oscillation criteria of solution for a second order...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2010, Article ID 171234, 6 pagesdoi:10.1155/2010/171234

Research ArticleOscillation Criteria of Solution for a Second OrderDifference Equation with Forced Term

Chen Huiqin1 and Jin Zhen2

1 School of Mathematics and Computer Sciences, Datong University, Datong, Shanxi 037009, China2 Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Correspondence should be addressed to Jin Zhen, [email protected]

Received 24 April 2010; Accepted 21 October 2010

Academic Editor: Guang Zhang

Copyright q 2010 C. Huiqin and J. Zhen. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We will consider oscillation criteria for the second order difference equation with forced termΔ(anΔ(xn + λxn−τ )) + qnxn−σ = rn (n ≥ 0). We establish sufficient conditions which guaranteethat every solution is oscillatory or eventually positive solutions converge to zero.

In the last thirty years, there has been an increasing interest in the study of oscillation andasymptotic behavior of solutions of second order difference equations (see [1–11]). In [1],Arul and Thandapani considered the equation

Δ(pnφ(Δxn)

)+ f(n, xn+1) = 0, n = 0, 1, 2, . . . , (1)

and gave some sufficient conditions for the existence of positive solutions. In [3], Sakerconsidered the equation

Δ(pnΔxn

)+ qnf(xn−σ) = 0, n = 0, 1, 2, . . . , (2)

and gave some sufficient conditions which guarantee that every solution is oscillatory.Following this trend, we are concerned with oscillation criteria of solutions for a second orderdifference equation with forced term

Δ(anΔ(xn + λxn−τ)) + qnxn−σ = rn, n = 1, 2, . . . , (3)

2 Discrete Dynamics in Nature and Society

where {an} is a positive sequence, {qn} is a nonnegative sequence and not identically zerofor all large n, {rn} is a real sequence, λ is a real number, and σ, τ are nonnegative integers,μ = max{σ, τ}.

A solution {xn} of (3) is said to be eventually positive if xn > 0 for all large n andeventually negative if xn < 0 for all large n. Equation (3) is said to be oscillatory if it is neithereventually positive nor eventually negative.

In order to obtain our conclusions, we first give two lemmas.

Lemma 0.1. If difference inequality

Δ(anΔzn) + qnzn−σ ≤ rn, n > 0, (4)

is oscillation, then difference equation

Δ(anΔzn) + qnzn−σ = rn, n > 0, (5)

is oscillation.

Otherwise, if (5) has eventually positive solution, then (4) has eventually positivesolution; this is contradictory.

Lemma 0.2. Suppose that {xn} is an eventually positive solution of (3), λ ≥ 0, and

(i)∑∞

n=1(1/an) = +∞,

(ii)∑∞

n=1 qn = +∞,

(iii)∑∞

n=1 rn < ∞.

Set zn = xn + λxn−τ . Then zn > 0 and limn→∞anΔzn = 0

Proof. Suppose that {xn} is an eventually positive solution of (3), then there exists n1 > μ,such that xn > 0, xn−τ > 0, and xn−σ > 0 for n > n1, then zn > 0 for n > n1. By summing up (3)from n1 to n, we obtain

an+1Δzn+1 − an1Δzn1 +n∑

s=n1

qsxs−σ =n∑

s=n1

rs. (6)

From (6), we know that limn→∞∑n

n=n1qsxs−σ = α, where α is a positive limited number or

α = +∞. Thus limn→∞anΔzn = β, β is a limited number or β = −∞.If β < c < 0 (c is a constant), then there exist n2 ≥ n1, anΔzn ≤ c for n ≥ n2, so that

zn+1 ≤ zn2 + cn∑

s=n2

1as

, (7)

which is contrary to zn > 0.

Discrete Dynamics in Nature and Society 3

If β > 0, then there exist n3 ≥ n1, anΔzn > β/2 for n ≥ n3; hence,

zn+1 ≥ zn3 +β

2

n∑

s=n3

1as

−→ +∞ (n −→ ∞), (8)

therefore, limn→∞xn = ∞, limn→∞xn−σ = ∞; thus, there exist n4 ≥ n3, xn ≥ M, and xn−σ ≥ M(M > 0) for n ≥ n4. By summing up (3) from n4 to, we obtain

an+1Δzn+1 − an4Δzn4 +Mn∑

s=n4

qs ≤n∑

s=n4

rs. (9)

As n → ∞, the right-hand side of (9) is bounded, but the left-hand side of (9) tends to ∞;this is contradictory.

Then β = 0; thus limn→∞anΔzn = 0. This completes the proof.

By means of Lemma 0.2, we obtain the following.

Theorem 0.3. If conditions (i), (ii), and (iii) hold and {xn} is an eventually positive solution of (3),then limn→∞xn = 0.

Proof. Making use of (6) and the conclusion of Lemma 0.2, we know

limn→∞

n∑

s=n1

qsxs−σ = α (0 < α < +∞), (10)

so limn→∞xn = 0. If not, suppose that limn→∞xn = l > 0, then there exist n5 > n1, xn ≥ l/2 > 0for n > n5. Now substitute xn ≥ l/2 > 0 for xn in (6), we obtain a contrary. This completes theproof.

Theorem 0.4. If conditions (i), (ii), and (iii) hold, let

wn =∞∑

s=n

1as

∞∑

t=srt, n > 0, (11)

and if {wn} is oscillation, then (3) is oscillation.

Proof. Suppose that {xn} is an eventually positive solution of (3), then there exist n1 > μ,xn > 0, xn−τ > 0, and xn−σ > 0 for n ≥ n1. From (6), we have

an+1Δzn+1 − an1Δzn1 <n∑

s=n1

rs. (12)

Letting n → ∞ and making use of Lemma 0.2, we get

−an1Δzn1 <∞∑

s=n1

rs (13)


or

−anΔzn <∞∑

s=nrs (n > n1). (14)

By summing up (14) from n1 to n, we obtain

zn1 − zn+1 <n∑

s=n1

1as

∞∑

t=srt. (15)

In view of Theorem 0.3, we know that limn→∞xn = 0, then there exists a sequence {nk}, suchthat limk→∞nk = ∞, limk→∞xnk−σ = 0, and limk→∞xnk = 0; by means of (15), we have

zn1 − znk+1 <nk∑

s=n1

1as

∞∑

t=srt, (16)

so

0 < zn <∞∑

s=n

1as

∞∑

t=srt. (17)

This shows that {wn} is nonoscillatory, which is a contradiction. This completes the proof.

The oscillation of {wn} is only the sufficient condition for the oscillation of (3). Thefollowing examples will illustrate this point.

Example 0.5. Consider the difference equation

Δ(1nΔ(xn + xn−1)

)+

3n + 1

xn =3

(n + 2)(n + 1), n ≥ 1. (18)

Here, wn =∑∞

s=n s(1/(s − 1) + 1/s + 1/(s + 1)) > 0 is nonoscillatory, and the other conditions(i), (ii), and (iii) are satisfied. Equation (18) has the nonoscillatory solution xn = (1/n) →0 (n → ∞).


Δ(1nΔ(xn + xn−1)

)+(n − 4)(2n + 1)(n + 2)(n − 1)

xn−4 = (−1)n 2n + 1n(n + 1)

, n ≥ 1. (19)

Here,wn =∑∞

s=n(−1)s is oscillatory, and the conditions (i), (ii), and (iii) are satisfied. Equation(19) is oscillation.


Δ(1nΔ(xn + 2xn−1)

)+(

(−1)nn(n + 1)

+2

n + 1+2n

)xn−4 =

1n(n + 1)

, n ≥ 1. (20)

Discrete Dynamics in Nature and Society 5

Here, wn =∑∞

s=n 1 > 0 is nonoscillatory, and the other conditions (i), (ii), (iii) are satisfied.But (20) has the oscillatory solution xn = (−1)n.

Remarks:

(1) When λ = 0, Theorems 0.3 and 0.4 still hold.

(2) As an = 1, Lemma 0.2, Theorems 0.3, and 0.4 still hold. In Theorem 0.4,

wn =∞∑

s=n

∞∑

t=srt, n > 0. (21)

It has been discussed that λ ≥ 0. We have the following conclusion as λ < 0. Set

zn = xn + λxn−τ . (22)

If {xn} is an eventually positive solution of (3), then there exist T > μ, zn < xn for n > T . Thus,

Δ(anΔzn) + qnzn−σ ≤ rn. (23)

Therefore, we obtain the following

Theorem 0.8. As λ < 0, if difference inequality (4) is oscillation, then difference equation (3) isoscillation.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grantno. 60771026), the Program for international cooperation of Shan’xi Province (Grantno. 2010081005), and the Natural Science Foundation of Shan’xi Province (Grant no.2010011007).

References

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[2] S. R. Grace and H. A. El-Morshedy, “Oscillation and nonoscillation theorems for certain second-orderdifference equations with forcing term,” Journal of Mathematical Analysis and Applications, vol. 216, no.2, pp. 614–625, 1997.

[3] S. H. Saker, “Oscillation of second order nonlinear delay difference equations,” Bulletin of the KoreanMathematical Society, vol. 40, no. 3, pp. 489–501, 2003.

[4] S. S. Cheng and H. J. Li, “A comparison theorem for asymptotically monotone solutions of nonlineardifference equations,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 21, no. 4, pp. 299–302,1993.

[5] S. S. Cheng, T. C. Yan, and H. J. Li, “Oscillation criteria for second order difference equation,”Funkcialaj Ekvacioj, vol. 34, no. 2, pp. 223–239, 1991.

[6] S. H. Saker, “Kamenev-type oscillation criteria for forced Emden-Fowler superlinear differenceequations,” Electronic Journal of Differential Equations, vol. 2002, no. 86, pp. 1–9, 2002.


[7] M. H. Abu-Risha, “Oscillation of second-order linear difference equations,” Applied MathematicsLetters, vol. 13, no. 1, pp. 129–135, 2000.

[8] W.-T. Li, “Oscillation theorems for second-order nonlinear difference equations,” Mathematical andComputer Modelling, vol. 31, no. 6-7, pp. 71–79, 2000.

[9] E. Thandapani and K. Ravi, “Oscillation of second-order half-linear difference equations,” AppliedMathematics Letters, vol. 13, no. 2, pp. 43–49, 2000.

[10] S. H. Saker, “New oscillation criteria for second-order nonlinear neutral delay difference equations,”Applied Mathematics and Computation, vol. 142, no. 1, pp. 99–111, 2003.

[11] Z.-R. Liu, W.-D. Chen, and Y.-H. Yu, “Oscillation criteria for second order nonlinear delay differenceequations,” Kyungpook Mathematical Journal, vol. 39, no. 1, pp. 127–132, 1999.

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