- Open Access
Oscillation criteria for second-order nonlinear difference equations of Euler type
© Yamaoka; licensee Springer 2012
- Received: 18 August 2012
- Accepted: 30 November 2012
- Published: 19 December 2012
The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation
where is continuous on ℝ and satisfies the signum condition if . The obtained results are best possible in a certain sense. Proof is given by means of the Riccati technique and phase plane analysis of a system. A discrete version of the Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results.
Here the forward difference operator Δ is defined as and .
A nontrivial solution of (1.1) is said to be oscillatory if for every positive integer N there exists such that . Otherwise, it is said to be nonoscillatory, that is, the solution is nonoscillatory if it is either eventually positive or eventually negative.
where , , , are arbitrary constants and z satisfies (for the proof, see [1–3]). Hence, all nontrivial solutions of equation (1.3) are oscillatory if , and otherwise they are nonoscillatory (see the Appendix). In other words, is the lower bound for all nontrivial solutions of equation (1.3) to be oscillatory. Such a number is generally called the oscillation constant. Other results on the oscillation constant for difference equations can be found in [4–7] and the references cited therein.
for sufficiently large. A natural question now arises. What is an oscillation constant for equation (1.1) where satisfies (1.6) for sufficiently large? The purpose of this paper is to answer the question. Our main results are stated as follows.
for sufficiently large. Then all nontrivial solutions of equation (1.1) are oscillatory.
for or , sufficiently large. Then equation (1.1) has a nonoscillatory solution.
is often considered instead of (1.3) (for example, see [4, 6, 7]), because Sturm’s separation and comparison theorems can be applied to equation (1.9). However, it is not easy to find an exact solution of equation (1.9). On the other hand, equation (1.3) has the general solution, and therefore, we can get more precise information for discrete analogues of equation (1.4). In this paper, we consider the nonlinear term for equation (1.1) as instead of to use exact solutions of linear difference equations.
This paper is organized as follows. In Section 2, we give general solutions of a discrete version of the Riemann-Weber generalization of Euler differential equation and decide an oscillation constant for the discrete equation. In Section 3, we complete the proof of Theorem 1.1 by means of the Riccati technique. In Section 4, using phase plane analysis, we prove Theorem 1.2.
2 General solutions of linear difference equations
where the function is positive and satisfies . Note that as . Here if and are positive functions, the notation as means that . Then we have the following result.
Hence, is a general solution of (2.1).
are linearly independent solutions of equation (2.1), and therefore, is a general solution of (2.1). □
If , then all nontrivial solutions of equation (2.1) are oscillatory.
If , then all nontrivial solutions of equation (2.1) are nonoscillatory.
Proof We consider only the case that because the other case can be proved easily.
because as . Thus, we conclude that is oscillatory.
as , we see that all nontrivial solutions of equation (2.1) are nonoscillatory. □
3 Oscillation theorem
To begin with, we prepare some lemmas which are useful for proving oscillation criteria, Theorem 1.1.
Lemma 3.1 Assume (1.2) and suppose that equation (1.1) has a positive solution. Then the solution is increasing for n sufficiently large and it tends to ∞ as .
We first show that for . By way of contradiction, we suppose that there exists such that . Then, using (3.1), we have for , and therefore, we can find such that . Using (3.1) again, we get for . Hence, we obtain as , which is a contradiction to the assumption that is positive for . Thus, is increasing for .
as . This contradicts the assumption that is bounded from above. Thus, we have . The proof is now complete. □
has a positive solution. Then the solution is nonincreasing and tends to as .
as . This is a contradiction to the assumption that is positive for . □
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1 By way of contradiction, we suppose that equation (1.1) has a nonoscillatory solution . Then we may assume, without loss of generality, that is eventually positive. Let R be a large number satisfying the assumption (1.7) for . From Lemma 3.1, is increasing and , and therefore, there exists such that and for .
for . From Lemma 3.2, we see that as , because is positive and satisfies (3.2) for .
Thus, the assertion is also true for .
which is a contradiction to (3.3). □
4 Nonoscillation theorem
To prove Theorem 1.2, we need the following results.
Proof We prove only the case , because the other cases are carried out in the same manner. Let . Then we have , and therefore, we obtain . Hence, we conclude that . □
for , where is nondecreasing with respect to for each fixed n. Then for .
Proof We use mathematical induction on n. It is clear that the assertion is true for . Assume that for . Since is nondecreasing with respect to x for each fixed p, we have . Thus, the assertion is also true for . This completes the proof. □
We are now ready to prove Theorem 1.2.
which is a contradiction. This completes the proof. □
Appendix: Euler-Cauchy difference equations
In this appendix, we show that the oscillation constant for Euler-Cauchy difference equation (1.3) is , that is, we prove the following result.
If , then all nontrivial solutions of equation (1.3) are oscillatory.
If , then all nontrivial solutions of equation (1.3) are nonoscillatory.
and therefore, is oscillatory.
Thus, we conclude that all nontrivial solutions of equation (1.3) are nonoscillatory. □
The author thanks the referees for their valuable suggestions that helped to improve this manuscript.
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