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Oscillation criteria for secondorder nonlinear difference equations of Euler type
Advances in Difference Equations volume 2012, Article number: 218 (2012)
Abstract
The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the secondorder nonlinear difference equation
where f(x) is continuous on ℝ and satisfies the signum condition xf(x)>0 if x\ne 0. 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 RiemannWeber generalization of EulerCauchy differential equation plays an important role in proving our results.
MSC:39A12, 39A21.
1 Introduction
We consider the secondorder nonlinear difference equation
where f(x) is a realvalued continuous function satisfying
Here the forward difference operator Δ is defined as \mathrm{\Delta}x(n)=x(n+1)x(n) and {\mathrm{\Delta}}^{2}x(n)=\mathrm{\Delta}(\mathrm{\Delta}x(n)).
A nontrivial solution x(n) of (1.1) is said to be oscillatory if for every positive integer N there exists n\ge N such that x(n)x(n+1)\le 0. Otherwise, it is said to be nonoscillatory, that is, the solution x(n) is nonoscillatory if it is either eventually positive or eventually negative.
When f(x)=\lambda x, equation (1.1) becomes the linear difference equation
which is called the EulerCauchy difference equation. It is known that (1.3) has the general solution
where {K}_{1}, {K}_{2}, {K}_{3}, {K}_{4} are arbitrary constants and z satisfies {z}^{2}z+\lambda =0 (for the proof, see [1–3]). Hence, all nontrivial solutions of equation (1.3) are oscillatory if \lambda >1/4, and otherwise they are nonoscillatory (see the Appendix). In other words, 1/4 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.
Equation (1.3) is a discrete analogue of the EulerCauchy differential equation
It is well known that an oscillation constant for equation (1.4) is also 1/4 (see [8]). The oscillation constant for equation (1.4) plays an important role in the oscillation problem for linear, halflinear and nonlinear differential equations. For example, those results can be found in [8–15]. In particular, using phase plane analysis of Liénard system, Sugie and Kita [12] considered the secondorder nonlinear differential equation
and gave a pair of an oscillation theorem and a nonoscillation theorem (see [[12], Theorems 3.1 and 4.1]). We note that their results are proved by using exact solutions of the RiemannWeber version of Euler differential equation
By their results, we can show that an oscillation constant for equation (1.5) is 1/4 provided
for x sufficiently large. A natural question now arises. What is an oscillation constant for equation (1.1) where f(x) satisfies (1.6) for x sufficiently large? The purpose of this paper is to answer the question. Our main results are stated as follows.
Theorem 1.1 Assume (1.2) and suppose that there exists λ with \lambda >1/4 such that
for x sufficiently large. Then all nontrivial solutions of equation (1.1) are oscillatory.
Theorem 1.2 Assume (1.2) and suppose that
for x>0 or x<0, x sufficiently large. Then equation (1.1) has a nonoscillatory solution.
Remark 1.1 As a discrete analogue of EulerCauchy differential equation (1.4), the linear difference equation
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 f(x(n)) instead of f(x(n+1)) 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 RiemannWeber 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
Consider the secondorder linear difference equation
where the function l(n) is positive and satisfies \mathrm{\Delta}l(n)=2/(2n+1). Note that l(n)\sim logn as n\to \mathrm{\infty}. Here if a(n) and b(n) are positive functions, the notation a(n)\sim b(n) as n\to \mathrm{\infty} means that {lim}_{n\to \mathrm{\infty}}a(n)/b(n)=1. Then we have the following result.
Proposition 2.1 Equation (2.1) has the general solution
where {K}_{1}, {K}_{2}, {K}_{3}, {K}_{4} are arbitrary constants and z is the root of the characteristic equation
Proof Put
Then \phi (n) and \psi (n) are the solutions of equation (2.1). We prove only the case that \phi (n) is a solution of equation (2.1), because the other case is carried out in the same manner. Here, we compute \mathrm{\Delta}\phi (n) and {\mathrm{\Delta}}^{2}\phi (n). Then we have
Since z satisfies (2.2), \phi (n) is a solution of equation (2.1). We also see that \phi (n) and \psi (n) are linearly independent if \lambda \ne 1/4. In fact, the Casoratian W(n) of \phi (n) and \psi (n) is given by
Hence, {K}_{1}\phi (n)+{K}_{2}\psi (n) is a general solution of (2.1).
We next consider the case that \lambda =1/4. Then (2.2) has the double root 1/2. Hence, by a direct computation, we can show that
are linearly independent solutions of equation (2.1), and therefore, {K}_{3}\tilde{\phi}(n)+{K}_{4}\tilde{\psi}(n) is a general solution of (2.1). □
To establish the oscillation constant for equation (2.1), we need the following lemma which is a corollary of the discrete l’Hospital rule (for example, see [16]).
Lemma 2.1 Let a(n) and b(n) be defined for n\ge {n}_{0}. Suppose that b(n) is positive and satisfies
If a(n)\sim b(n) as n\to \mathrm{\infty}, then
as n\to \mathrm{\infty}.
Proposition 2.2 The oscillation constant for equation (2.1) is 1/4. To be precise, equation (2.1) can be classified into two types as follows.

(i)
If \lambda >1/4, then all nontrivial solutions of equation (2.1) are oscillatory.

(ii)
If \lambda \le 1/4, then all nontrivial solutions of equation (2.1) are nonoscillatory.
Proof We consider only the case that \lambda \ne 1/4 because the other case can be proved easily.
In case \lambda >1/4, equation (2.2) has the conjugate roots z=(1\pm i\alpha )/2, where \alpha =\sqrt{4\lambda 1}. Hence, by Proposition 2.1 and Euler’s formula, the real solution of equation (2.1) can be written as
where r(j) and \theta (j) satisfy 0<\theta (j)<\pi /2,
for {n}_{0}\le j\le n1. If ({K}_{5},{K}_{6})=(0,0), then x(n) is the trivial solution. On the other hand, if ({K}_{5},{K}_{6})\ne (0,0), then
where {K}_{7}=\sqrt{{K}_{5}^{2}+{K}_{6}^{2}}, sin{K}_{8}={K}_{5}/{K}_{7} and cos{K}_{8}={K}_{6}/{K}_{7}. Since
as n\to \mathrm{\infty}, we obtain \theta (n)\sim tan\theta (n)\sim \alpha /(2nl(n))\sim \alpha /(2nlogn) as n\to \mathrm{\infty}. Using Lemma 2.1, we have
as n\to \mathrm{\infty}, because
as n\to \mathrm{\infty}. We note that, for any sufficiently large p\in \mathbb{N}, there exists n\in \mathbb{N} such that
because \theta (n)\searrow 0 as n\to \mathrm{\infty}. Thus, we conclude that x(n) is oscillatory.
We next consider the case that \lambda <1/4. Put
where z satisfies (2.2). Then, without loss of generality, we may assume that z>1/2. From Proposition 2.1, the solution of equation (2.1) can be represented as
for some {K}_{1}\in \mathbb{R} and {K}_{2}\in \mathbb{R}. Since
as n\to \mathrm{\infty}, 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 n\to \mathrm{\infty}.
Proof Let x(n) be a positive solution of equation (1.1). Then there exists {n}_{0}\in \mathbb{N} such that x(n)>0 for n\ge {n}_{0}. Hence, by (1.2) we have
for n\ge {n}_{0}.
We first show that \mathrm{\Delta}x(t)>0 for n\ge {n}_{0}. By way of contradiction, we suppose that there exists {n}_{1}\ge {n}_{0} such that \mathrm{\Delta}x({n}_{1})\le 0. Then, using (3.1), we have \mathrm{\Delta}x(n)<\mathrm{\Delta}x({n}_{1})\le 0 for n>{n}_{1}, and therefore, we can find {n}_{2}>{n}_{1} such that \mathrm{\Delta}x({n}_{2})<0. Using (3.1) again, we get \mathrm{\Delta}x(n)\le \mathrm{\Delta}x({n}_{2})<0 for n\ge {n}_{2}. Hence, we obtain x(n)\le \mathrm{\Delta}x({n}_{2})(n{n}_{2})+x({n}_{2})\to \mathrm{\infty} as n\to \mathrm{\infty}, which is a contradiction to the assumption that x(n) is positive for n\ge {n}_{0}. Thus, x(n) is increasing for n\ge {n}_{0}.
We next suppose that x(n) is bounded from above. Then there exists L>0 such that {lim}_{n\to \mathrm{\infty}}x(n)=L. Since f(x) is continuous on ℝ, we have {lim}_{n\to \mathrm{\infty}}f(x(n))=f(L), and therefore, there exists {n}_{3}\ge {n}_{0} such that 0<f(L)/2<f(x(n)) for n\ge {n}_{3}. Hence, we have
for n>m\ge {n}_{3}. Taking the limit of this inequality as n\to \mathrm{\infty}, we get \mathrm{\Delta}x(m)\ge f(L)/2m for m\ge {n}_{3}, and therefore, we obtain
as m\to \mathrm{\infty}. This contradicts the assumption that x(n) is bounded from above. Thus, we have {lim}_{n\to \mathrm{\infty}}x(n)=\mathrm{\infty}. The proof is now complete. □
Lemma 3.2 Suppose that the difference inequality
has a positive solution. Then the solution is nonincreasing and tends to 1/2 as n\to \mathrm{\infty}.
Proof Let w(n) be a positive solution of (3.2). Then there exists {n}_{0}\in \mathbb{N} such that w(n)>0 for n\ge {n}_{0}. Hence, we see that w(n) is nonincreasing because w(n) satisfies
for n\ge {n}_{0}. Thus, we can find \alpha \ge 0 such that w(n)\searrow \alpha as n\to \mathrm{\infty}. If \alpha \ne 1/2, then there exists {n}_{1}\ge {n}_{0} such that w(n)1/2>\alpha 1/2/2 for n\ge {n}_{1}. Since w(n) is nonincreasing, there exists {n}_{2}\ge {n}_{1} such that w(n)<n for n\ge {n}_{2}. Hence, we have
for n\ge {n}_{2}, and therefore, we get
as n\to \mathrm{\infty}. This is a contradiction to the assumption that w(n) is positive for n\ge {n}_{0}. □
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 x(n). Then we may assume, without loss of generality, that x(n) is eventually positive. Let R be a large number satisfying the assumption (1.7) for x\ge R. From Lemma 3.1, x(n) is increasing and {lim}_{n\to \mathrm{\infty}}x(n)=\mathrm{\infty}, and therefore, there exists {n}_{0}\in \mathbb{N} such that x(n)\ge R and \mathrm{\Delta}x(n)>0 for n\ge {n}_{0}.
We define
Then, using (1.7), we have
for n\ge {n}_{0}. From Lemma 3.2, we see that w(n)\searrow 1/2 as n\to \mathrm{\infty}, because w(n) is positive and satisfies (3.2) for n\ge {n}_{0}.
Since \lambda >1/4, we can find {\epsilon}_{0}>0 such that
Then we see that there exists {n}_{1}>{n}_{0} such that w(n)\le 1/2+{\epsilon}_{0} for n\ge {n}_{1}, that is, x(n) satisfies
for n\ge {n}_{1}. Hence, we have
for n\ge {n}_{1}. Since log(1+z)\le z for z>1, we get
for n\ge {n}_{1}. Recall that l(n) satisfies \mathrm{\Delta}l(n)=2/(2n+1)\sim 1/n as n\to \mathrm{\infty}. Using Lemma 2.1, we see that
as n\to \mathrm{\infty}. Hence, there exists {n}_{2}\ge {n}_{1} such that
for n\ge {n}_{2}. We therefore conclude that
for n\ge {n}_{2}.
Let v(n) be the function satisfying v({n}_{2})=w({n}_{2})>0 and v(n+1)=F(n,v(n)), where the function F:\mathbb{N}\times [0,\mathrm{\infty})\to \mathbb{R} defined by
Using mathematical induction on n, we show that the function v(n) is well defined and satisfies v(n)\ge w(n)>0 for n\ge {n}_{2}. It is clear that the assertion is true for n={n}_{2}. Assume that the assertion is true for n=p. Then v(p+1)=F(p,v(p)) exists because v(p)>0. Since
F(p,v) is nondecreasing with respect to v\in [0,\mathrm{\infty}) for each fixed p. Hence, together with (3.4), we have
Thus, the assertion is also true for n=p+1.
Letting
we can easily see that y(n) is a positive solution of the difference equation
Hence, from Proposition 2.2, we have
which is a contradiction to (3.3). □
4 Nonoscillation theorem
In this section, we give a sufficient condition for equation (1.1) to have a nonoscillatory solution. Let x(n) be a solution of equation (1.1) and put y(n)=n\mathrm{\Delta}x(n)x(n). Then we have
and therefore, we can transform (1.1) into the system
To prove Theorem 1.2, we need the following results.
Lemma 4.1 Let (x(n),y(n)) be a nontrivial solution of system (4.1). If (x(n),y(n))\in {D}_{k}, then (x(n+1),y(n+1))\in {D}_{k}\cup {D}_{k+1} for k=1,2,3,4, where
and {D}_{5}={D}_{1}.
Proof We prove only the case k=1, because the other cases are carried out in the same manner. Let (x(n),y(n))\in {D}_{1}. Then we have n\mathrm{\Delta}x(n)=y(n)+x(n)\ge 0, and therefore, we obtain x(n+1)\ge x(n). Hence, we conclude that (x(n+1),y(n+1))\in {D}_{1}\cup {D}_{2}. □
Lemma 4.2 Suppose that \theta (n) and \phi (n) satisfy \theta ({n}_{0})=\phi ({n}_{0}) and
for {n}_{0}\le n<{n}_{1}, where F(n,x) is nondecreasing with respect to x\in \mathbb{R} for each fixed n. Then \theta (n)\ge \phi (n) for {n}_{0}\le n\le {n}_{1}.
Proof We use mathematical induction on n. It is clear that the assertion is true for n={n}_{0}. Assume that \theta (n)\ge \phi (n) for n=p<{n}_{1}. Since F(p,x) is nondecreasing with respect to x for each fixed p, we have \theta (p+1)\ge F(p,\theta (p))\ge F(p,\phi (p))=\phi (p+1). Thus, the assertion is also true for n=p+1. This completes the proof. □
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2 We give only the proof of the case that
for x>0 sufficiently large. The proof is by contradiction. Suppose that all nontrivial solutions of equation (1.1) are oscillatory. Let (x(n),y(n)) be the solution of system (4.1) satisfying the initial condition
Then x(n) is a nontrivial oscillatory solution of equation (1.1) and
and therefore, (x(n),y(n)) cannot stay in \tilde{D}\subset {D}_{1}. By Lemma 4.1, there exists {n}_{1}>{n}_{0} such that
Hence, we see that
for {n}_{0}\le n<{n}_{1}, and therefore, we have x(n+1)\ge \{1+1/(2n)\}x(n) for {n}_{0}\le n<{n}_{1}. Thus, we get
for {n}_{0}\le n<{n}_{1}. Hence, we obtain
for {n}_{0}\le n<{n}_{1}. We define \theta (n)=y(n)/x(n). Then, using (1.8), we have
for {n}_{0}\le n<{n}_{1}. Note that \theta (n) satisfies
We compare the function \theta (n) with a solution
of the equation
It follows from Lemma 4.2 that \phi (n)\le \theta (n) for {n}_{0}\le n\le {n}_{1} because \phi ({n}_{0})=\theta ({n}_{0}). Hence, together with (4.2) and (4.3), we have
which is a contradiction. This completes the proof. □
Appendix: EulerCauchy difference equations
In this appendix, we show that the oscillation constant for EulerCauchy difference equation (1.3) is 1/4, that is, we prove the following result.
Proposition A.1 Equation (1.3) can be classified into two types as follows.

(i)
If \lambda >1/4, then all nontrivial solutions of equation (1.3) are oscillatory.

(ii)
If \lambda \le 1/4, then all nontrivial solutions of equation (1.3) are nonoscillatory.
Proof The general solution of equation (1.3) is given by
where {K}_{1}, {K}_{2}, {K}_{3}, {K}_{4} are arbitrary constants and z satisfies
(for the proof, see [1–3]). Hence, we consider only the case that \lambda \ne 1/4 because we can easily check that all nontrivial solutions of equation (1.3) are nonoscillatory if \lambda =1/4.
In case \lambda >1/4, equation (A.2) has the conjugate roots z=(1\pm i\alpha )/2, where \alpha =\sqrt{4\lambda 1}. Hence, by (A.1) and Euler’s formula, the real solution of equation (1.3) can be written as
where r(j) and \theta (j) satisfy 0<\theta (j)<\pi /2,
for {n}_{0}\le j\le n1. If ({K}_{5},{K}_{6})\ne (0,0), then
where {K}_{7}=\sqrt{{K}_{5}^{2}+{K}_{6}^{2}}, sin{K}_{8}={K}_{5}/{K}_{7} and cos{K}_{8}={K}_{6}/{K}_{7}. Since
as n\to \mathrm{\infty}, we obtain \theta (n)\sim tan\theta (n) as n\to \mathrm{\infty}. Using Lemma 2.1, we have
as n\to \mathrm{\infty}. Hence, for any sufficiently large p\in \mathbb{N}, there exists n\in \mathbb{N} such that
and therefore, x(n) is oscillatory.
We next consider the case that \lambda <1/4. Put
where z satisfies (A.2). Then, without loss of generality, we may assume that z>1/2. From (A.1), the solution of equation (1.3) can be represented as
for some {K}_{1}\in \mathbb{R} and {K}_{2}\in \mathbb{R}. By Stirling’s formula for the gamma function, we see that
as t\to \mathrm{\infty}, where Γ is the gamma function (as to Stirling’s formula, for example, see [17]). Hence, we have
as n\to \mathrm{\infty}, and therefore, we obtain
Thus, we conclude that all nontrivial solutions of equation (1.3) are nonoscillatory. □
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Yamaoka, N. Oscillation criteria for secondorder nonlinear difference equations of Euler type. Adv Differ Equ 2012, 218 (2012). https://doi.org/10.1186/168718472012218
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DOI: https://doi.org/10.1186/168718472012218