Volume 2012, Issue 1 285051

Research Article

Open Access

Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Zhonghai Guo,

Zhonghai Guo

Department of Mathematics, Xinzhou Teachers University, Shanxi Xinzhou 034000, China xztc.edu.cn

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Xiaoliang Zhou,

Corresponding Author

Xiaoliang Zhou

[email protected]

Department of Mathematics, Guangdong Ocean University, Guangdong Zhanjiang 524088, China gdou.edu.cn

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Wu-Sheng Wang,

Wu-Sheng Wang

Department of Mathematics, Hechi University, Guangxi Yizhou 546300, China hcnu.edu.cn

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Zhonghai Guo,

Zhonghai Guo

Department of Mathematics, Xinzhou Teachers University, Shanxi Xinzhou 034000, China xztc.edu.cn

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Xiaoliang Zhou,

Corresponding Author

Xiaoliang Zhou

[email protected]

Department of Mathematics, Guangdong Ocean University, Guangdong Zhanjiang 524088, China gdou.edu.cn

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Wu-Sheng Wang,

Wu-Sheng Wang

Department of Mathematics, Hechi University, Guangxi Yizhou 546300, China hcnu.edu.cn

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First published: 13 August 2012

https://doi.org/10.1155/2012/285051

Citations: 3

Academic Editor: Alfredo Bellen

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Abstract

We study the following second-order super-half-linear impulsive differential equations with delay [r(t)φ_γ(x′(t))]′ + p(t)φ_γ(x(t − σ)) + q(t)f(x(t − σ)) = e(t), t ≠ τ_k, x(t⁺) = a_kx(t), x′(t⁺) = b_kx′(t), t = τ_k, where t ≥ t₀ ∈ ℝ, φ_*(u) = |u|^*−1u, σ is a nonnegative constant, {τ_k} denotes the impulsive moments sequence with τ₁ < τ₂ < ⋯<τ_k < ⋯, lim _k→∞τ_k = ∞, and τ_k+1 − τ_k > σ. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

1. Introduction

We consider the following second-order super-half-linear impulsive differential equations with delay

(1.1)

where t ≥ t₀ ∈ ℝ, φ_*(u) = |u|^*−1u, σ is a nonnegative constant, {τ_k} denotes the impulsive moments sequence with τ₁ < τ₂ < ⋯<τ_k < ⋯, lim _k→∞τ_k = ∞, and τ_k+1 − τ_k > σ.

Let J ⊂ ℝ be an interval, we define

(1.2)

For given t₀ and ϕ ∈ PLC([t₀ − σ, t₀], ℝ), we say x ∈ PLC([t₀ − σ, ∞), ℝ) is a solution of (1.1) with initial value ϕ if x(t) satisfies (1.1) for t ≥ t₀ and x(t) = ϕ(t) for t ∈ [t₀ − σ, t₀].

A solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory.

Impulsive differential equation is an adequate mathematical apparatus for the simulation of processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth. Because it has more richer theory than its corresponding nonimpulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).

In the last decades, there has been an increasing interest in obtaining sufficient conditions for oscillation and/or nonoscillation of different classes impulsive differential equations with delay (constant or variable), see, for example, [1–9] and the references cited therein.

In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers, see [10–15].

However, for the impulsive equations almost all of interval oscillation results in the existing literature were established only for the case of “without delay,” in other words, for the case of “with delay” the study on the interval oscillation is very scarce. To the best of our knowledge, Huang and Feng [16] gave the first research in this direction recently. They considered the second-order delay differential equations with impulses

(1.3)

and established some interval oscillation criteria which developed some known results for the equation without delay or impulses [13, 17, 18].

Motivated mainly by [16], in this paper, we study the interval oscillation of the delay impulsive equation (1.1). By using classical inequalities, Riccati transformation, and two classes of functions (introduced first by Philos [19]), we establish some interval oscillation criteria which generalize and improve some known results of [13, 16–18]. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

2. Main Results

Throughout the paper, we always assume that the following conditions hold:

(A₁ r(t) ∈ C([t₀, ∞), (0, ∞)) and is nondecreasing, p(t), q(t), e(t) ∈ PLC([t₀, ∞), ℝ);
A₂ γ is a quotient of odd positive integers a_k, b_k are real constants satisfying b_k ≥ a_k > 0, k = 1,2, …;
A₃ f ∈ C(ℝ, ℝ), xf(x) > 0 and there exist some positive constants η and α such that f(x)/φ_α(x) > η for all x ≠ 0 with α ≥ γ.

We introduce the following notations at intervals [c₁, d₁] and [c₂, d₂]

(2.1)

For two constants c, d ∉ {τ_k} with c < d, k(c) < k(d) and a function φ ∈ C([c, d], ℝ), we define an operator Q : C([c, d], ℝ) → ℝ by

(2.2)

where

if s > t.

In the discussion of the impulse moments of x(t) and x(t − σ), we need to consider the following cases for k(c_j) < k(d_j),

(S1)
and ; (S2) and ;
(S3)
and ; (S4) and , and the cases for k(c_j) = k(d_j);
; (2) ; (3) .

Combining (S*) with (*), we can get 12 cases. In order to save space, throughout the paper, we study (1.1) under the case of combination of (S1) with (1) only. The discussions for other cases are similar and omitted.

The following preparatory lemmas will be useful to prove our theorems. The first is derived from [20] and the second from [21].

Lemma 2.1. Let λ and δ be positive real numbers with λ > δ. Then

(2.3)

for all A, B ≥ 0 and X > 0.

Lemma 2.2. Suppose X and Y are nonnegative, then

(2.4)

where equality holds if and if X = Y.

Let γ > 0, B ≥ 0, A > 0 and y ≥ 0. Put

(2.5)

It follows from Lemma 2.2 that

(2.6)

Lemma 2.3. Assume that for any T ≥ t₀, there exists c_j, d_j ∉ {τ_k}, j = 1, 2, such that T < c₁ < d₁ ≤ c₂ < d₂ and

(2.7)

If x(t) is a nonoscillatory solution of (1.1), then there exist the following estimations of x(t − σ)/x(t):

(2.8)

where i = k(c_j), …, k(d_j) − 1, j = 1, 2.

Proof. Without loss of generality, we assume that x(t) > 0 and x(t − σ) > 0 for t ≥ t₀. In this case the selected interval of t is [c₁, d₁]. From (1.1) and (2.7), we obtain

(2.9)

Hence r(t)φ_γ(x′(t)) is nonincreasing on the interval [c₁, d₁]∖{τ_k}.

Case (a) (if τ_i + σ < t ≤ τ_i+1, then (t − σ, t)⊂(τ_i, τ_i+1]). Thus there is no impulsive moment in (t − σ, t). For any s ∈ (t − σ, t), we have

(2.10)

Since

, the function φ_γ(·) is an increasing function and r(s)φ_γ(x′(s)) is non-increasing on (τ_i, τ_i+1), we have

(2.11)

From (2.11) and the conditions r(s) > 0 and r(s) is nondecreasing, we have

(2.12)

Thus

(2.13)

Integrating both sides of above inequality from t − σ to t, we obtain

(2.14)

Case (b) (if τ_i < t < τ_i + σ, then τ_i − σ < t − σ < τ_i < t < τ_i + σ). There is an impulsive moment τ_i in (t − σ, t). For any t ∈ (τ_i, τ_i + σ), we have

(2.15)

Using the impulsive condition of (1.1) and the monotone properties of r(t), φ_γ(·) and r(t)φ_γ(x′(t)), we get

(2.16)

Since x(τ_i) > 0, we have

(2.17)

In addition,

(2.18)

Similar to the analysis of (2.11)–(2.14), we have

(2.19)

From (2.17) and (2.19) and note that the monotone properties of φ_γ(·) and r(t), we get

(2.20)

In view of (A₃), we have

(2.21)

On the other hand, using similar analysis of (2.11)–(2.19), we get

(2.22)

Integrating (2.22) from t − σ to τ_i, where t ∈ (τ_i, τ_i + σ), we have

(2.23)

From (2.21) and (2.23), we obtain

(2.24)

Case (c) (). Since , then . So, there is no impulsive moment in (t − σ, t). Similar to (2.14) of Case (a), we have

(2.25)

Case (d) (). Since , then . Hence, there is an impulsive moment in (t − σ, t). Making a similar analysis of Case (b), we obtain

(2.26)

When x(t) < 0, we can choose interval [c₂, d₂] to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Theorem 2.4. Assume that for any T ≥ t₀, there exists c_j, d_j ∉ {τ_k}, j = 1,2, such that T < c₁ < d₁ ≤ c₂ < d₂ and (2.7) holds. If there exists w_j(t) ∈ Ω_j(c_j, d_j) (j = 1,2) such that, for k(c_j) < k(d_j),

(2.27)

and for k(c_j) = k(d_j),

(2.28)

where

, then (1.1) is oscillatory.

We define

(2.29)

Differentiating u(t) and in view of (1.1) we obtain, for t ≠ t_k,

(2.30)

Putting A = ηq(t), B = |e(t)|, X = x(t − σ), λ = α, and δ = γ, by Lemma 2.1, we see that

(2.31)

where R(t) = p(t) + αγ^−γ/α(α − γ) ^γ/α−1(ηq(t)) ^γ/α(t) | e(t)|^1−γ/α.

First, we consider the case k(c₁) < k(d₁).

In this case, we assume impulsive moments in [c₁, d₁] are . Choosing a w₁(t) ∈ Ω₁(c₁, d₁), multiplying both sides of (2.31) by and then integrating it from c₁ to d₁, we obtain

(2.32)

where

. Using the integration by parts on the left side of above inequality and noting the condition w₁(c₁) = w₁(d₁) = 0, we obtain

(2.33)

Letting

, B = (γ + 1) | w₁′(t)|, A = γ/r^1/γ(t) and using (2.6), for the integrand function in above inequality we have that

(2.34)

Meanwhile, for t = τ_k, k = 1, 2, …, we have

(2.35)

Hence

(2.36)

Therefore, we get

(2.37)

On the other hand, for t ∈ (τ_i−1, τ_i]⊂[c₁, d₁], i = k(c₁) + 2, …, k(d₁), we have

(2.38)

In view of x(τ_i−1) > 0, and note that the monotone properties of φ_γ(·), r(t)φ_γ(x′(t)), and r(t), we obtain

(2.39)

This is

(2.40)

Let

, it follows

(2.41)

Similar analysis on

, we can get

(2.42)

Then from (2.41), (2.42), and (A₂), we have

(2.43)

where

and

. From (2.37) and (2.43) and applying Lemma 2.3, we obtain

(2.44)

This contradicts (2.27).

Next we consider the case k(c₁) = k(d₁). By the condition (1) we know there is no impulsive moment in [c₁, d₁]. Multiplying both sides of (2.31) by and integrating it from c₁ to d₁, we obtain

(2.45)

Using the integration by parts on the left-hand side and noting the condition w₁(c₁) = w₁(d₁) = 0, we obtain

(2.46)

It follows that

(2.47)

Letting

, and y = |u(t)| and applying the inequality (2.6), we get

(2.48)

Using same way as Case (a) we get

(2.49)

From (2.48) and (2.49) we obtain

(2.50)

This contradicts our assumption (2.28).

When x(t) < 0, we can choose interval [c₂, d₂] to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Remark 2.5. When γ = 1, p(t) = 0, f(x) = |x|^α−1x and the delay term σ = 0, (1.1) reduces to that studied by Liu and Xu [13]. Therefore our Theorem 2.4 generalizes Theorem 2.1 of [13].

Remark 2.6. When γ = α = 1, r(t) = 1 and p(t) = 0, (1.1) reduces to the (1.3) studied by Huang and Feng [16]. Therefore our Theorem 2.4 extends Theorem 2.1 of [16].

Remark 2.7. When a_k = b_k = 1, for all k = 1,2, …, the impulses in (1.1) disappear, Theorem 2.4 reduces to the main results of [17, 18].

In the following we will establish a Kemenev type interval oscillation criteria for (1.1) by the ideas of Philos [19] and Kong [22].

Let D = {(t, s) : t₀ ≤ s ≤ t}, H₁, H₂ ∈ C¹(D, ℝ), then a pair function H₁, H₂ is said to belong to a function set ℋ, defined by (H₁, H₂) ∈ ℋ, if there exists h₁, h₂ ∈ L_loc(D, ℝ) satisfying the following conditions:

A₄ H₁(t, t) = H₂(t, t) = 0, H₁(t, s) > 0, H₂(t, s) > 0 for t > s;
A₅ (∂/∂t)H₁(t, s) = h₁(t, s)H₁(t, s), (∂/∂s)H₂(t, s) = h₂(t, s)H₂(t, s).

We assume there exist c_j, d_j, δ_j ∉ {τ_k, k = 1,2, …} (j = 1,2) which satisfy T < c₁ < δ₁ < d₁ ≤ c₂ < δ₂ < d₂ for any T ≥ t₀. Noticing whether or not there are impulsive moments of x(t) in [c_j, δ_j] and [δ_j, d_j], we should consider the following four cases, namely,

(S1)
k(c_j) < k(δ_j) < k(d_j); (S2) k(c_j) = k(δ_j) < k(d_j);
(S3)
k(c_j) < k(δ_j) = k(d_j); (S4) k(c_j) = k(δ_j) = k(d_j).

Moreover, in the discussion of the impulse moments of x(t − σ), it is necessary to consider the following two cases:

; (2) .

In the following theorem, we only consider the case of combination of (S1) with (1). For the other cases, similar conclusions can be given and their proofs will be omitted here.

For convenience in the expressing blow, we define

(2.51)

where

(j = 1,2) and R(t) = p(t) + αγ^−γ/α(α − γ) ^γ/α−1η^γ/αq^γ/α(t) | e(t)|^1−γ/α.

Theorem 2.8. Assume that for any T ≥ t₀, there exists c_j, d_j ∉ {τ_k}, j = 1,2, such that T < c₁ < d₁ ≤ c₂ < d₂ and if there exists a pair of (H₁, H₂) ∈ ℋ such that

(2.52)

then (1.1) is oscillatory.

Proof. Assume, to the contrary, that x(t) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t) > 0 and x(t − σ) > 0 for t ≥ t₀. In this case the interval of t selected for the following discussion is [c₁, d₁]. Using the same proof as in Theorem 2.4, we can get (2.31). Multiplying both sides of (2.31) by H₁(t, c₁) and integrating it from c₁ to δ₁, we have

(2.53)

where

. Noticing impulsive moments

are in [c₁, δ₁] and using the integration by parts on the left-hand side of above inequality, we obtain

(2.54)

Substituting (2.54) into (2.53), we have

(2.55)

Letting A = γ/r^1/γ(t), B = |h₁(t, c₁)|, y = |u(t)| and using (2.6) to the right side of above inequality, we have

(2.56)

Because there are different integration intervals in (2.56), we need to divide the integration interval [c₁, δ₁] into several subintervals for estimating the function x(t − σ)/x(t). Using Lemma 2.2, (2.6), we get estimation for the left-hand side of above inequality as follows,

(2.57)

From (2.56) and (2.57), we have

(2.58)

Multiplying both sides of (2.31) by H₂(d₁, t) and using similar analysis to the above, we can obtain

(2.59)

Dividing (2.58) and (2.59) by H₁(δ₁, c₁) and H₂(d₁, δ₁), respectively, and adding them, we get

(2.60)

On the other hand, similar to (2.43), we have

(2.61)

From (2.60) and (2.61), we can obtain a contradiction to the condition (2.52).

When x(t) < 0, we can choose interval [c₂, d₂] to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Remark 2.9. Let H₁(t, s) = H₂(t, s) = H(t, s), and , the conditions (A₄), (A₅) can be changed into

A₆ H(t, t) = 0, H(t, s) > 0, for t > s;
A₇ , .

We know that (A₆) and (A₇) are the main assumptions used in [13, 16] to obtain Kemenev type oscillation criteria. Therefore, Theorem 2.8 is a generalization of Theorem 2.3 in [13] and Theorem 2.5 in [16].

3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.

Example 3.1. Consider the following equation

(3.1)

where τ_k,i = 2kπ + (i − 1)(π/2) + (−1) ⁽ⁱ⁻¹⁾(π/8), i = 1,2, k = 1,2, …, t ≥ 0, γ = 3, α = 6, ν₁ and ν₂ are positive constants.

For any T > 0, we can choose large n₀ ∈ ℕ such that

(3.2)

There are impulsive moments τ_n,1 = 2nπ + π/8 in [c₁, d₁] and τ_n,2 = 2nπ + 3π/8 in [c₂, d₂]. From τ_n,2 − τ_n,1 = π/4 > π/12 and τ_n+1,1 − τ_n,2 = 13π/8 > π/12 for all n > n₀, we know that condition τ_k+1 − τ_k > σ is satisfied. Moreover, we also see the conditions (S1) and (2.7) are satisfied.

Let w(t) = sin 12t. It is easy to get that . In view of as k(c₁) + 1 > k(d₁) − 1, by a simple calculation, the left side of (2.27) is the following

(3.3)

On the other hand, we have

(3.4)

Thus condition (2.27) is satisfied for t ∈ [c₁, d₁] if

(3.5)

Similarly, for t ∈ [c₂, d₂] we can get the following condition

(3.6)

which ensures (2.27). Hence, by Theorem 2.4, if (3.5) and (3.6) hold, (3.1) is oscillatory. It is easy to see that (3.5) and (3.6) may be satisfied when ν₁ or ν₂ is large enough. Particularly, let a_n,i = b_n,i, for i = 1,2 and n = 1,2, …, condition (3.5) and (3.6) become a simple form

(3.7)

Example 3.2. Consider the following equation

(3.8)

where τ_k: τ_n,1 = 8n + 1/2, τ_n,2 = 8n + 3/2, τ_n,3 = 8n + 9/2, τ_n,4 = 8n + (11/2) (n = 0,1, 2, …), γ = 1, α = 2, σ = 2/3, b_k ≥ a_k > 0, μ₁, μ₂ (>0) is a constant. Clearly, τ_k+1 − τ_k > σ. In addition, let

(3.9)

For any t₀ > 0, we choose n large enough such that t₀ < 8n and let [c₁, d₁] = [8n, 8n + 2], [c₂, d₂] = [8n + 4,8n + 6], δ₁ = 8n + 1 and δ₂ = 8n + 5. Then p(t), q(t), and e(t) on [c₁, d₁] and [c₂, d₂] satisfy (2.7). Let H₁(t, s) = H₂(t, s) = (t − s) ³, then h₁(t, s) = −h₂(t, s) = 3/(t − s). By simple calculation, we get

(3.10)

Then the left side of the inequality (2.52) is

(3.11)

Because M₁ = M₂ = 1, and , it is easy to get that the right side of the inequality (2.52) for j = 1 is

(3.12)

Thus (2.52) is satisfied for j = 1 if

(3.13)

When j = 2, with the similar argument above we get that the left side of inequality (2.52) is

(3.14)

and the right side of the inequality (2.52) is

(3.15)

Therefore (2.52) is satisfied for j = 2 if

(3.16)

Hence, by Theorem 2.8, (3.8) is oscillatory if (3.13) and (3.16) hold. Particularly, let a_n,i = b_n,i, for i = 1,2, 3,4, condition (3.13) and (3.16) become a simple form

(3.17)

Acknowledgments

The authors thank the anonymous reviewers for their detailed and insightful comments and suggestions for improvement of paper. This work was supported by the NNSF of China (11161018), and the NSF of Guangdong Province (10452408801004217).

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Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Abstract

1. Introduction

2. Main Results

3. Examples

Acknowledgments

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Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Abstract

1. Introduction

2. Main Results

3. Examples

Acknowledgments

References

Citing Literature

References

Related

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