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Oscillation of Second Order Nonlinear Difference Equation with Continuous Variable
Journal of Mathematical Analysis and Applications 255, 349–357 (2001) doi:10.1006/jmaa.2000.7271, available online at http://www.idealibrary.com on
Oscillation of Second Order Nonlinear Difference Equation with Continuous Variable1 Zhenguo Zhang2 and Ping Bi College of Mathematics and Information Science, Hebei Teachers’ University, Shijiazhuang 050016, People’s Republic of China
and Jianfeng Chen The 54th Research Institute (Electronics) of Ministry Information Industry, Shijiazhuang 050081, People’s Republic of China Submitted by William F. Ames Received January 28, 2000
In this paper, we are mainly concerned with the second order nonlinear difference equation with continuous variable. Here, by using the iterated integral transformations, generalized Riccati transformations, and integrating factors, we give some oscillatory criteria for this equation. © 2001 Academic Press Key Words: continuous variable; oscillation; integral transformations; generalized Riccati transformations.
1. INTRODUCTION We consider the second order nonlinear difference equation with continuous variable 2τ xt + f t xt − σ = 0
(1.1)
1 Research was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Hebei Province. 2 E-mail: [email protected].
349 0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
350
zhang, bi, and chen
≥ where τ xt = xt + τ − xt, τ σ > 0, f ∈ Ct0 ∞ × R R, f tu u pt > 0 for u = 0 p ∈ CR R+ . A function xt is called the solution of (1.1) if xt = ϕt ϕ ∈ Ct0 − max τ σ t0 R and it satisfies (1.1) when t ≥ t0 . A solution xt is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. Recently, there has been an increasing interest in the study of the oscillation of the first order difference equations with continuous variable such as [1–5]. In this paper, we are mainly concerned with the second order nonlinear difference equation with continuous variable. By using the iterated integral transformations, generalized Riccati transformations, and integrating factors, we give some oscillatory criteria for (1.1).
2. MAIN RESULTS Theorem 1. Assume that qt = mint≤s≤t+2τ ps and for any t ≥ t0 , there exists a t ≥ t such that ∞
qt + iτ = ∞
(2.1)
i=0
Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. Let x(t) be an eventually positive solution of (1.1). Set ut =
t+τ
t
ds
s
s+τ
xθdθ
Thus ut > 0 u t = 2τ xt ≤ 0 u t > 0 eventually. Then (1.1) becomes u t + f t xt − σ = 0 u t + ptxt − σ ≤ 0
(2.2)
Integrating (2.2) in the region D=
t ≤ s ≤ t + τ s ≤ u ≤ s + τ
we have 2τ ut + qtut − σ ≤ 0
(2.3)
oscillation of difference equations
351
Define zt = τ ut/ut − σ. Since u t > 0, it follows that τ ut = ut + τ − ut > 0 zt > 0. Then τ zt =
ut + τ − σ2τ ut − τ ut + ττ ut − σ ut + τ − σut − σ
≤ −qt − z 2 t + τ ≤ 0 Therefore τ zt + iτ + qt + iτ + z 2 t + i + 1τ ≤ 0
i = 0 1 2 (2.4)
Summing (2.4) from 0 to k − 1 zt + kτ − zt +
k−1
qt + iτ
i=0
k−1
z 2 t + i + 1τ ≤ 0
i=0
Hence we have k−1
qt + iτ < zt
i=0
Then ∞
qt + iτ ≤ zt < ∞
i=0
which contradicts (2.1). This completes the proof. Theorem 2. Assume that there exists a positive continuous function At with τ At ≤ 0 and for any t ≥ t0 , there exists a t ≥ t such that ∞ τ At + iτ 2 At + iτ qt + iτ + 2At + iτ i=0 τ At + i − 1τ + τ = ∞ (2.5) 2At + i − 1τ Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. Let xt be a eventually positive solution of (1.1). From the proof of Theorem 1, there exists ut > 0 such that τ ut > 0 and 2τ ut + qtut − σ ≤ 0 Define
zt = At
At − τ τ ut − τ ut − σ 2At − τ
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zhang, bi, and chen
Then zt > 0 and τ At zt +τ At +τ ut +τ −σ2τ ut−τ ut +ττ ut −σ +At ut +τ −σut −σ τ At −τ −τ 2At −τ At τ ut +τ 2 ≤ τ +qt zt +τ−At At +τ ut +τ −σ τ At −τ +τ 2At −τ 2 τ At τ At −τ Atz 2 t +τ = −At qt+ +τ − 2At 2At −τ A2 t +τ
τ zt =
Therefore
τ At + iτ 2 τ zt + iτ + At + iτ qt + iτ + 2At + iτ τ At + i − 1τ At + iτz 2 t + i + 1τ +τ + ≤ 0 (2.6) 2At + i − 1τ A2 t + i + 1τ
Summing (2.6) from 0 to k − 1, we have k−1 τ At + iτ 2 At + iτ qt + iτ + zt + kτ − zt + 2At + iτ i=0 k−1 At + iτz 2 t + i + 1τ τ At + i − 1τ +τ + ≤ 0 2At + i − 1τ A2 t + i + 1τ i=0 Thus we get
τ At + iτ 2 At + iτ qt + iτ + 2At + iτ i=0 τ At + i − 1τ ≤ zt < ∞ +τ 2At + i − 1τ
∞
which contradicts (2.5). The proof is completed. Remark 1. If (2.1) or (2.5) does not hold, by using similar methods in the papers [6, 7], we can obtain some meticulous conditions for the oscillation.
oscillation of difference equations Theorem 3. Assume that qt ¯ = mint≤s≤t+2τ ps/τ2 , ∞ qs ¯ ds = ∞ t
353
(2.7)
Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. From the proof of Theorem 1, there exists ut > 0 such that u t > 0 and 2τ ut + τ2 qtut ¯ − σ ≤ 0 Define vt =
t+τ
t
ds
s+τ
s
(2.8)
uθdθ > 0
Since u t > 0, we have vt ≤ τ2 ut + 2τ vt − σ − 2τ ≤ τ2 ut − σ, and v t = 2τ ut ≤ 0. Thus v t > 0. Therefore (2.8) becomes v t + qtvt ¯ − σ − 2τ ≤ 0 Let wt = v t/vt − σ − 2τ. Then w t =
vt − 2τ − σv t − v tv t − σ − 2τ v2 t − 2τ − σ
≤ −qt ¯ −
v tv t − σ − 2τ v2 t − 2τ − σ
Since v t ≤ 0 v t > 0, it follows that 2 v t w t ≤ −qt ¯ − ≤ −qt ¯ − w2 t vt − 2τ − σ Integrating (2.9) from t to T , we have wT − wt ≤ −
T t
qs ¯ ds −
T t
Thus we get
T
t ∞ t
qs ¯ ds ≤ wt qs ¯ ds ≤ ws < ∞
which contradicts (2.7). This completes the proof.
w2 t ds
(2.9)
354
zhang, bi, and chen
Theorem 4. Assume that there exists a function At with A ∈ C t0 ∞ 0 ∞ such that A t ≤ 0 and ∞ A s2 A s ds = ∞ (2.10) As qs ¯ + + 4As2 2As t Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. By the proof of Theorem 3, there exists a function vt such that v t > 0 and v t + qtvt ¯ − σ − 2τ ≤ 0 Let wt = At v t/vt − 2τ − σ − A t/2At > 0 Then v t A t w t = A t − vt − 2τ − σ 2At v tvt − 2τ − σ − v tv t − 2τ − σ A t + At − 2 v t − 2τ − σ 2At 2 A t A t v t ≤ wt − At qt ¯ + + At 2At vt − σ − 2τ A t A t wt 2 A t = + wt − At qt ¯ + + At 2At At 2At 2 A t w2 t A t = −At qt ¯ + − + 2At 2At At Thus wT − wt ≤ −
− So t
∞
T t
t
T
As qs ¯ +
A s 2As
2
+
A s 2As
ds
w2 s ds As
A s 2 A s ds ≤ wt < ∞ As qs ¯ + + 2As 2As
which contradicts (2.10). The proof is completed. Theorem 5. assume that 0 < Q1 t =
In addition to the assumptions of Theorem 4, further
∞ t
A s 2 A s ds < ∞ (2.11) As qs ¯ + + 2As 2As
oscillation of difference equations
355
s Q12 s Q1 θ exp 2 dθ ds = ∞ As t t Aθ Then every solution of (1.1) is oscillatory.
(2.12)
but
∞
Proof. Suppose to the contrary. From the proof of Theorem 4, there exists a function wt > 0 which eventually satisfies ∞ w2 s wt ≥ Q1 t + ds As t ∞ 2 Set yt = t w s/As ds. Then y t = −w2 t/At < 0 and s s w2 s Q1 θ Q1 θ y s exp 2 dθ = − exp 2 dθ (2.13) As t Aθ t Aθ Integrating (2.13) from t to T , then T s T 2ysQ s Q1 θ Q1 θ 1 yT exp 2 dθ − yt − exp 2 dθ ds Aθ As t t t Aθ s Q θ T w2 s 1 exp 2 dθ ds =− As t t Aθ Therefore, T exp 2 s Q θ/Aθdθ
1 t yt ≥ w2 s − 2ysQ1 s ds As t Note that wt ≥ Q1 t + yt and Q1 t > 0 yt > 0. Hence w2 t − 2Q1 tyt ≥ Q12 t + y 2 t Thus we have s s ∞ Q2 s ∞ y 2 s Q1 θ Q1 θ 1 yt ≥ exp 2 dθ ds + exp 2 dθ ds As As t t Aθ t t Aθ which contradicts (2.12). The proof is complete. We construct the following sequence Qi t of functions for i = 1 2 m. Qi t is defined as ∞ A s 2 A s Q1 t = As qs ¯ + + ds 2As 2As t s ∞ Q2 s Q1 θ 1 Q2 t = exp 2 dθ ds (2.14) As t t Aθ ··· s ∞ Q2 s Qm−1 θ m−1 exp 2 Qm t = dθ ds As Aθ t t We will have the following theorem by using similar methods in the proof of Theorem 5.
356
zhang, bi, and chen
Theorem 6. If Qi t in (2.14) exists, for i = 1 2 3 m0 − 1 but Qm0 does not exist, where m0 > 0 is a positive integer, then every solution of (1.1) is oscillatory. Example 1. Consider the difference equation 1 2τ xt + xt − σ = 0 t
(2.15)
where τ > 0 σ > 0, pt = 1t , qt = mint≤s≤t+2τ ps =
1 , t+2τ
and
∞
1 = ∞ t + 2τ + iτ i=1
By Theorem 1, every solution of (2.15) is oscillatory, or
∞ t
1 ds = ∞ s + 2ττ2
By Theorem 3, every solution of (2.15) is oscillatory. Example 2. Consider the difference equation 2τ xt +
λ xt − σ = 0 t − 2τ2
t > τ
(2.16)
where τ > 0 σ > 0, qt ¯ = λ/t 2 τ2 . Choose At ≡ 1 in Theorem 6,
∞
∞
λ λ ds = 2 2 s2 τ τ t t ∞ λ2 s λ exp 2 Q2 t = dθ ds 2 τ4 s2 t t τ θ λ2 ∞ 1 = ds 2λ 21− λ2 τ τ 4 t τ2 t s Q1 t =
where λ ≥ τ2 /2, t
1 s
21−
λ τ2
ds = ∞
By Theorem 6, every solution of (2.15) is oscillatory. Remark 2 If (2.7),(2.10), and (2.12) do not hold, by using similar methods in the paper [9], we can obtain some meticulous conditions for the oscillation.
oscillation of difference equations
357
REFERENCES 1. I. P. Stavroulakis, Oscillation of delay difference equations, Comput. Math. Appl. 29 (1995), 83–88. 2. Y. Zhang and J. Yan, Oscillation criteria for difference equations with continuous arguments, Acta Math. Sinica 38 (1995), 406–411. 3. Yuzhu Zhang, Jurang Yan, and Aimin Zhao, Oscillation criteria for a difference equation, Indian J. Pure Appl. Math. 28 (1997), 1241–1249. 4. B. G. Zhang, Jun Yan, and S. K. Choi, Oscillation for difference equations with continuous variable, Comput. Math. Appl. 36 (1998), 11–18. 5. Jurang Yan and Fengqin Zhang, Oscillation for system of delay difference equation, J. Math. Anal. Appl. 230 (1999), 223–231. 6. Z. G. Zhang and Q. L. Li, Oscillation theorems for second-order advanced functional difference equations, Comput. Math. Appl. 36, No. 6 (1998), 11–18. 7. Z. G. Zhang and J. L. Zhang, Oscillation criteria for second-order advanced difference equations with “Summation Small” coefficient, Comput. Math. Appl. 138, No. 1 (1999), 25–31. 8. Z. G. Zhang and Yu Yuanhong, Oscillation of solutions for a class of nonlinear second order difference equations, Math. J. Res. Exposition 19, No. 4 (1999), 699–703. 9. J. Yan, The oscillation properties for the solutions of the second order differential equation with “internal small” coefficient, Acta Math. Sinica 30 (1987), 209–215.
Oscillation of Second Order Nonlinear Difference Equation with Continuous Variable1 Zhenguo Zhang2 and Ping Bi College of Mathematics and Information Science, Hebei Teachers’ University, Shijiazhuang 050016, People’s Republic of China
and Jianfeng Chen The 54th Research Institute (Electronics) of Ministry Information Industry, Shijiazhuang 050081, People’s Republic of China Submitted by William F. Ames Received January 28, 2000
In this paper, we are mainly concerned with the second order nonlinear difference equation with continuous variable. Here, by using the iterated integral transformations, generalized Riccati transformations, and integrating factors, we give some oscillatory criteria for this equation. © 2001 Academic Press Key Words: continuous variable; oscillation; integral transformations; generalized Riccati transformations.
1. INTRODUCTION We consider the second order nonlinear difference equation with continuous variable 2τ xt + f t xt − σ = 0
(1.1)
1 Research was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Hebei Province. 2 E-mail: [email protected].
349 0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
350
zhang, bi, and chen
≥ where τ xt = xt + τ − xt, τ σ > 0, f ∈ Ct0 ∞ × R R, f tu u pt > 0 for u = 0 p ∈ CR R+ . A function xt is called the solution of (1.1) if xt = ϕt ϕ ∈ Ct0 − max τ σ t0 R and it satisfies (1.1) when t ≥ t0 . A solution xt is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. Recently, there has been an increasing interest in the study of the oscillation of the first order difference equations with continuous variable such as [1–5]. In this paper, we are mainly concerned with the second order nonlinear difference equation with continuous variable. By using the iterated integral transformations, generalized Riccati transformations, and integrating factors, we give some oscillatory criteria for (1.1).
2. MAIN RESULTS Theorem 1. Assume that qt = mint≤s≤t+2τ ps and for any t ≥ t0 , there exists a t ≥ t such that ∞
qt + iτ = ∞
(2.1)
i=0
Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. Let x(t) be an eventually positive solution of (1.1). Set ut =
t+τ
t
ds
s
s+τ
xθdθ
Thus ut > 0 u t = 2τ xt ≤ 0 u t > 0 eventually. Then (1.1) becomes u t + f t xt − σ = 0 u t + ptxt − σ ≤ 0
(2.2)
Integrating (2.2) in the region D=
t ≤ s ≤ t + τ s ≤ u ≤ s + τ
we have 2τ ut + qtut − σ ≤ 0
(2.3)
oscillation of difference equations
351
Define zt = τ ut/ut − σ. Since u t > 0, it follows that τ ut = ut + τ − ut > 0 zt > 0. Then τ zt =
ut + τ − σ2τ ut − τ ut + ττ ut − σ ut + τ − σut − σ
≤ −qt − z 2 t + τ ≤ 0 Therefore τ zt + iτ + qt + iτ + z 2 t + i + 1τ ≤ 0
i = 0 1 2 (2.4)
Summing (2.4) from 0 to k − 1 zt + kτ − zt +
k−1
qt + iτ
i=0
k−1
z 2 t + i + 1τ ≤ 0
i=0
Hence we have k−1
qt + iτ < zt
i=0
Then ∞
qt + iτ ≤ zt < ∞
i=0
which contradicts (2.1). This completes the proof. Theorem 2. Assume that there exists a positive continuous function At with τ At ≤ 0 and for any t ≥ t0 , there exists a t ≥ t such that ∞ τ At + iτ 2 At + iτ qt + iτ + 2At + iτ i=0 τ At + i − 1τ + τ = ∞ (2.5) 2At + i − 1τ Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. Let xt be a eventually positive solution of (1.1). From the proof of Theorem 1, there exists ut > 0 such that τ ut > 0 and 2τ ut + qtut − σ ≤ 0 Define
zt = At
At − τ τ ut − τ ut − σ 2At − τ
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zhang, bi, and chen
Then zt > 0 and τ At zt +τ At +τ ut +τ −σ2τ ut−τ ut +ττ ut −σ +At ut +τ −σut −σ τ At −τ −τ 2At −τ At τ ut +τ 2 ≤ τ +qt zt +τ−At At +τ ut +τ −σ τ At −τ +τ 2At −τ 2 τ At τ At −τ Atz 2 t +τ = −At qt+ +τ − 2At 2At −τ A2 t +τ
τ zt =
Therefore
τ At + iτ 2 τ zt + iτ + At + iτ qt + iτ + 2At + iτ τ At + i − 1τ At + iτz 2 t + i + 1τ +τ + ≤ 0 (2.6) 2At + i − 1τ A2 t + i + 1τ
Summing (2.6) from 0 to k − 1, we have k−1 τ At + iτ 2 At + iτ qt + iτ + zt + kτ − zt + 2At + iτ i=0 k−1 At + iτz 2 t + i + 1τ τ At + i − 1τ +τ + ≤ 0 2At + i − 1τ A2 t + i + 1τ i=0 Thus we get
τ At + iτ 2 At + iτ qt + iτ + 2At + iτ i=0 τ At + i − 1τ ≤ zt < ∞ +τ 2At + i − 1τ
∞
which contradicts (2.5). The proof is completed. Remark 1. If (2.1) or (2.5) does not hold, by using similar methods in the papers [6, 7], we can obtain some meticulous conditions for the oscillation.
oscillation of difference equations Theorem 3. Assume that qt ¯ = mint≤s≤t+2τ ps/τ2 , ∞ qs ¯ ds = ∞ t
353
(2.7)
Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. From the proof of Theorem 1, there exists ut > 0 such that u t > 0 and 2τ ut + τ2 qtut ¯ − σ ≤ 0 Define vt =
t+τ
t
ds
s+τ
s
(2.8)
uθdθ > 0
Since u t > 0, we have vt ≤ τ2 ut + 2τ vt − σ − 2τ ≤ τ2 ut − σ, and v t = 2τ ut ≤ 0. Thus v t > 0. Therefore (2.8) becomes v t + qtvt ¯ − σ − 2τ ≤ 0 Let wt = v t/vt − σ − 2τ. Then w t =
vt − 2τ − σv t − v tv t − σ − 2τ v2 t − 2τ − σ
≤ −qt ¯ −
v tv t − σ − 2τ v2 t − 2τ − σ
Since v t ≤ 0 v t > 0, it follows that 2 v t w t ≤ −qt ¯ − ≤ −qt ¯ − w2 t vt − 2τ − σ Integrating (2.9) from t to T , we have wT − wt ≤ −
T t
qs ¯ ds −
T t
Thus we get
T
t ∞ t
qs ¯ ds ≤ wt qs ¯ ds ≤ ws < ∞
which contradicts (2.7). This completes the proof.
w2 t ds
(2.9)
354
zhang, bi, and chen
Theorem 4. Assume that there exists a function At with A ∈ C t0 ∞ 0 ∞ such that A t ≤ 0 and ∞ A s2 A s ds = ∞ (2.10) As qs ¯ + + 4As2 2As t Then every solution of (1.1) is oscillatory. Proof. Suppose to the contrary. By the proof of Theorem 3, there exists a function vt such that v t > 0 and v t + qtvt ¯ − σ − 2τ ≤ 0 Let wt = At v t/vt − 2τ − σ − A t/2At > 0 Then v t A t w t = A t − vt − 2τ − σ 2At v tvt − 2τ − σ − v tv t − 2τ − σ A t + At − 2 v t − 2τ − σ 2At 2 A t A t v t ≤ wt − At qt ¯ + + At 2At vt − σ − 2τ A t A t wt 2 A t = + wt − At qt ¯ + + At 2At At 2At 2 A t w2 t A t = −At qt ¯ + − + 2At 2At At Thus wT − wt ≤ −
− So t
∞
T t
t
T
As qs ¯ +
A s 2As
2
+
A s 2As
ds
w2 s ds As
A s 2 A s ds ≤ wt < ∞ As qs ¯ + + 2As 2As
which contradicts (2.10). The proof is completed. Theorem 5. assume that 0 < Q1 t =
In addition to the assumptions of Theorem 4, further
∞ t
A s 2 A s ds < ∞ (2.11) As qs ¯ + + 2As 2As
oscillation of difference equations
355
s Q12 s Q1 θ exp 2 dθ ds = ∞ As t t Aθ Then every solution of (1.1) is oscillatory.
(2.12)
but
∞
Proof. Suppose to the contrary. From the proof of Theorem 4, there exists a function wt > 0 which eventually satisfies ∞ w2 s wt ≥ Q1 t + ds As t ∞ 2 Set yt = t w s/As ds. Then y t = −w2 t/At < 0 and s s w2 s Q1 θ Q1 θ y s exp 2 dθ = − exp 2 dθ (2.13) As t Aθ t Aθ Integrating (2.13) from t to T , then T s T 2ysQ s Q1 θ Q1 θ 1 yT exp 2 dθ − yt − exp 2 dθ ds Aθ As t t t Aθ s Q θ T w2 s 1 exp 2 dθ ds =− As t t Aθ Therefore, T exp 2 s Q θ/Aθdθ
1 t yt ≥ w2 s − 2ysQ1 s ds As t Note that wt ≥ Q1 t + yt and Q1 t > 0 yt > 0. Hence w2 t − 2Q1 tyt ≥ Q12 t + y 2 t Thus we have s s ∞ Q2 s ∞ y 2 s Q1 θ Q1 θ 1 yt ≥ exp 2 dθ ds + exp 2 dθ ds As As t t Aθ t t Aθ which contradicts (2.12). The proof is complete. We construct the following sequence Qi t of functions for i = 1 2 m. Qi t is defined as ∞ A s 2 A s Q1 t = As qs ¯ + + ds 2As 2As t s ∞ Q2 s Q1 θ 1 Q2 t = exp 2 dθ ds (2.14) As t t Aθ ··· s ∞ Q2 s Qm−1 θ m−1 exp 2 Qm t = dθ ds As Aθ t t We will have the following theorem by using similar methods in the proof of Theorem 5.
356
zhang, bi, and chen
Theorem 6. If Qi t in (2.14) exists, for i = 1 2 3 m0 − 1 but Qm0 does not exist, where m0 > 0 is a positive integer, then every solution of (1.1) is oscillatory. Example 1. Consider the difference equation 1 2τ xt + xt − σ = 0 t
(2.15)
where τ > 0 σ > 0, pt = 1t , qt = mint≤s≤t+2τ ps =
1 , t+2τ
and
∞
1 = ∞ t + 2τ + iτ i=1
By Theorem 1, every solution of (2.15) is oscillatory, or
∞ t
1 ds = ∞ s + 2ττ2
By Theorem 3, every solution of (2.15) is oscillatory. Example 2. Consider the difference equation 2τ xt +
λ xt − σ = 0 t − 2τ2
t > τ
(2.16)
where τ > 0 σ > 0, qt ¯ = λ/t 2 τ2 . Choose At ≡ 1 in Theorem 6,
∞
∞
λ λ ds = 2 2 s2 τ τ t t ∞ λ2 s λ exp 2 Q2 t = dθ ds 2 τ4 s2 t t τ θ λ2 ∞ 1 = ds 2λ 21− λ2 τ τ 4 t τ2 t s Q1 t =
where λ ≥ τ2 /2, t
1 s
21−
λ τ2
ds = ∞
By Theorem 6, every solution of (2.15) is oscillatory. Remark 2 If (2.7),(2.10), and (2.12) do not hold, by using similar methods in the paper [9], we can obtain some meticulous conditions for the oscillation.
oscillation of difference equations
357
REFERENCES 1. I. P. Stavroulakis, Oscillation of delay difference equations, Comput. Math. Appl. 29 (1995), 83–88. 2. Y. Zhang and J. Yan, Oscillation criteria for difference equations with continuous arguments, Acta Math. Sinica 38 (1995), 406–411. 3. Yuzhu Zhang, Jurang Yan, and Aimin Zhao, Oscillation criteria for a difference equation, Indian J. Pure Appl. Math. 28 (1997), 1241–1249. 4. B. G. Zhang, Jun Yan, and S. K. Choi, Oscillation for difference equations with continuous variable, Comput. Math. Appl. 36 (1998), 11–18. 5. Jurang Yan and Fengqin Zhang, Oscillation for system of delay difference equation, J. Math. Anal. Appl. 230 (1999), 223–231. 6. Z. G. Zhang and Q. L. Li, Oscillation theorems for second-order advanced functional difference equations, Comput. Math. Appl. 36, No. 6 (1998), 11–18. 7. Z. G. Zhang and J. L. Zhang, Oscillation criteria for second-order advanced difference equations with “Summation Small” coefficient, Comput. Math. Appl. 138, No. 1 (1999), 25–31. 8. Z. G. Zhang and Yu Yuanhong, Oscillation of solutions for a class of nonlinear second order difference equations, Math. J. Res. Exposition 19, No. 4 (1999), 699–703. 9. J. Yan, The oscillation properties for the solutions of the second order differential equation with “internal small” coefficient, Acta Math. Sinica 30 (1987), 209–215.