Oscillation of Second-Order Sublinear Impulsive Differential Equations

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: 𝑥′′(𝑡)+𝑝(𝑡)|𝑥(𝜏(𝑡))|^𝛼−1𝑥(𝜏(𝑡))=0, 𝑡≠𝜃_𝑘; Δ𝑥′(𝑡)|_{𝑡=𝜃𝑘}+𝑞_𝑘|𝑥(𝜏(𝜃_𝑘))|^𝛼−1𝑥(𝜏(𝜃_𝑘))=0; Δ𝑥(𝑡)|_{𝑡=𝜃𝑘}=0, (0<𝛼<1) by considering the cases 𝜏(𝑡)≤𝑡 and 𝜏(𝑡)=𝑡, 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 form𝑥Δ^𝑥(_𝑡)+𝑝(𝑡)||𝑥(𝜏(𝑡))||^𝛼−1𝑥(𝜏(𝑡))=0,𝑡≠𝜃_𝑘,(𝑡)||_{𝑡=𝜃𝑘}+𝑞_𝑘||Δ𝑥𝑥(𝜏𝑡)||𝜃_𝑘_{𝑡=𝜃𝑘}||^𝛼−1𝑥𝜏𝜃_𝑘=0,=0,(1.1) where 0<𝛼<1, 𝑡≥𝑡₀, and 𝑘≥𝑘₀ for some 𝑡₀∈ℝ₊ and 𝑘₀∈ℕ, {𝜃_𝑘} is a strictly increasing unbounded sequence of positive real numbers, Δ𝑧(𝑡)||_𝑡=𝜃∶=𝑧𝜃⁺−𝑧(𝜃⁻),𝑧𝜃^∓∶=lim_{𝑡→𝜃∓}𝑧(𝑡).(1.2)

Let PLC(𝐽,𝑅) denote the set of all real-valued functions 𝑢 defined on 𝐽 such that 𝑢 is continuous for all 𝑡∈𝐽 except possibly at 𝑡=𝜃_𝑘 where 𝑢(𝜃^±𝑘) exists and 𝑢(𝜃_𝑘)∶=𝑢(𝜃^−𝑘).

We assume in the sequel that (a)𝑝∈PLC([𝑡₀,∞),ℝ), (b){𝑞_𝑘} is a sequence of real numbers, (c)𝜏∈𝐶([𝑡₀,∞),ℝ₊), 𝜏(𝑡)≤𝑡, lim_𝑡→∞𝜏(𝑡)=∞.

By a solution of (1.1) on an interval 𝐽⊂[𝑡₀,∞), we mean a function 𝑥(𝑡) which is defined on 𝐽 such that 𝑥,𝑥^,𝑥^∈PLC(𝐽) and which satisfies (1.1). Because of the requirement Δ𝑥(𝑡)|_{𝑡=𝜃𝑘}=0 every solution of (1.1) is necessarily continuous.

As usual we assume that (1.1) has solutions which are nontrivial for all large 𝑡. 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 delay𝑥^(𝑡)+𝑝(𝑡)||𝑥(𝜏(𝑡))||^𝛼−1𝑥(𝜏(𝑡))=0,0<𝛼<1,(1.3) and without delay𝑥^+𝑝(𝑡)|𝑥|^𝛼−1𝑥=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(𝑡)≥0, then a necessary and sufficient condition for every solution of (1.3) to be oscillatory is that ^∞[𝜏(𝑡)]^𝛼𝑝(𝑡)𝑑𝑡=∞.(1.5)

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

Theorem 1.2. If ^∞𝑡^𝛽𝑝(𝑡)𝑑𝑡=∞(1.6) for some 𝛽∈[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 𝐴_𝑖 is continuous on [𝑎,𝑏], then ^𝑏𝑎_{𝑠≤𝜃𝑖<𝑏}𝐴_𝑖(𝑠)𝑑𝑠=_{𝑎≤𝜃𝑖<𝑏}^𝜃𝑖_𝑎𝐴_𝑖(𝑠)𝑑𝑠.(1.7)

Lemma 1.4. Fix𝐽=[𝑎,𝑏], let 𝑢,𝜆∈𝐶(𝐽,ℝ₊), ℎ∈𝐶(ℝ₊,ℝ₊), and 𝑐∈ℝ₊, and let {𝜆_𝑘} a sequence of positive real numbers. If 𝑢(𝐽)⊂𝐼⊂ℝ₊ and 𝑢(𝑡)≤𝑐+^𝑡𝑎𝜆(𝑠)ℎ(𝑢(𝑠))𝑑𝑠+_{𝑎<𝜃𝑘<𝑡}𝜆_𝑘ℎ𝑢𝜃_𝑘,𝑡∈𝐽,(1.8) then 𝑢(𝑡)≤𝐺⁻¹⎧⎪⎨⎪⎩𝐺(𝑐)+^𝑡𝑎𝜆(𝑠)𝑑𝑠+_{𝑎<𝜃𝑘<𝑡}𝜆_𝑘⎫⎪⎬⎪⎭,𝑡∈[𝑎,𝛽),(1.9) where 𝐺(𝑢)=^𝑢𝑢₀𝑑𝑥ℎ(𝑥),𝑢,𝑢⁰𝛽=sup⎧⎪⎨⎪⎩𝜈∈𝐽∶𝐺(𝑐)+_𝑡𝑎∈𝐼,𝜆(𝑠)𝑑𝑠+_{𝑎<𝜃𝑘<𝑡}𝜆_𝑘∈𝐺(𝐼),𝑎≤𝑡≤𝜈⎫⎪⎬⎪⎭.(1.10)

2. The Main Results

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

Theorem 2.1. If ^∞[𝜏(𝑡)]^𝛼||𝑝(𝑡)||𝑑𝑡+^∞𝜏𝜃^𝑘^𝛼||𝑞_𝑘||<∞,(2.1) then (1.1) has a solution 𝑥(𝑡) satisfying lim_𝑡→∞𝑥(𝑡)𝑡=𝑎≠0.(2.2)

Proof. Choose 𝑡₁≥max{1,𝑡₀}. In view of Lemma 1.3 by integrating (1.1) twice from 𝑡₀ to 𝑡, we obtain 𝑥(𝑡)=−𝑥𝑡_1𝑡𝑡1−𝑥^𝑡₁𝑡−𝑡₁−_{𝑡1≤𝜃𝑘<𝑡}𝑞_𝑘||𝑥𝜏𝜃_𝑘||^𝛼−1𝑥𝜏𝜃_𝑘𝑡−𝜃_𝑘(𝑡−𝑠)𝑝(𝑠)||𝑥(𝜏(𝑠))||^𝛼−1𝑥((𝜏(𝑠)))𝑑𝑠,𝑡≥𝑡₁.(2.3) Set 𝑢(𝑡)=𝑐+_{𝑡1≤𝜃𝑘<𝑡}||𝑞_𝑘||||𝑥𝜏𝜃_𝑘||^𝛼+^𝑡𝑡₁||𝑝(𝑠)||||𝑥(𝜏(𝑠))||^𝛼𝑑𝑠,𝑡≥𝑡₁,(2.4) where 𝑐=|𝑥(𝑡₁)|+|𝑥^(𝑡₁)|. Then ||𝑥(𝑡)||≤𝑡𝑢(𝑡),𝑡≥𝑡₁.(2.5) Let 𝑡₂≥𝑡₁ be such that 𝜏(𝑡)≥𝑡₁ for all 𝑡≥𝑡₂. Replacing 𝑡 by 𝜏(𝑡) in (2.5) and using the increasing character of 𝑢(𝑡), we see that ||𝑥(𝜏(𝑡))||≤𝜏(𝑡)𝑢(𝑡),𝑡≥𝑡₂.(2.6) From (2.4), we also see that 𝑢^Δ(𝑡𝑢)(=𝑡)||||𝑝(_{𝑡=𝜃𝑘}𝑡)||||𝑥(𝜏(𝑡))||^𝛼,𝑡≠𝜃_𝑘,(2.7)=||𝑞_𝑘||||𝑥𝜏𝜃_𝑘||^𝛼(2.8) for 𝑡≥𝑡₂ and 𝜃_𝑘≥𝑡₂. Now, in view of (2.6) and (2.8), an integration of (2.7) from 𝑡₂ to 𝑡 leads to 𝑢(𝑡)≤𝑐+^𝑡𝑡₂||𝑝(𝑠)||[𝜏(𝑠)]^𝛼[𝑢(𝑠)]^𝛼𝑑𝑠+_{𝑡2≤𝜃𝑘<𝑡}||𝑞_𝑘||𝜏𝜃_𝑘^𝛼𝑢𝜃_𝑘^𝛼.(2.9) Applying Lemma 1.4 with ℎ(𝑥)=𝑥^𝛼,𝜆(𝑠)=||𝑝(𝑠)||[𝜏(𝑠)]^𝛼,𝜆_𝑘=||𝑞_𝑘||𝜏𝜃_𝑘^𝛼,(2.10) we easily see that 𝑢(𝑡)≤𝐺⁻¹⎧⎪⎨⎪⎩𝐺(𝑐)+^𝑡𝑡₂||𝑝(𝑠)||[𝜏(𝑠)]^𝛼𝑑𝑠+_{𝑡2≤𝜃𝑘<𝑡}||𝑞_𝑘||𝜏𝜃_𝑘^𝛼⎫⎪⎬⎪⎭.(2.11) Since 𝐺(𝑢)=𝑢^1−𝛼1−𝛼−1𝑢−^01−𝛼𝛼,𝐺⁻¹(𝑢)=(1−𝛼)𝑢+𝑢^01−𝛼^1/(1−𝛼),(2.12) the inequality (2.11) becomes 𝑢(𝑡)≤⎡⎢⎢⎣𝑐^1−𝛼+(1−𝛼)^𝑡𝑡₁||𝑝(𝑠)||[𝜏(𝑠)]^𝛼𝑑𝑠+(1−𝛼)_{𝑡1≤𝜃𝑘<𝑡}||𝑞_𝑘||𝜏𝜃_𝑘^𝛼⎤⎥⎥⎦^1/(1−𝛼),(2.13) from which, on using (2.1), we have 𝑢(𝑡)≤𝑐₁,𝑡≥𝑡₂,(2.14) where 𝑐₁=⎡⎢⎢⎣𝑐^1−𝛼+(1−𝛼)^∞𝑡₁||𝑝(𝑠)||[𝜏(𝑠)]^𝛼𝑑𝑠+(1−𝛼)_{𝑡1≤𝜃𝑘<∞}||𝑞_𝑘||𝜏𝜃_𝑘^𝛼⎤⎥⎥⎦^1/(1−𝛼).(2.15) In view of (2.5), (2.6), and (2.14) we see that ||𝑥(𝑡)||≤𝑐₁𝑡,||𝑥(𝜏(𝑡))||≤𝑐₁𝜏(𝑡),𝑡≥𝑡₂.(2.16)
To complete the proof it suffices to show that 𝑥^(𝑡) approaches a nonzero limit as 𝑡 tends to ∞. To see this we integrate (1.1) from 𝑡₂ to 𝑡 to get𝑥^(𝑡)=𝑥^𝑡₁−^𝑡𝑡₂𝑝(𝑠)||𝑥(𝜏(𝑠))||^𝛼−1𝑥(𝜏(𝑠))𝑑𝑠−_{𝑡2≤𝜃𝑘<𝑡}𝑞_𝑘||𝑥𝜏𝜃_𝑘||^𝛼−1𝑥𝜏𝜃_𝑘.(2.17) Employing (2.16) we have ^∞𝑡₂||𝑝(𝑠)𝑥(𝜏(𝑠))||^𝛼𝑑𝑠≤𝑐^𝛼1^∞𝑡₂||𝑝(𝑠)||[𝜏(𝑠)]^𝛼𝑑𝑠<∞,_{𝑡2≤𝜃𝑘<∞}||𝑞_𝑘𝑥𝜏𝜃_𝑘||^𝛼≤𝑐^𝛼1_𝑡2≤_𝜃𝑘<∞||𝑞_𝑘||𝜏𝜃_𝑘^𝛼<∞.(2.18) Therefore, lim_𝑡→∞𝑥^(𝑡)=𝐿 exists. Clearly, we can make 𝐿≠0 by requiring that 𝑥^𝑡₂>𝑐^𝛼1⎡⎢⎢⎣^∞𝑡₂||𝑝(𝑠)||[𝜏(𝑠)]^𝛼𝑑𝑠+_{𝑡2≤𝜃𝑘<∞}||𝑞_𝑘||𝜏𝜃_𝑘^𝛼⎤⎥⎥⎦,(2.19) which is always possible by arranging 𝑡₂.

Theorem 2.2. Suppose that 𝑝 and {𝑞_𝑘} are nonnegative. Then every solution of (1.1) is oscillatory if and only if ^∞[𝜏(𝑡)]^𝛼𝑝(𝑡)𝑑𝑡+^∞𝜏𝜃^𝑘^𝛼𝑞_𝑘=∞.(2.20)

Proof. Let (2.20) fail to hold. Then, by Theorem 2.1 we see that there is a solution 𝑥(𝑡) 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 𝑥(𝑡) of (1.1). We may assume that 𝑥(𝑡) is eventually positive; the case 𝑥(𝑡) being eventually negative is similar. Clearly, there exists 𝑡₁≥𝑡₀ such that 𝑥(𝜏(𝑡))>0 for all 𝑡≥𝑡₁. From (1.1), we have that𝑥^(𝑡)≤0for𝑡≥𝑡₁,𝑡≠𝜃_𝑘.(2.21) Thus, 𝑥^(𝑡) is decreasing on every interval not containing 𝑡=𝜃_𝑘. From the impulse conditions in (1.1), we also have Δ𝑥^(𝜃_𝑘)≤0. Therefore, we deduce that 𝑥^(𝑡) is nondecreasing on [𝑡₁,∞).
We may claim that 𝑥^(𝑡) is eventually positive. Because if 𝑥^(𝑡)<0 eventually, then 𝑥(𝑡) becomes negative for large values of 𝑡. This is a contradiction.
It is now easy to show that𝑥(𝑡)≥𝑡−𝑡₁𝑥^(𝑡),𝑡≥𝑡₁.(2.22) Therefore, 𝑥(𝑡)≥𝑡2𝑥^(𝑡),𝑡≥𝑡₂=2𝑡₁.(2.23) Let 𝑡₃≥𝑡₂ be such that 𝜏(𝑡)≥𝑡₂ for 𝑡≥𝑡₃. Using (2.23) and the nonincreasing character of 𝑥^(𝑡), we have 𝑥(𝜏(𝑡))≥𝜏(𝑡)2𝑥^(𝑡),𝑡≥𝑡₃,(2.24) and so, by (1.1), 𝑥^(𝑡)+2^−𝛼𝑝(𝑡)[𝜏(𝑡)]^𝛼𝑥^(𝑡)^𝛼≤0,𝑡≠𝜃_𝑘.(2.25) Dividing (2.25) by [𝑥^(𝑡)]^𝛼 and integrating from 𝑡₃ to 𝑡, we obtain _𝑡3≤_{𝜃𝑘<𝑡}𝑥^𝜃_𝑘^1−𝛼+𝑥_−𝑥^𝜃_𝑘−𝑞_𝑘𝑥𝜏𝜃_𝑘^𝛼^1−𝛼(𝑡)^1−𝛼−𝑥^𝑡₃^1−𝛼+(1−𝛼)2^−𝛼^𝑡𝑡₃[𝜏(𝑡)]^𝛼𝑝(𝑠)𝑑𝑠≤0(2.26) which clearly implies that _𝑡3≤_{𝜃𝑘<𝑡}𝑎_𝑘+(1−𝛼)2^−𝛼^𝑡𝑡₃[𝜏(𝑡)]^𝛼𝑝(𝑠)𝑑𝑠≤𝑥^𝑡₃^1−𝛼,(2.27) where 𝑎_𝑘=𝑥^𝜃_𝑘^1−𝛼1−1−𝑞^𝑘𝑥𝜏𝜃_𝑘^𝛼𝑥^𝜃_𝑘^1−𝛼.(2.28) Since 1−(1−𝑢)^1−𝛼≥(1−𝛼)𝑢 for 𝑢∈(0,∞) and 0<𝛼<1, by taking 𝑢=𝑞^𝑘𝑥𝜏𝜃_𝑘^𝛼𝑥^𝜃_𝑘,(2.29) we see from (2.28) that 𝑎_𝑘≥(1−𝛼)𝑞^𝑘𝑥𝜏𝜃_𝑘^𝛼𝑥^𝜃_𝑘^𝛼.(2.30) But, (2.24) gives 𝑥𝜏𝜃_𝑘≥𝜏𝜃^𝑘2𝑥^𝜏𝜃_𝑘≥𝜏𝜃^𝑘2𝑥^𝜃_𝑘,(2.31) and hence 𝑎_𝑘≥(1−𝛼)2^−𝛼𝜏𝜃_𝑘^𝛼𝑞_𝑘.(2.32)
Finally, (2.27) and (2.32) result in^∞𝑡₃[𝜏(𝑡)]^𝛼𝑝(𝑡)𝑑𝑡+_{𝑡3<𝜃𝑘<∞}𝜏𝜃_𝑘^𝛼𝑞_𝑘<∞,(2.33) which contradicts (2.20). The proof is complete.

Example 2.3. Consider the impulsive delay differential equation 𝑥Δ^𝑥(𝑡_)+(𝑡−1)⁻²||𝑥(𝑡−1)||^−1/2𝑥(𝑡−1)=0,𝑡≠𝑘,(𝑡)||_𝑡=𝑘+(𝑘−1Δ)𝑥(⁻¹𝑡||𝑥)||(𝑘_𝑡=𝑘−1)||^−1/2𝑥(𝑘−1)=0,=0,(2.34) where 𝑡≥2 and 𝑖≥2.
We see that 𝜏(𝑡)=𝑡−1, 𝛼=1/2, 𝑝(𝑡)=(𝑡−1)⁻², and 𝑞_𝑘=(𝑘−1)⁻¹, 𝜃_𝑘=𝑘. Since^∞(𝑡−1)^−3/2𝑑𝑡+^∞(𝑘−1)^−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, since^∞(𝑡−1)^−5/2𝑑𝑡<∞,(2.36) the equation 𝑥^(𝑡)+(𝑡−1)⁻²||𝑥(𝑡−1)||^−1/2𝑥(𝑡−1)=0(2.37) has a nonoscillatory solution by Theorem 1.1.

Let us now consider (1.1) when 𝜏(𝑡)=𝑡. That is,𝑥^Δ+𝑥𝑝_(𝑡)|𝑥|^𝛼−1𝑥=0,𝑡≠𝜃_𝑘,||Δ_{𝑡=𝜃𝑘}𝑥+||𝑞_{𝑘𝑡=𝜃𝑘}|𝑥|^𝛼−1𝑥=0,=0,(2.38) where 0<𝛼<1 and 𝑝𝑞_𝑘 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(𝑡) and {𝑞_𝑘}.

Theorem 2.4. If ^∞𝑡^𝛽𝑝(𝑡)𝑑𝑡+^∞𝜃^𝛽𝑘𝑞_𝑘=∞(2.39) for some 𝛽∈[0,𝛼], then every solution of (2.38) is oscillatory.

Proof. Assume on the contrary that (2.38) has a nonoscillatory solution 𝑥(𝑡) such that 𝑥(𝑡)>0 for all 𝑡≥𝑡₀ for some 𝑡₀≥0. The proof is similar when 𝑥(𝑡) is eventually negative. We set 𝑤(𝑡)=𝑡⁻¹𝑥(𝑡)^1−𝛼,𝑡≥𝑡₀.(2.40) It is not difficult to see that 𝑤^(𝑡)=(𝛼−1)𝑡^𝛼−2[𝑥(𝑡)]^1−𝛼+(1−𝛼)𝑡^𝛼−1[𝑥(𝑡)]^−𝛼𝑥^(𝑡),𝑡≠𝜃_𝑘,(2.41) and hence Δ𝑤′||_{𝑡=𝜃𝑘}=(1−𝛼)𝑞_𝑘𝜃^𝑘𝛼−1.(2.42) From (2.41), we have 𝑡^{𝛽−1−𝛼}𝑡²𝑤^(𝑡)^=−(1𝛼−(1𝛼−)𝑡^𝛽𝛼𝑥)𝑡^(𝑡_𝛽−2)𝑥^−𝛼(𝑡)𝑥^−𝛼−1(𝑡)𝑡𝑥^(𝑡)−𝑥(𝑡)²,(2.43) and so 𝑡^{𝛽−1−𝛼}𝑡²𝑤^(𝑡)^≤(1−𝛼)𝑡^𝛽𝑝(𝑡),𝑡≠𝜃_𝑘.(2.44) In view of (2.42), by a straightforward integration of (2.44), we have ^𝑡𝑡₀𝑠^{𝛽−1−𝛼}𝑠²𝑤^(𝑠)^𝑑𝑠=𝑠^{𝛽−1−𝛼}𝑠²𝑤^(𝑠)||^𝑡𝑡₀−_{𝑡0≤𝜃𝑘<𝑡}Δ𝑡^{𝛽−𝛼+1}𝑤^(𝑡)||_{𝑡=𝜃𝑘}=−𝑡^𝑡𝑡₀(_{𝛽−𝛼+1}𝛽−1−𝛼)𝑠^𝛽−𝛼𝑤^(𝑠)𝑑𝑠𝑤^(𝑡)−𝑡^{0𝛽−𝛼+1}𝑤^𝑡₀−_{𝑡0≤𝜃𝑘<𝑡}(1−𝛼)𝑞_𝑘𝜃^𝛽𝑘−(𝛽−𝛼−1)𝑠^𝛽−𝛼𝑤(𝑠)||^𝑡𝑡₀+(𝛽−𝛼)(𝛽−𝛼−1)^𝑡𝑡₀𝑠^{𝛽−1−𝛼}𝑤(𝑠)𝑑𝑠,(2.45) which combined with (2.44) leads to 𝑡^{𝛽−𝛼+1}𝑤′(𝑡)≤+𝑡(1−^{0𝛽−𝛼+1}𝑤𝛼)′⎡⎢⎢⎣𝑡₀−_{𝑡0≤𝜃𝑘<𝑡}(𝛽−𝛼+1)𝑡^0𝛽−𝛼𝑤𝑡₀𝜃^𝛽𝑘𝑞_𝑘+^𝑡𝑡₀𝑠^𝛽𝑝(𝑠)𝑑𝑠⎤⎥⎥⎦.(2.46) Finally, by using (2.39) in the last inequality, we see that there is a 𝑡₁>𝑡₀ such that 𝑤′(𝑡)≤−𝑡^{𝛼−𝛽−1},𝑡≥𝑡₁,(2.47) which, however, implies that 𝑤(𝑡)→−∞ as 𝑡→∞, a contradiction with 𝑥(𝑡)>0. The proof is complete.

Example 2.5. Consider the impulsive differential equation 𝑥Δ^+𝑥_𝑡^−7/3|𝑥|^−1/2𝑥=0,𝑡≠𝑘,||_𝑡=𝑘+Δ𝑥𝑘||^−1/6_𝑡=𝑘|𝑥|^−1/2𝑥=0,=0,(2.48) where 𝑡≥1 and 𝑖≥1.
We have that 𝑝(𝑡)=𝑡^7/3, 𝛼=1/2, and 𝑞_𝑘=𝑘^−1/6, 𝜃_𝑘=𝑘. Taking 𝛽=1/3 we see from (2.38) that^∞𝑡⁻²𝑑𝑡+^∞𝑘^−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, since^∞𝑡⁻²𝑑𝑡<∞,(2.50) the equation 𝑥^+𝑡^−7/3|𝑥|^−1/2𝑥=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).