.1) when 𝜏(𝑑)=𝑑. That is,π‘₯ξ…žξ…ž+𝑝(𝑑)|π‘₯|π›Όβˆ’1π‘₯=0,π‘‘β‰ πœƒπ‘˜,Ξ”π‘₯ξ…ž||𝑑=πœƒπ‘˜+π‘žπ‘˜|π‘₯|π›Όβˆ’1||π‘₯=0,Ξ”π‘₯𝑑=πœƒπ‘˜=0,(2.38) where 0<𝛼<1 and π‘π‘žπ‘˜ are given by (a) and (b).

The following theorem is an extension of Theorem 1.2. Note that no sign condition is imposed on 𝑝(𝑑) and {π‘žπ‘˜}.

Theorem 2.4. If ξ€œβˆžπ‘‘π›½π‘(𝑑)𝑑𝑑+βˆžξ“πœƒπ›½π‘˜π‘žπ‘˜=∞(2.39) for some π›½βˆˆ[0,𝛼], then every solution of (2.38) is oscillatory.

Proof. Assume on the contrary that (2.38) has a nonoscillatory solution π‘₯(𝑑) such that π‘₯(𝑑)>0 for all 𝑑β‰₯𝑑0 for some 𝑑0β‰₯0. The proof is similar when π‘₯(𝑑) is eventually negative. We set 𝑑𝑀(𝑑)=βˆ’1ξ€Έπ‘₯(𝑑)1βˆ’π›Ό,𝑑β‰₯𝑑0.(2.40) It is not difficult to see that π‘€ξ…ž(𝑑)=(π›Όβˆ’1)π‘‘π›Όβˆ’2[]π‘₯(𝑑)1βˆ’π›Ό+(1βˆ’π›Ό)π‘‘π›Όβˆ’1[]π‘₯(𝑑)βˆ’π›Όπ‘₯ξ…ž(𝑑),π‘‘β‰ πœƒπ‘˜,(2.41) and hence ||Δ𝑀′𝑑=πœƒπ‘˜=(1βˆ’π›Ό)π‘žπ‘˜πœƒπ‘˜π›Όβˆ’1.(2.42) From (2.41), we have π‘‘π›½βˆ’1βˆ’π›Όξ€·π‘‘2π‘€ξ…ž(𝑑)ξ…ž=(1βˆ’π›Ό)𝑑𝛽π‘₯ξ…žξ…ž(𝑑)π‘₯βˆ’π›Ό(𝑑)βˆ’π›Ό(1βˆ’π›Ό)π‘‘π›½βˆ’2π‘₯βˆ’π›Όβˆ’1(𝑑)𝑑π‘₯ξ…ž(𝑑)βˆ’π‘₯(𝑑)2,(2.43) and so π‘‘π›½βˆ’1βˆ’π›Όξ€·π‘‘2π‘€ξ…ž(𝑑)ξ…žβ‰€(1βˆ’π›Ό)𝑑𝛽𝑝(𝑑),π‘‘β‰ πœƒπ‘˜.(2.44) In view of (2.42), by a straightforward integration of (2.44), we have ξ€œπ‘‘π‘‘0π‘ π›½βˆ’1βˆ’π›Όξ€·π‘ 2π‘€ξ…ž(𝑠)ξ…žπ‘‘π‘ =π‘ π›½βˆ’1βˆ’π›Όπ‘ 2π‘€ξ…ž||(𝑠)𝑑𝑑0βˆ’ξ“π‘‘0β‰€πœƒπ‘˜<π‘‘Ξ”ξ€·π‘‘π›½βˆ’π›Ό+1π‘€ξ…ž(ξ€Έ||𝑑)𝑑=πœƒπ‘˜βˆ’ξ€œπ‘‘π‘‘0(π›½βˆ’1βˆ’π›Ό)π‘ π›½βˆ’π›Όπ‘€ξ…ž(𝑠)𝑑𝑠=π‘‘π›½βˆ’π›Ό+1π‘€ξ…ž(𝑑)βˆ’π‘‘0π›½βˆ’π›Ό+1π‘€ξ…žξ€·π‘‘0ξ€Έβˆ’ξ“π‘‘0β‰€πœƒπ‘˜<𝑑(1βˆ’π›Ό)π‘žπ‘˜πœƒπ›½π‘˜βˆ’ξ€Ίπ‘ (π›½βˆ’π›Όβˆ’1)π›½βˆ’π›Όπ‘€ξ€»||(𝑠)𝑑𝑑0ξ€œ+(π›½βˆ’π›Ό)(π›½βˆ’π›Όβˆ’1)𝑑𝑑0π‘ π›½βˆ’1βˆ’π›Όπ‘€(𝑠)𝑑𝑠,(2.45) which combined with (2.44) leads to π‘‘π›½βˆ’π›Ό+1𝑀′(𝑑)≀𝑑0π›½βˆ’π›Ό+1𝑑𝑀′0ξ€Έβˆ’(π›½βˆ’π›Ό+1)𝑑0π›½βˆ’π›Όπ‘€ξ€·π‘‘0ξ€ΈβŽ‘βŽ’βŽ’βŽ£ξ“+(1βˆ’π›Ό)𝑑0β‰€πœƒπ‘˜<π‘‘πœƒπ›½π‘˜π‘žπ‘˜+ξ€œπ‘‘π‘‘0π‘ π›½βŽ€βŽ₯βŽ₯⎦.𝑝(𝑠)𝑑𝑠(2.46) Finally, by using (2.39) in the last inequality, we see that there is a 𝑑1>𝑑0 such that 𝑀′(𝑑)β‰€βˆ’π‘‘π›Όβˆ’π›½βˆ’1,𝑑β‰₯𝑑1,(2.47) which, however, implies that 𝑀(𝑑)β†’βˆ’βˆž as π‘‘β†’βˆž, a contradiction with π‘₯(𝑑)>0. The proof is complete.

Example 2.5. Consider the impulsive differential equation π‘₯ξ…žξ…ž+π‘‘βˆ’7/3|π‘₯|βˆ’1/2π‘₯=0,π‘‘β‰ π‘˜,Ξ”π‘₯ξ…ž||𝑑=π‘˜+π‘˜βˆ’1/6|π‘₯|βˆ’1/2||π‘₯=0,Ξ”π‘₯𝑑=π‘˜=0,(2.48) where 𝑑β‰₯1 and 𝑖β‰₯1.
We have that 𝑝(𝑑)=𝑑7/3, 𝛼=1/2, and π‘žπ‘˜=π‘˜βˆ’1/6, πœƒπ‘˜=π‘˜. Taking 𝛽=1/3 we see from (2.38) thatξ€œβˆžπ‘‘βˆ’2𝑑𝑑+βˆžξ“π‘˜βˆ’1/3=∞.(2.49) Since the conditions of Theorem 2.4 are satisfied, every solution of (2.48) is oscillatory.
Note that if the impulses are absent, then, sinceξ€œβˆžπ‘‘βˆ’2𝑑𝑑<∞,(2.50) the equation π‘₯ξ…žξ…ž+π‘‘βˆ’7/3|π‘₯|βˆ’1/2π‘₯=0(2.51) is oscillatory by Theorem 1.2.

Acknowledgment

This work was partially supported by METU-B AP (project no: 01-01-2011-003).

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