Oscillation by Impulses for a Second Order Neutral Delay Differential Equations

Oscillation by Impulses for a Second Order Neutral Delay Differential Equations

R. Sakthivel

PG and Research Department of Mathematics, Pachaiyappa’s College, Chennai, Tamilnadu, INDIA.

email:[email protected].

(Received on: September 28, 2018) ABSTRACT

In this paper, we consider the following second order impulsive neutral delay differential equation

{[𝑟(𝑡) (𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)))^′]^′+ 𝑞(𝑡)𝑥(𝜎(𝑡)) = 0, 𝑡 ≥ 𝑡0, 𝑡 ≠ 𝑡𝑘 ,

𝑥(𝑡𝑘) = 𝐼𝑘(𝑥(𝑡_𝑘⁻)), 𝑥^′(𝑡𝑘) = 𝐽𝑘(𝑥^′(𝑡_𝑘⁻)), 𝑘 = 1,2,3, … (𝐸) where 0 ≤ 𝑡0< 𝑡1< 𝑡2< ⋯ < 𝑡𝑘… with lim

k→∞𝑡𝑘= ∞, some interesting results are obtained, which illustrate the impulses play a very important role in giving rise to the oscillations of equations.

AMS Subject Classification: 34C10, 34A37.

Keywords: Oscillation, Impulsive, Neutral Delay Differential Equations.

1. INTRODUCTION

The oscillatory behavior of delay and neutral delay differential equation has been studied by many mathematicians, see for example

, and the references cited therein. In recent years, there has been an increasing interest on the oscillatory behavior of second order delay and neutral delay differential equations with impulse action, see for example

and the references cited therein. In this paper some interesting results established, which shows that all solutions of equation oscillate when proper impulse controls are imposed.

In

, L. P. Gimenses and M. Federson studies the oscillatory behavior of the following

delay differential equation with impulses

{

𝑥

(𝑡) + 𝑓(𝑡, 𝑥(𝑡), 𝑥

(𝑡)) + 𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏)) = 0, 𝑡 ≥ 𝑡

, 𝑡 ≠ 𝑡

𝑥(𝑡

) = 𝐼

(𝑥(𝑡

)), 𝑥

(𝑡

) = 𝐽

(𝑥

(𝑡

)), 𝑘 = 1,2,3, …, 𝑥(𝑡) = 𝜙(𝑡), 𝑡

− 𝜏 ≤ 𝑡 ≤ 𝑡

,

where 0 ≤ 𝑡

< 𝑡

< 𝑡

< ⋯ < 𝑡

… with lim

k→∞

𝑡

= ∞.

In

, E. M. Bonoto, L.P. Gimenses and M. Federson prove an oscillation theorem for the second order neutral differential equation with impulses

{

[𝑟(𝑡) (𝑥(𝑡) + 𝑝(𝑡)(𝑥(𝑡 − 𝜏)))

]

+ 𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝛿)) = 0, 𝑡 ≥ 𝑡

, 𝑡 ≠ 𝑡

, 𝑥(𝑡

) = 𝐼

(𝑥(𝑡

)), 𝑥

(𝑡

) = 𝐽

(𝑥

(𝑡

)), 𝑘 = 1,2,3, …, 𝑥(𝑡) = 𝜙(𝑡), 𝑡

− 𝜎 ≤ 𝑡 ≤ 𝑡

, where 𝛿, 𝜏 are positive constants, 0 ≤ 𝑡

< 𝑡

< 𝑡

< ⋯ < 𝑡

… with lim

k→∞

𝑡

= ∞ and 𝑡

− 𝑡

> 𝜎, where 𝜎 = max{𝜎, 𝜏}.

In fact, it is known that some particular cases of (E) oscillate without the presence of impulses. Our main results, namely Theorem 1 and Theorem 2, give conditions under which system (E) remains oscillatory. In order to obtain such results, we employ some ideas from

and

.

Our paper is organized as follows. In Section 2, we present a lemma, that plays an important role in the proof of main results in Section 3. In Section 4, an interesting example is also given.

2. PRELIMINARIES

Consider the second order neutral delay impulsive differential equation of the form { [𝑟(𝑡) (𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)))

]

+ 𝑞(𝑡)𝑥(𝜎(𝑡)) = 0, 𝑡 ≥ 𝑡

, 𝑡 ≠ 𝑡

,

𝑥(𝑡

) = 𝐼

(𝑥(𝑡

)), 𝑥

(𝑡

) = 𝐽

(𝑥

(𝑡

)), 𝑘 = 1,2,3, … (1) where 0 ≤ 𝑡

< 𝑡

< 𝑡

< ⋯ < 𝑡

… with lim

k→∞

𝑡

= ∞.

Throughout this paper, we always assume that:

(i) 𝑟 ∈ 𝐶(𝑅, (0, +∞)), 𝑝 ∈ 𝑃𝐶

([𝑡

, ∞), 𝑅

), 𝑞 ∈ 𝐶([𝑡

, +∞), (0, +∞)), 𝑞(𝑡) > 0;

(ii) 𝜎, 𝜏 ∈ 𝑃𝐶

([𝑡

, ∞), 𝑅), 𝜏(𝑡) ≤ 𝑡, 𝜎(𝑡) ≤ 𝑡, 𝜏

(𝑡) = 𝜏

> 0, 𝜎

(𝑡) > 0,

t→∞

lim 𝜏(𝑡) = lim

t→∞

𝜎(𝑡) = ∞, 𝜏(𝜎(𝑡)) = 𝜎(𝜏(𝑡)) Where 𝜏

is a constant ;

(iii) 𝐼

, 𝐽

∈ 𝐶(𝑅, 𝑅) and there exist a positive constant 𝑎

, 𝑎

, 𝑏

, 𝑐

such that 𝑎

≤

≤ 𝑎

, 𝐽

(𝑥) = 𝑐

(𝑥) for all 𝑥 ≠ 0, 𝑘 = 1,2,3, … ;

(iv) 𝑝(𝑡) and 𝑝

(𝑡) are right continuous on (𝑡

, 𝑡

) with left lateral limits 𝑝(𝑡

) =

𝑐𝑘

𝑝(𝑡

) and 𝑝

(𝑡

) =

𝑐𝑘

𝑝

(𝑡

), for all 𝑘 ∈ 𝑁.

Let 𝐽 ⊂ 𝑅 be an interval, we define 𝑃𝐶(𝐽, 𝑅) = {𝑥 ∶ 𝐽 → 𝑅; x(t) is continuous everywhere

except some 𝑡

’s at which 𝑥(𝑡

) and 𝑥(𝑡

) exist and 𝑥(𝑡

) = 𝑥(𝑡

)};

𝑃𝐶

(𝐽, 𝑅) = {𝑥 ∈ 𝑃𝐶(𝐽, 𝑅): x(t) is continuously differentiable everywhere except some 𝑡

at which 𝑥

(𝑡

) and 𝑥

(𝑡

) exist and 𝑥

(𝑡

) = 𝑥

(𝑡

)}.

We start by presenting a lemma which is borrowed from [10] replacing the left continuity by the right continuity of 𝑚(𝑡) and 𝑚

(𝑡) at 𝑡

for all 𝑘 ∈ 𝑁.

Lemma 1. Suppose

(i) the sequence {𝑡

}

satisfies 0 ≤ 𝑡

< 𝑡

< 𝑡

< ⋯ < 𝑡

… with lim

𝑘→∞

𝑡

= ∞;

(ii) 𝑚, 𝑚

∶ 𝑅

→ 𝑅 are right continuous on 𝑅

\{𝑡

∶ 𝑘 ∈ 𝑁}, there exist the lateral limits 𝑚(𝑡

), 𝑚

(𝑡

), 𝑚(𝑡

), 𝑚

(𝑡

) and 𝑚(𝑡

) = 𝑚(𝑡

), 𝑘 = 1,2,3, … ;

(iii) for 𝑘 = 1,2,3, … and 𝑡 ≠ 𝑡

, we have

𝑚

(𝑡) ≤ 𝑝(𝑡)𝑚(𝑡) + 𝑞(𝑡), 𝑡 ≠ 𝑡

, (2)𝑚(𝑡

) ≤ 𝛼

𝑚(𝑡

) + 𝛽

, (3) where 𝑝, 𝑞 ∈ 𝐶(𝑅

, 𝑅), 𝛼

and 𝛽

are real constants with 𝛼

≥ 0. Then the following inequality holds

𝑚(𝑡) ≤ 𝑚(𝑡

) ∏ 𝛼

exp (∫ 𝑝(𝑠)𝑑𝑠

0

)

𝑡₀<𝑡_𝑘<𝑡

+ ∫ ∏

exp (∫ 𝑝(𝑢)𝑑𝑢

)

0

𝑞(𝑠)𝑑𝑠 +

∑ ∏ 𝛼

exp (∫ 𝑝(𝑠)𝑑𝑠

𝑘

)

𝑡𝑘<𝑡𝑗<𝑡

𝑡₀<𝑡_𝑘<𝑡

𝛽

, 𝑡 ≥ 𝑡

. (4)

For notation convenience, let

𝑦(𝑡) = 𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)). (5) Lemma 2. Let 𝑥(𝑡) be a solution of equation (1). Suppose that there exists some 𝑇 ≥ 𝑡

such that 𝑥(𝑡) > 0 for 𝑡 ≥ 𝑇. If 𝑐

≥ 1 and

(𝐻

) lim

t→∞

∫ 1

𝑟(𝑠) [ ∏ 𝑐

max {𝑎

, 𝑐

}

𝑡_𝑗0<𝑡_𝑘≤𝑠

]

𝑡

𝑡_𝑗

𝑑𝑠 = +∞,

then 𝑦

(𝑡

) ≥ 0 and 𝑦

(𝑡) = [𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡))] ≥ 0 for 𝑡 ∈ [𝑡

, 𝑡

) where 𝑡

≥ 𝑇 and 𝑘 ∈ 𝑁, 𝑦(𝑡) is defined by (5).

Proof. Suppose 𝑥(𝑡) > 0 for 𝑡 ≥ 𝑇. At first we shall prove that 𝑦

(𝑡

) ≥ 0 for any 𝑡

≥ 𝑇 and 𝑘 ∈ 𝑁. If it is not true, then there exist some 𝑗

such that 𝑡

≥ 𝑇, 𝑦

(𝑡

) < 0. Let 𝑦

(𝑡

) = −𝛼 where 𝛼 > 0.

From equation (5) and (iii), we have

𝑦

(𝑡

) = 𝑥

(𝑡

) + 𝑝

(𝑡

)𝑥(𝜏(𝑡

)) + 𝑝(𝑡

)𝑥

(𝜏(𝑡

))𝜏

(𝑡

)

= 𝐽

(𝑥

(𝑡

)) + 𝑐

𝑝

(𝑡

)𝑥(𝜏(𝑡

))

+ 𝑐

𝑝(𝑡

)𝑥

(𝜏(𝑡

))𝜏

(𝑡

) = 𝑐

𝑥

(𝑡

) + 𝑐

𝑝

(𝑡

)𝑥(𝜏(𝑡

)) + 𝑐

𝑝(𝑡

)𝑥

(𝜏(𝑡

))𝜏

(𝑡

) = 𝑐

𝑦

(𝑡

),

for all 𝑘 ∈ 𝑁.

Since 𝑥(𝑡) > 0 for 𝑡 ≥ 𝑇, without loss of generality we assume that 𝑥(𝜎(𝑡)) > 0 and 𝑥(𝜏(𝑡)) > 0, for all 𝑡 ≥ 𝑇.

𝑦(𝑡) = 𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)) > 0 for 𝑡 ≥ 𝑇.

From (1), for 𝑡 ∈ [𝑡

, 𝑡

), 𝑘 ∈ 𝑁, 𝑡

≥ 𝑇 we have

(𝑟(𝑡)𝑦

(𝑡))

= −𝑞(𝑡)𝑥(𝜎(𝑡)) ≤ 0. (6) Hence 𝑟(𝑡)𝑦

(𝑡) is non increasing function on each interval [𝑡

, 𝑡

), 𝑘 ∈ 𝑁 such that 𝑡

> 𝑇.

We now consider the impulsive differential inequality

(𝑟(𝑡)𝑦^′(𝑡))^′≤ 0, 𝑡 > 𝑡_𝑗0, 𝑡 ≠ 𝑡_𝑘, 𝑘 = 𝑗₀+ 1, 𝑗₀+ 2, … 𝑦^′(𝑡_𝑘) = 𝑐_𝑘𝑦^′(𝑡_𝑘⁻), 𝑘 = 𝑗₀+ 1, 𝑗₀+ 2 , …

Let 𝑚(𝑡) = 𝑟(𝑡)𝑦

(𝑡), then

𝑚^′(𝑡) ≤ 0, 𝑡 > 𝑡_𝑗0, 𝑡 ≠ 𝑡_𝑘, 𝑘 = 𝑗₀+ 1, 𝑗₀+ 2, … 𝑚(𝑡_𝑘) = 𝑐_𝑘𝑚(𝑡_𝑘⁻), 𝑘 = 𝑗₀+ 1, 𝑗₀+ 2 , …

By Lemma 1, we have

𝑚(𝑡) ≤ 𝑚(𝑡

) ∏ 𝑐

𝑡_𝑗0<𝑡_𝑘<𝑡

𝑖. 𝑒., 𝑦

(𝑡) ≤ 𝑟(𝑡

) 𝑦

(𝑡

)

𝑟(𝑡) ∏ 𝑐

.

𝑡_𝑗0<𝑡_𝑘<𝑡

Then, using the fact that

𝑦(𝑡

) = 𝑥(𝑡

) + 𝑝(𝑡

)𝑥(𝜏(𝑡

)) = 𝐼

(𝑥(𝑡

)) + 𝑐

𝑝(𝑡

)𝑥(𝜏(𝑡

))

≤ 𝑎

𝑥(𝑡

) + 𝑐

𝑝(𝑡

)𝑥(𝜏(𝑡

))

≤ max{𝑎

, 𝑐

}𝑦(𝑡

), 𝑘 = 𝑗

+ 1, 𝑗

+ 2, … Applying Lemma 1, we have

𝑦(𝑡) ≤ 𝑦(𝑡

) ∏ max{𝑎

, 𝑐

} + ∫ ∏ max {𝑎

, 𝑐

} [ 𝑟(𝑡

)𝑦

(𝑡

)

𝑟(𝑠) ∏ 𝑐

𝑡_𝑗0<𝑡𝑘<𝑠

]

𝑠<𝑡𝑘<𝑡 𝑡 𝑡_𝑗0 𝑡_𝑗0<𝑡𝑘<𝑡

𝑑𝑠

≤ ∏ max(𝑎

, 𝑐

) [𝑦(𝑡

) − 𝛼𝑟(𝑡

) ∫ 1

𝑟(𝑠) ∏ 𝑐

max{𝑎

, 𝑐

} 𝑑𝑠

𝑡_𝑗0<𝑡_𝑘≤𝑠 𝑡

𝑡_𝑗0

]

𝑡_𝑗0<𝑡_𝑘<𝑡

, and since 𝑦(𝑡) > 0 for 𝑡 ≥ 𝑇, the last inequality contradicts (𝐻

). Therefore 𝑦

(𝑡

) ≥ 0 for all 𝑡

, 𝑡

≥ 𝑇. Since 𝑟(𝑡)𝑦

(𝑡) is non-increasing on [𝑡

, 𝑡

), it is clear that

𝑦

(𝑡) ≥ 𝑟(𝑡

)𝑦

(𝑡

)

𝑟(𝑡) ≥ 0 for 𝑡 ∈ [𝑡

, 𝑡

), 𝑡 ≥ 𝑇.

Thus the proof of Lemma 2 is complete.

3. MAIN RESULTS

In this section, we present some sufficient conditions for the oscillation of all solutions of equations (1).

Theorem 1. Assume that condition (𝐻

) hold. If 𝑐

≥ 1 and

t→∞

lim ∫ ∏ 𝑐

𝑞(𝑠)[1 − 𝑝(𝜎(𝑠))]𝑑𝑠 = ∞. (7)

𝑠<𝑡𝑘<𝑡 𝑡 𝑡_𝑗0

then all solutions of equation (1) oscillates.

Proof. Let 𝑥(𝑡) be a non-oscillatory solution of equation (1). Without loss of generality we may assume that 𝑥(𝑡) > 0, 𝑡 ≥ 𝑇. By Lemma 2, 𝑦(𝑡) ≥ 0, 𝑦

(𝑡) ≥ 0 for 𝑡 ∈ [𝑡

, 𝑡

), where 𝑡

≥ 𝑇 and 𝑘 ∈ 𝑁.

By equation (6), we have

(𝑟(𝑡)𝑦

(𝑡))

+ 𝑞(𝑡)𝑥(𝜎(𝑡)) ≤ 0.

Since 𝑦

(𝑡) ≥ 0 for 𝑡 ∈ [𝑡

, 𝑡

]. Hence we have

𝑟(𝑡)𝑦

(𝑡) ≤ 𝑟(𝜎(𝑡))𝑦

(𝜎(𝑡)) 𝑦

(𝜎(𝑡))

𝑦

(𝑡) ≥ 𝑟(𝑡) 𝑟(𝜎(𝑡)) Now

𝑥(𝑡) = 𝑦(𝑡) − 𝑝(𝑡)𝑥(𝜏(𝑡))𝑥(𝜎(𝑡)) = 𝑦(𝜎(𝑡)) − 𝑝(𝜎(𝑡))𝑥 (𝜏(𝜎(𝑡)))

≥ 𝑦(𝜎(𝑡)) − 𝑝(𝜎(𝑡))𝑦 (𝜏(𝜎(𝑡))) ≥ [1 − 𝑝(𝜎(𝑡))]𝑦(𝜎(𝑡)) which together with equation (6), leads to

(𝑟(𝑡)𝑦

(𝑡))

≤ −𝑞(𝑡)𝑥(𝜎(𝑡)) ≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))]𝑦(𝜎(𝑡)) (𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡))

≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))]

Define

𝑤(𝑡) = 𝑟(𝑡)𝑦

(𝑡) 𝑦(𝜎(𝑡))

Then 𝑤(𝑡

) ≥ 0 (𝑘 = 1,2,3, … . ), 𝑤(𝑡) ≥ 0 for 𝑡 ≥ 𝑡

. If 𝑡 = 𝑡

𝑤

(𝑡) = [𝑟(𝑡)𝑦

(𝑡)]

𝑦(𝜎(𝑡)) − 𝑟(𝑡)𝑦

(𝑡)𝑦

(𝜎(𝑡))𝜎

(𝑡) [𝑦(𝜎(𝑡))]

≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))] − 𝑤(𝑡)𝑦

(𝜎(𝑡))𝜎

(𝑡) 𝑦(𝜎(𝑡))

≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))] − 𝑤

(𝑡)𝜎

(𝑡)

𝑟(𝜎(𝑡)) . ≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))].

At 𝑡 = 𝑡

,

𝑤(𝑡

) = 𝑟(𝑡

)𝑦

(𝑡

)

𝑦(𝜎(𝑡

)) = 𝑟(𝑡

)𝑐

𝑦

(𝑡

)

𝑦(𝜎(𝑡

)) 𝑤(𝑡

) = 𝑐

𝑤(𝑡

).

Therefore

𝑤

(𝑡) ≤ −𝑞(𝑡)[1 − 𝑝(𝜎(𝑡))], 𝑡 ≠ 𝑡

, 𝑘 = 𝑗

+ 1, 𝑗

+ 2, … . . 𝑤(𝑡

) = 𝑐

𝑤(𝑡

), 𝑡 ≠ 𝑡

. Therefore by Lemma 1, we have

𝑤(𝑡) ≤ 𝑤(𝑡

) ∏ 𝑐

− ∫ ∏ 𝑐

𝑞(𝑠)[1 − 𝑝(𝜎(𝑠))]𝑑𝑠.

𝑠<𝑡_𝑘<𝑡 𝑡

𝑡_𝑗0 𝑡_𝑗0<𝑡_𝑘<𝑡

Since 𝑤(𝑡) > 0 for 𝑡 > 𝑇, the last inequality contradicts with (7). This completes the proof.

Theorem 2. Assume that 𝜎(𝑡) ≤ 𝜏(𝑡) and (𝐻

) holds. If there exists ℎ ∈ 𝑃𝐶

([𝑡

, ∞), 𝑅

) such that

t→∞

lim ∫ ∏ 𝑐

[𝑄(𝑠)ℎ(𝑠) − 1

4 (1 + 1

𝜏

) [ℎ

(𝑠)]

𝑟(𝜎(𝑠)) ℎ(𝑠)𝜎

(𝑠) ]

𝑠<𝑡_𝑘<𝑡 𝑡

𝑡_𝑗0

𝑑𝑠 = ∞, (8) where 𝑄(𝑡) = min{𝑞(𝑡), 𝑞(𝜏(𝑡))} holds, then every solution 𝑥 of (1) is oscillatory.

Proof. Let x be a non oscillatory solution of equation (1). Without loss of generality we can assume 𝑥(𝑡) > 0 for some 𝑇 ≥ 𝑡

.

Recall that 𝑦(𝑡) = 𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)). By Lemma 2, 𝑦(𝑡) > 0, 𝑦

(𝑡) > 0 for 𝑡 ∈ [𝑡

, 𝑡

), where 𝑡

≥ 𝑇 and 𝑘 ∈ 𝑁.

Define a Riccati transformation,

𝑤(𝑡) = ℎ(𝑡) 𝑟(𝑡)𝑦

(𝑡) 𝑦(𝜎(𝑡)) . Then 𝑤(𝑡

) ≥ 0 (𝑘 = 1,2,3, … ), 𝑤(𝑡) ≥ 0 for 𝑡 ≥ 𝑡

. 𝑤

(𝑡) = ℎ

(𝑡)(𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡)) + ℎ(𝑡)(𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡)) − ℎ(𝑡)𝑟(𝑡)𝑦

(𝑡)𝑦

(𝜎(𝑡))𝜎

(𝑡) [𝑦(𝜎(𝑡))]

= ℎ

(𝑡)

ℎ(𝑡) 𝑤(𝑡) + ℎ(𝑡) (𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡)) − 𝑤(𝑡) 𝑦

(𝜎(𝑡))𝜎

(𝑡) [𝑦(𝜎(𝑡))] . Since 𝑦

(𝜎(𝑡))𝑟(𝜎(𝑡)) ≥ 𝑟(𝑡)𝑦

(𝑡), and

𝑤

(𝑡) ≤ ℎ

(𝑡)

ℎ(𝑡) 𝑤(𝑡) + ℎ(𝑡) (𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡)) − 𝑤(𝑡) 𝑟(𝑡)𝑦

(𝑡)𝜎

(𝑡) 𝑟(𝜎(𝑡))𝑦(𝜎(𝑡))

≤ ℎ

(𝑡)

ℎ(𝑡) 𝑤(𝑡) + ℎ(𝑡) (𝑟(𝑡)𝑦

(𝑡))

𝑦(𝜎(𝑡)) − 𝑤

(𝑡) 𝜎

(𝑡) ℎ(𝑡)𝑟(𝜎(𝑡)) . Similarly, we introduce another Riccati transformation,

𝑣(𝑡) = ℎ(𝑡) 𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡)) 𝑦(𝜎(𝑡)) . Then 𝑣(𝑡

) ≥ 0 (𝑘 = 1,2,3, … ), 𝑣(𝑡) ≥ 0 for 𝑡 ≥ 𝑡

. 𝑣

(𝑡) = ℎ

(𝑡) 𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡))

𝑦(𝜎(𝑡)) + ℎ(𝑡) [𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡))]

𝑦(𝜎(𝑡))

− ℎ(𝑡) 𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡))𝑦

(𝜎(𝑡))𝜎

(𝑡) [𝑦(𝜎(𝑡))]

= ℎ

(𝑡)

ℎ(𝑡) 𝑣(𝑡) + ℎ(𝑡) (𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡)))

𝑦(𝜎(𝑡)) − 𝑣(𝑡) 𝑦

(𝜎(𝑡))𝜎

(𝑡)

𝑦(𝜎(𝑡)) .

Since 𝑟(𝑡)𝑦

(𝑡) ≤ 0, 𝑦

(𝑡) > 0, 𝜎(𝑡) ≤ 𝜏(𝑡) and 𝑦

(𝜎(𝑡))𝑟(𝜎(𝑡)) ≥ 𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡)).

𝑣^′(𝑡) ≤ℎ^′(𝑡)

ℎ(𝑡)𝑣(𝑡) +ℎ(𝑡) (𝑟(𝜏(𝑡))𝑦^′(𝜏(𝑡)))^′

𝑦(𝜎(𝑡)) − 𝑣(𝑡)𝑟(𝜏(𝑡))𝑦^′(𝜏(𝑡))𝜎^′(𝑡) 𝑟(𝜎(𝑡))𝑦(𝜎(𝑡))

≤ℎ^′(𝑡)

ℎ(𝑡)𝑣(𝑡) +ℎ(𝑡) (𝑟(𝜏(𝑡))𝑦^′(𝜏(𝑡)))^′

𝑦(𝜎(𝑡)) − 𝑣²(𝑡) 𝜎^′(𝑡) ℎ(𝑡)𝑟(𝜎(𝑡)).

Now

𝑤^′(𝑡) + 1

𝜏₀²𝑣^′(𝑡) ≤ℎ^′(𝑡)

ℎ(𝑡)𝑤(𝑡) +ℎ(𝑡)(𝑟(𝑡)𝑦^′(𝑡))^′

𝑦(𝜎(𝑡)) − 𝑤²(𝑡) 𝜎^′(𝑡)

ℎ(𝑡)𝑟(𝜎(𝑡))+ 1 𝜏₀²

ℎ^′(𝑡) ℎ(𝑡) 𝑣(𝑡) + 1

𝜏₀²

ℎ(𝑡) (𝑟(𝜏(𝑡))𝑦^′(𝜏(𝑡)))^′

𝑦(𝜎(𝑡)) − 1

𝜏₀²𝑣²(𝑡) 𝜎^′(𝑡) ℎ(𝑡)𝑟(𝜎(𝑡)).

Using the definition of 𝑦(𝑡) and applying (1) we get,

(𝑟(𝑡)𝑦

(𝑡))

+ 𝑞(𝑡)𝑥(𝜎(𝑡)) + 𝑞(𝜏(𝑡))𝑥 (𝜎(𝜏(𝑡))) + 1

𝜏

(𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡)))

= 0, for 𝑡 ≠ 𝑡

, 𝑘 ∈ ℕ and thus

(𝑟(𝑡)𝑦

(𝑡))

+ 𝑄(𝑡)𝑦(𝜎(𝑡)) + 1

𝜏

(𝑟(𝜏(𝑡))𝑦

(𝜏(𝑡)))

≤ 0, 𝑡 ≠ 𝑡

, 𝑘 ∈ ℕ

𝜏₀²𝑣^′(𝑡) ≤ℎ^′(𝑡)

ℎ(𝑡)𝑤(𝑡) − 𝑤²(𝑡) 𝜎^′(𝑡)

ℎ(𝑡)𝑟(𝜎(𝑡))+ 1 𝜏₀²

ℎ^′(𝑡)

ℎ(𝑡)𝑣(𝑡) − 1 𝜏₀²

𝑣²(𝑡)𝜎^′(𝑡)

ℎ(𝑡)𝑟(𝜎(𝑡))− 𝑄(𝑡)ℎ(𝑡)

≤[ℎ^′(𝑡)]²𝑟(𝜎(𝑡)) 4ℎ(𝑡)𝜎^′(𝑡) + 1

4𝜏₀²

[ℎ^′(𝑡)]²𝑟(𝜎(𝑡))

ℎ(𝑡)𝜎^′(𝑡) − 𝑄(𝑡)ℎ(𝑡) [𝑤(𝑡) + 1 𝜏₀²𝑣(𝑡)]

′

≤ − [𝑄(𝑡)ℎ(𝑡) −1 4(1 + 1

𝜏₀²) [ℎ^′(𝑡)]²𝑟(𝜎(𝑡)) ℎ(𝑡)𝜎^′(𝑡) ].

Now, 𝑤(𝑡

) + 1

𝜏

𝑣(𝑡

) = ℎ(𝑡

)𝑟(𝑡

)𝑦

(𝑡

) 𝑦(𝜎(𝑡

)) + 1

𝜏

ℎ(𝑡

)𝑟(𝜏(𝑡

))𝑦

(𝜏(𝑡

)) 𝑦(𝜎(𝑡

))

= ℎ(𝑡

)𝑟(𝑡

)𝑐

𝑦

(𝑡

) 𝑦 (𝜎(𝑡

)) + 1

𝜏

ℎ(𝑡

)𝑟(𝜏(𝑡

))𝑐

𝑦

(𝜏(𝑡

)) 𝑦 (𝜎(𝑡

))

= 𝑐

[𝑤(𝑡

) + 1

𝜏

𝑣(𝑡

)] , 𝑘 = 𝑗

+ 1, 𝑗

+ 2, ….

By Lemma 1, 𝑤(𝑡) + 1

𝜏

𝑣(𝑡) ≤ [𝑤(𝑡

) + 1

𝜏

𝑣(𝑡

)] ∏ 𝑐

𝑡_𝑗0<𝑡_𝑘<𝑡

− ∫ ∏ 𝑐

[𝑄(𝑠)ℎ(𝑠) − 1

4 (1 + 1

𝜏

) [ [ℎ

(𝑠)]

𝑟(𝜎(𝑠)) ℎ(𝑠)𝜎

(𝑠) ] 𝑑𝑠

𝑠<𝑡𝑘<𝑡 𝑡

𝑡₀

which contradicts with (8). This completes the proof.

Corollary 3. Suppose that (𝐻

) holds. If ℎ(𝑡) = 1 in Theorem 2, then

t→∞

lim ∫ ∏ 𝑐

𝑄(𝑠)𝑑𝑠 = ∞, (9)

𝑠<𝑡_𝑘<𝑡 𝑡

𝑡_𝑗0

where 𝑄 is defined as in Theorem 2, then every solution of (1) oscillates.

Corollary 4. Suppose that (𝐻

) holds. If ℎ(𝑡) = 𝑡 in Theorem 2 and

t→∞

lim ∫ ∏ 𝑐

[𝑠𝑄(𝑠) − 1

4 (1 + 1

𝜏

) 𝑟(𝜎(𝑠))

𝑠𝜎

(𝑠) ] 𝑑𝑠 = ∞, (10)

𝑠<𝑡_𝑘<𝑡 𝑡

𝑡_𝑗0

where 𝑄 is defined as in Theorem 2, then every solution of (1) oscillates.

4 Examples

Example:1 Consider the following second order impulsive type neutral delay differential equation

{

[𝑥(𝑡) + (1 − 1

𝑡 ) 𝑥(𝑡 − 𝜋)]

′′

+ 𝑡𝑥(𝑡) = 0, 𝑡 ≥ 1, 𝑡 ≠ 2

, 𝑘 = 1,2,3, … 𝑥(2

) = ( 𝑘 + 1

𝑘 ) 𝑥((2

)

), 𝑥

(2

) = ( 𝑘 + 1

𝑘 ) 𝑥

((2

)

), 𝑘 = 1,2,3, … .

(11)

we have 𝑟(𝑡) = 1, 𝑎

= 𝑐

=

, 𝑝(𝑡) = 1 −

, 𝑞(𝑡) = 𝑡, 𝑡

= 2

, 𝑡

= 1, 𝜏(𝑡) = 𝑡 − 𝜋 and 𝜎(𝑡) = 𝑡.

Obviously,

t→∞lim∫ 1

𝑟(𝑠)[ ∏ 𝑐𝑘

max (𝑎𝑘, 𝑐𝑘)

𝑡𝑗0<𝑡𝑘<𝑠

] 𝑑𝑠

𝑡 𝑡_𝑗0

= lim

t→∞∫ 1

𝑟(𝑠)[ ∏ 𝑐_𝑘

max (𝑐𝑘, 𝑐𝑘)

𝑡𝑗0<𝑡𝑘<𝑠

]

𝑡

𝑡_𝑗0 𝑑𝑠

= lim

t→∞∫ 𝑑𝑠

𝑡

𝑡_𝑗0 = +∞.

Thus hypotheses (𝐻

) and (𝐻

) of Lemma 2 satisfied. Now

t→∞lim∫ ∏ 𝑐𝑘𝑞(𝑠)[1 − 𝑝(𝜎(𝑠))]𝑑𝑠

𝑡₀<𝑡_𝑘<𝑡 𝑡 𝑡₀

= lim

t→∞∫ ∏ 𝑘 + 1 𝑘 𝑑𝑠

𝑡₀<𝑡_𝑘<𝑡 𝑡 1

= lim

t→∞∫ ∏ 𝑘 + 1

𝑡₀<𝑡_𝑘<𝑡 𝑘

𝑡₁

1 𝑑𝑠 + ∫ ∏ 𝑘 + 1

𝑡₀<𝑡_𝑘<𝑡 𝑘

𝑡₂

𝑡₁ 𝑑𝑠 + ⋯ = ∞.

Therefore, by Theorem 1, every solution of equation (11) oscillates.

Example: 2 Consider the following second order impulsive type neutral delay differential equation

{

𝑡𝑥(𝑡) + 7𝑥(𝑡 − 1) + 𝑡𝑥 ( 𝑡

2 ) = 0, 𝑡 ≥ 1, 𝑡 ≠ 2

, 𝑘 = 1,2,3, … 𝑥(2

) = ( 𝑘 + 1

𝑘 ) 𝑥((2

)

), 𝑥

(2

) = ( 𝑘 + 1

𝑘 ) 𝑥

((2

)

), 𝑘 = 1,2,3, … .

(12)

We have,

and 𝜎(𝑡) =

2

. Obviously,

t→∞

lim ∫ 1

𝑟(𝑠) [ ∏ 𝑐

max (𝑎

, 𝑐

)

𝑡_𝑗0<𝑡_𝑘<𝑠

]

𝑡 𝑡_𝑗0

𝑑𝑠 = lim

t→∞

∫ 1

𝑠 [ ∏ 𝑐

max (𝑐

, 𝑐

)

𝑡_𝑗0<𝑡_𝑘<𝑠

]

𝑡 𝑡_𝑗0

𝑑𝑠

= lim

t→∞

∫ 1

𝑠 𝑑𝑠

𝑡

𝑡_𝑗0

= +∞.

Thus hypotheses (𝐻

) and (𝐻

) of Lemma 2 satisfied.

Now,

t→∞

lim ∫ ∏ 𝑐

𝑄(𝑠)𝑑𝑠 = ∞.

𝑡_𝑗0<𝑡_𝑘<𝑠 𝑡

𝑡_𝑗0

It follows from Corollary 3 that every solution of (12) is oscillatory.

REFERENCES

1. L. Berenzansky and E. Braverman, On oscillation of a second order impulsive delay differential equation, J. Math. Anal. Appl. 39, 217-225 (2000).

2. E.M. Bonotto, L.P. Gimenses and M. Federson, Oscillation for a second order neutral differential equation with impulses, Cadernos De Matematica 9, 169-190, October(2008).

3. Jin-Fa Cheng and Yu-Ming Chu, Oscillation of second order neutral impulsive differential equations, Journal of Inequalities and Applications, volume 2010, 29 pages.

4. L. P. Gimenses and Marica Federson, Oscillation by impulses for a second order delay differential equation, Cardenos De Matematica 06, 181-191, October (2005).

5. K. Gopalsamy and B.G. Zhang, On delay differential equations with impulses, J. Math.

Anal. Appl, 139, 110-122 (1989).

6. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, (1991).

7. Chengjun Guo and Zhiting Xu, On the oscillation of second order linear impulsive differential equations, Differntial Equtions and Applications, volume 2, No 3 (2010).

8. Zhenlai Han, Tongxing Li, Shurong Sun and Weisong Chen, On the oscillation of second

order neutral delay differential equations, Advances in Difference Equations, volume

2010, 8 pages.

9. Zhimin He and Weigao Ge, Oscillation of impulsive delay differential equations, Indian.

J. Pure. Appl. Math. 31(9), 1089-1101, September (2000).

10. Lakshmikantham. Y, Bainov. D, Simenov. P.S., Theory of Impulsive Differential Equations, World Scientific, Singapore 32 (1989).

11. Wu Xiu-Li, Chen si-Yang and Hong Ji, Oscillation of a class of second order nonlinear ODE with impulses, Appl. Math. Comput. 138, No 2-3, 181-188 (2003).

12. Xiaodi Li, Oscillation properties of second order delay differential - difference equations with impulses, Advances in Applied Mathematical Analysis, volume3 Number 1, 55-66 (2008).

13. Qiong Meng and Jurang Yan, Bounded oscillation for second order non- linear neutral delay differential equations in critical and non-critical cases, Nonlinear Analysis 64, 7543- 1561 (2006).

14. Mingshu Peng, Oscillation caused by impulses, J. Math. Anal. Appl. 255, 163-176. (2001).

15. Mingshu Peng and Weigao Ge, Oscillation criteria for second order non- linear differential equations with impulses, Computer and Mathematics With Applications 39, 217-255, (2000).

16. Mingshu Peng and R.P. Agarwal, Oscillation theorems of second order non- linear neutral delay differential equations under impulsive perturbations, Indian. J. Pure. Appl. Math 33(7), 1017-1029, July (2002).

17. Yuri.V. Rogovchenko, Oscillation of a second order nonlinear delay differential equations, Funkcialaj Ekvacioj 43, 1-29, (2000).

18. Qigui Yang, Lijun Yang and Siming Zhu, Internal criteria for oscillation of second order nonlinear neutral differential equations, Computers and Mathematics With Applications 46, 903-918 (2003).

19. Jurang Yan, Oscillation properties of a second order impulsive delay differential equation, Computers and Mathematics with Applications 47, 253-258 (2004).

20. Y. Zhang, A. Zhao and J. Yan, Oscillation criteria for impulsive delay differential equations, J. Math. Anal. Appl. 205, 461-470 (1997).

21. A. Zhao and J.Yan, Existence of positive solutions for delay differential equations with

impulses, J. Math. Anal. Appl. 210, 667-678 (1997).