Oscillation Criteria of Solution for a Second Order Difference Equation with Forced Term

Abstract

We will consider oscillation criteria for the second order difference equation with forced term Δ(𝑎_𝑛Δ(𝑥_𝑛+𝜆𝑥_𝑛−𝜏))+𝑞_𝑛𝑥_𝑛−𝜎=𝑟_𝑛 (𝑛≥0). We establish sufficient conditions which guarantee that every solution is oscillatory or eventually positive solutions converge to zero.

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In the last thirty years, there has been an increasing interest in the study of oscillation and asymptotic behavior of solutions of second order difference equations (see [1–11]). In [1], Arul and Thandapani considered the equationΔ𝑝_𝑛𝜙Δ𝑥_𝑛+𝑓𝑛,𝑥_𝑛+1=0,𝑛=0,1,2,…, (1) and gave some sufficient conditions for the existence of positive solutions. In [3], Saker considered the equationΔ𝑝_𝑛Δ𝑥_𝑛+𝑞_𝑛𝑓𝑥_𝑛−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 order difference equation with forced term Δ𝑎_𝑛Δ𝑥_𝑛+𝜆𝑥_𝑛−𝜏+𝑞_𝑛𝑥_𝑛−𝜎=𝑟_𝑛,𝑛=1,2,…, (3) where {𝑎_𝑛} is a positive sequence, {𝑞_𝑛} is a nonnegative sequence and not identically zero for all large 𝑛, {𝑟_𝑛} is a real sequence, 𝜆 is a real number, and 𝜎, 𝜏 are nonnegative integers, 𝜇=max{𝜎,𝜏}.

A solution {𝑥_𝑛} of (3) is said to be eventually positive if 𝑥_𝑛>0 for all large 𝑛 and eventually negative if 𝑥_𝑛<0 for all large 𝑛. Equation (3) is said to be oscillatory if it is neither eventually positive nor eventually negative.

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

Lemma 0.1. If difference inequality Δ𝑎_𝑛Δ𝑧_𝑛+𝑞_𝑛𝑧_𝑛−𝜎≤𝑟_𝑛,𝑛>0, (4) is oscillation, then difference equation Δ𝑎_𝑛Δ𝑧_𝑛+𝑞_𝑛𝑧_𝑛−𝜎=𝑟_𝑛,𝑛>0, (5) is oscillation.

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

Lemma 0.2. Suppose that {𝑥_𝑛} is an eventually positive solution of (3), 𝜆≥0, and(i)∑^∞_𝑛=1(1/𝑎_𝑛)=+∞, (ii)∑^∞_𝑛=1𝑞_𝑛=+∞, (iii)∑^∞_𝑛=1𝑟_𝑛<∞. Set 𝑧_𝑛=𝑥_𝑛+𝜆𝑥_𝑛−𝜏. Then 𝑧_𝑛>0 and lim_𝑛→∞𝑎_𝑛Δ𝑧_𝑛=0

Proof. Suppose that {𝑥_𝑛} is an eventually positive solution of (3), then there exists 𝑛₁>𝜇, such that 𝑥_𝑛>0, 𝑥_𝑛−𝜏>0, and 𝑥_𝑛−𝜎>0for 𝑛>𝑛₁, then 𝑧_𝑛>0 for 𝑛>𝑛₁. By summing up (3) from 𝑛₁ to 𝑛, we obtain 𝑎_𝑛+1Δ𝑧_𝑛+1−𝑎_𝑛1Δ𝑧_𝑛1+^𝑛_𝑠=𝑛1𝑞_𝑠𝑥_𝑠−𝜎=^𝑛_𝑠=𝑛1𝑟_𝑠. (6) From (6), we know that lim_𝑛→∞∑^𝑛_𝑛=𝑛1𝑞_𝑠𝑥_𝑠−𝜎=𝛼, where α is a positive limited number or 𝛼=+∞. Thus lim_𝑛→∞𝑎_𝑛Δ𝑧_𝑛=𝛽, β is a limited number or 𝛽=−∞.
If 𝛽<𝑐<0 (𝑐 is a constant), then there exist 𝑛₂≥𝑛₁, 𝑎_𝑛Δ𝑧_𝑛≤𝑐 for 𝑛≥𝑛₂, so that 𝑧_𝑛+1≤𝑧_𝑛2+𝑐^𝑛_𝑠=𝑛21𝑎_𝑠, (7) which is contrary to 𝑧_𝑛>0.
If 𝛽>0, then there exist 𝑛₃≥𝑛₁, 𝑎_𝑛Δ𝑧_𝑛>𝛽/2 for 𝑛≥𝑛₃; hence, 𝑧_𝑛+1≥𝑧_𝑛3+𝛽2^𝑛_𝑠=𝑛31𝑎_𝑠⟶+∞(𝑛⟶∞), (8) therefore, lim_𝑛→∞𝑥_𝑛=∞,lim_𝑛→∞𝑥_𝑛−𝜎=∞; thus, there exist 𝑛₄≥𝑛₃, 𝑥_𝑛≥𝑀, and 𝑥_𝑛−𝜎≥𝑀 (𝑀>0) for 𝑛≥𝑛₄. By summing up (3) from 𝑛₄ to, we obtain 𝑎_𝑛+1Δ𝑧_𝑛+1−𝑎_𝑛4Δ𝑧_𝑛4+𝑀^𝑛_𝑠=𝑛4𝑞_𝑠≤^𝑛_𝑠=𝑛4𝑟_𝑠. (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 lim_𝑛→∞𝑎_𝑛Δ𝑧_𝑛=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 {𝑥_𝑛} is an eventually positive solution of (3), then lim_𝑛→∞𝑥_𝑛=0.

Proof. Making use of (6) and the conclusion of Lemma 0.2, we know lim_𝑛→∞^𝑛_𝑠=𝑛1𝑞_𝑠𝑥_𝑠−𝜎=𝛼(0<𝛼<+∞), (10) so lim_𝑛→∞𝑥_𝑛=0. If not, suppose that lim_𝑛→∞𝑥_𝑛=𝑙>0, then there exist 𝑛₅>𝑛₁,  𝑥_𝑛≥𝑙/2>0 for 𝑛>𝑛₅. Now substitute 𝑥_𝑛≥𝑙/2>0 for 𝑥_𝑛 in (6), we obtain a contrary. This completes the proof.

Theorem 0.4. If conditions (i), (ii), and (iii) hold, let 𝑤_𝑛=^∞_𝑠=𝑛1𝑎_𝑠^∞_𝑡=𝑠𝑟_𝑡,𝑛>0, (11) and if {𝑤_𝑛} is oscillation, then (3) is oscillation.

Proof. Suppose that {𝑥_𝑛} is an eventually positive solution of (3), then there exist 𝑛₁>𝜇, 𝑥_𝑛>0, 𝑥_𝑛−𝜏>0, and 𝑥_𝑛−𝜎>0 for 𝑛≥𝑛₁. From (6), we have 𝑎_𝑛+1Δ𝑧_𝑛+1−𝑎_𝑛1Δ𝑧_𝑛1<^𝑛_𝑠=𝑛1𝑟_𝑠. (12) Letting 𝑛→∞ and making use of Lemma 0.2, we get −𝑎_𝑛1Δ𝑧_𝑛1<^∞_𝑠=𝑛1𝑟_𝑠 (13) or −𝑎_𝑛Δ𝑧_𝑛<^∞_𝑠=𝑛𝑟_𝑠𝑛>𝑛₁. (14) By summing up (14) from 𝑛₁ to 𝑛, we obtain 𝑧_𝑛1−𝑧_𝑛+1<^𝑛_𝑠=𝑛11𝑎_𝑠^∞𝑡=𝑠𝑟_𝑡. (15) In view of Theorem 0.3, we know that lim_𝑛→∞𝑥_𝑛=0, then there exists a sequence {𝑛_𝑘}, such that lim_𝑘→∞𝑛_𝑘=∞, lim_𝑘→∞𝑥_{𝑛𝑘−𝜎}=0, and lim_𝑘→∞𝑥_𝑛𝑘=0; by means of (15), we have 𝑧_𝑛1−𝑧_𝑛𝑘+1<^𝑛𝑘_𝑠=𝑛11𝑎_𝑠^∞𝑡=𝑠𝑟_𝑡, (16) so 0<𝑧_𝑛<^∞_𝑠=𝑛1𝑎_𝑠^∞𝑡=𝑠𝑟_𝑡. (17) This shows that {𝑤_𝑛} is nonoscillatory, which is a contradiction. This completes the proof.

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

Example 0.5. Consider the difference equation Δ1𝑛Δ𝑥_𝑛+𝑥_𝑛−1+3𝑥𝑛+1^𝑛=3(𝑛+2)(𝑛+1),𝑛≥1. (18) Here, 𝑤_𝑛=∑^∞_𝑠=𝑛𝑠(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 𝑥_𝑛=(1/𝑛)→0(𝑛→∞).

Example 0.6. Consider the difference equation Δ1𝑛Δ𝑥_𝑛+𝑥_𝑛−1+(𝑛−4)(2𝑛+1)()𝑥𝑛+2)(𝑛−1^𝑛−4=(−1)^𝑛2𝑛+1𝑛(𝑛+1),𝑛≥1. (19) Here, 𝑤_𝑛=∑^∞_𝑠=𝑛(−1)^𝑠 is oscillatory, and the conditions (i), (ii), and (iii) are satisfied. Equation (19) is oscillation.

Example 0.7. Consider the difference equation Δ1𝑛Δ𝑥_𝑛+2𝑥_𝑛−1+(−1)^𝑛𝑛()+2𝑛+1+2𝑛+1𝑛𝑥_𝑛−4=1𝑛(𝑛+1),𝑛≥1. (20) Here, 𝑤_𝑛=∑^∞_𝑠=𝑛1>0 is nonoscillatory, and the other conditions (i), (ii), (iii) are satisfied. But (20) has the oscillatory solution 𝑥_𝑛=(−1)^𝑛.

Remarks:
(1)When 𝜆=0, Theorems 0.3 and 0.4 still hold.(2)As 𝑎_𝑛=1, Lemma 0.2, Theorems 0.3, and 0.4 still hold. In Theorem 0.4,𝑤_𝑛=^∞∞_{𝑠=𝑛𝑡=𝑠}𝑟_𝑡,𝑛>0. (21) It has been discussed that 𝜆≥0. We have the following conclusion as 𝜆<0. Set 𝑧_𝑛=𝑥_𝑛+𝜆𝑥_𝑛−𝜏. (22) If {𝑥_𝑛} is an eventually positive solution of (3), then there exist 𝑇>𝜇, 𝑧_𝑛<𝑥_𝑛 for 𝑛>𝑇. Thus, Δ𝑎_𝑛Δ𝑧_𝑛+𝑞_𝑛𝑧_𝑛−𝜎≤𝑟_𝑛. (23) Therefore, we obtain the following

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

Acknowledgments

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