R E S E A R C H
Open Access
Oscillation criteria for second-order nonlinear
difference equations of Euler type
Naoto Yamaoka
**Correspondence:
[email protected] Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531, Japan
Abstract
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
2x(n) + 1
n(n+ 1)f
(
x(n))
= 0,wheref(x) is continuous onRand satisfies the signum conditionxf(x) > 0 ifx= 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 Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results.
MSC: 39A12; 39A21
Keywords: oscillation constant; Euler-Cauchy equation; nonlinear difference equations; Riccati technique; phase plane analysis
1 Introduction
We consider the second-order nonlinear difference equation
x(n) +
n(n+ )f
x(n)= , n≥n, (.)
wheref(x) is a real-valued continuous function satisfying
xf(x) > ifx= . (.)
Here the forward difference operatoris defined asx(n) =x(n+ ) –x(n) andx(n) =
(x(n)).
A nontrivial solutionx(n) of (.) is said to beoscillatoryif for every positive integerN
there existsn≥N such thatx(n)x(n+ )≤. Otherwise, it is said to benonoscillatory, that is, the solutionx(n) is nonoscillatory if it is either eventually positive or eventually negative.
Whenf(x) =λx, equation (.) becomes the linear difference equation
x(n) + λ
n(n+ )x(n) = , (.)
which is called the Euler-Cauchy difference equation. It is known that (.) has the general solution
x(n) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
K
n–
j=n +z
j
+K
n–
j=n
+ –z
j
ifλ=
, n–
j=n +
j
K+K
n–
k=n k+
ifλ= ,
whereK,K,K,Kare arbitrary constants andzsatisfiesz–z+λ= (for the proof, see
[–]). Hence, all nontrivial solutions of equation (.) are oscillatory ifλ> /, and other-wise they are nonoscillatory (see the Appendix). In other words, / is the lower bound for all nontrivial solutions of equation (.) to be oscillatory. Such a number is generally called theoscillation constant. Other results on the oscillation constant for difference equations can be found in [–] and the references cited therein.
Equation (.) is a discrete analogue of the Euler-Cauchy differential equation
x+ λ
tx= . (.)
It is well known that an oscillation constant for equation (.) is also / (see []). The oscillation constant for equation (.) plays an important role in the oscillation problem for linear, half-linear and nonlinear differential equations. For example, those results can be found in [–]. In particular, using phase plane analysis of Liénard system, Sugie and Kita [] considered the second-order nonlinear differential equation
x+
tf(x) = , (.)
and gave a pair of an oscillation theorem and a nonoscillation theorem (see [, Theo-rems . and .]). We note that their results are proved by using exact solutions of the Riemann-Weber version of Euler differential equation
x+
t
+
λ
(logt)
x= .
By their results, we can show that an oscillation constant for equation (.) is / provided
f(x) = +
λ
(logx)
x (.)
for|x|sufficiently large. A natural question now arises. What is an oscillation constant for equation (.) wheref(x) satisfies (.) for|x|sufficiently large? The purpose of this paper is to answer the question. Our main results are stated as follows.
Theorem . Assume(.)and suppose that there existsλwithλ> /such that f(x)
x ≥
+
λ
(logx) (.)
Theorem . Assume(.)and suppose that
f(x)
x ≤
+
(logx) (.)
for x> or x< ,|x|sufficiently large.Then equation(.)has a nonoscillatory solution.
Remark . As a discrete analogue of Euler-Cauchy differential equation (.), the linear difference equation
x(n) + λ
n(n+ )x(n+ ) = (.)
is often considered instead of (.) (for example, see [, , ]), because Sturm’s separa-tion and comparison theorems can be applied to equasepara-tion (.). However, it is not easy to find an exact solution of equation (.). On the other hand, equation (.) has the gen-eral solution, and therefore, we can get more precise information for discrete analogues of equation (.). In this paper, we consider the nonlinear term for equation (.) asf(x(n)) instead off(x(n+ )) to use exact solutions of linear difference equations.
This paper is organized as follows. In Section , 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 , we complete the proof of Theorem . by means of the Riccati technique. In Section , using phase plane analysis, we prove Theorem ..
2 General solutions of linear difference equations
Consider the second-order linear difference equation
x(n) +
n(n+ )
+
λ
l(n)l(n+ )
x(n) = , (.)
where the functionl(n) is positive and satisfiesl(n) = /(n+ ). Note thatl(n)∼lognas
n→ ∞. Here ifa(n) andb(n) are positive functions, the notationa(n)∼b(n) asn→ ∞
means thatlimn→∞a(n)/b(n) = . Then we have the following result.
Proposition . Equation(.)has the general solution
x(n) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
K
n–
j=n +
j+ z jl(j)
+K
n–
j=n +
j+
–z jl(j)
ifλ=
, n–
j=n +
j+
jl(j)
K+K
n–
k=n
(k+ )l(k) +
ifλ= ,
where K,K,K,Kare arbitrary constants and z is the root of the characteristic equation
Proof Put is a solution of equation (.), because the other case is carried out in the same manner. Here, we computeϕ(n) andϕ(n). Then we have direct computation, we can show that
˜
general solution of (.).
Lemma . Let a(n)and b(n)be defined for n≥n.Suppose that b(n)is positive and
sat-isfies
∞
n=n
b(n) =∞.
If a(n)∼b(n)as n→ ∞,then
n
j=n
a(j)∼ n
j=n
b(j)
as n→ ∞.
Proposition . The oscillation constant for equation(.)is/.To be precise,equation
(.)can be classified into two types as follows.
(i) Ifλ> /,then all nontrivial solutions of equation(.)are oscillatory. (ii) Ifλ≤/,then all nontrivial solutions of equation(.)are nonoscillatory.
Proof We consider only the case thatλ= / because the other case can be proved easily. In caseλ> /, equation (.) has the conjugate rootsz= (±iα)/, whereα=√λ– . Hence, by Proposition . and Euler’s formula, the real solution of equation (.) can be written as
x(n) =K n–
j=n
r(j)
cos
n–
j=n
θ(j)
+K n–
j=n
r(j)
sin
n–
j=n
θ(j)
,
wherer(j) andθ(j) satisfy <θ(j) <π/,
r(j)cosθ(j) = + j+
jl(j) and r(j)sinθ(j) =
α
jl(j)
forn≤j≤n– . If (K,K) = (, ), thenx(n) is the trivial solution. On the other hand, if
(K,K)= (, ), then
x(n) =K n–
j=n
r(j)
sin
n–
j=n
θ(j) +K
,
whereK=
K
+K,sinK=K/KandcosK=K/K. Since
tanθ(n) = α
nl(n) +l(n) + →
as n→ ∞, we obtain θ(n)∼ tanθ(n)∼ α/(nl(n))∼ α/(nlogn) as n → ∞. Using
Lemma ., we have
n–
j=n
θ(j)∼ n–
j=n
α
jlogj∼
α
asn→ ∞, because
n–
j=n
α
jlogj∼
α
n
n
dx xlogxdx=
α
log(logn) –log(logn)
→ ∞
asn→ ∞. We note that, for any sufficiently largep∈N, there existsn∈Nsuch that
pπ≤
n–
j=n
θ(j) +K< (p+ )π
becauseθ(n) asn→ ∞. Thus, we conclude thatx(n) is oscillatory. We next consider the case thatλ< /. Put
ϕ(n) = n–
j=n +
j+ z jl(j)
and ψ(n) = n–
j=n +
j+
–z jl(j)
,
wherezsatisfies (.). Then, without loss of generality, we may assume thatz> /. From Proposition ., the solution of equation (.) can be represented as
x(n) =Kϕ(n) +Kψ(n) =ϕ(n)
K+K
ψ(n)
ϕ(n)
for someK∈RandK∈R. Since
ψ(n)
ϕ(n) = n–
j=n
– z–
jl(j) +l(j)/ +z
≤exp
– n–
j=n
z–
jl(j) +l(j)/ +z
→
asn→ ∞, we see that all nontrivial solutions of equation (.) are nonoscillatory.
3 Oscillation theorem
To begin with, we prepare some lemmas which are useful for proving oscillation criteria, Theorem ..
Lemma . Assume(.)and suppose that equation(.)has a positive solution.Then the solution is increasing for n sufficiently large and it tends to∞as n→ ∞.
Proof Letx(n) be a positive solution of equation (.). Then there existsn∈Nsuch that
x(n) > forn≥n. Hence, by (.) we have
x(n) = –
n(n+ )f
x(n)< (.)
forn≥n.
We first show thatx(t) > forn≥n. By way of contradiction, we suppose that there
existsn≥n such thatx(n)≤. Then, using (.), we havex(n) <x(n)≤ for
n>n, and therefore, we can findn>nsuch thatx(n) < . Using (.) again, we get
n→ ∞, which is a contradiction to the assumption thatx(n) is positive forn≥n. Thus,
x(n) is increasing forn≥n.
We next suppose thatx(n) is bounded from above. Then there existsL> such that limn→∞x(n) =L. Sincef(x) is continuous onR, we havelimn→∞f(x(n)) =f(L), and
there-fore, there existsn≥nsuch that <f(L)/ <f(x(n)) forn≥n. Hence, we have
x(m) =x(n) + n–
j=m
j(j+ )f
x(j)>f(L)
n–
j=m
j(j+ )=
f(L)
m–
n
forn>m≥n. Taking the limit of this inequality asn→ ∞, we getx(m)≥f(L)/mfor
m≥n, and therefore, we obtain
x(m)≥x(n) +
f(L)
m–
k=n
k→ ∞
asm→ ∞. This contradicts the assumption thatx(n) is bounded from above. Thus, we havelimn→∞x(n) =∞. The proof is now complete.
Lemma . Suppose that the difference inequality
w(n) +
n+w(n) w(n) –
≤ (.)
has a positive solution.Then the solution is nonincreasing and tends to/as n→ ∞.
Proof Letw(n) be a positive solution of (.). Then there existsn∈Nsuch thatw(n) >
forn≥n. Hence, we see thatw(n) is nonincreasing becausew(n) satisfies
w(n)≤–
n+w(n) w(n) –
≤
forn≥n. Thus, we can findα≥ such thatw(n)αasn→ ∞. Ifα= /, then there
existsn≥nsuch that|w(n) – /|>|α– /|/ forn≥n. Sincew(n) is nonincreasing,
there existsn≥nsuch thatw(n) <nforn≥n. Hence, we have
w(n)≤–
n+w(n) w(n) –
≤– n
α– /
forn≥n, and therefore, we get
w(n)≤w(n) –
α– /
n–
j=n
j →–∞
asn→ ∞. This is a contradiction to the assumption thatw(n) is positive forn≥n.
We are now ready to prove Theorem ..
eventually positive. LetRbe a large number satisfying the assumption (.) for|x| ≥R.
Then, using (.), we have
forn≥n. Recall thatl(n) satisfiesl(n) = /(n+ )∼/nasn→ ∞. Using Lemma .,
we see that
l(n)∼ n–
j=n
j
asn→ ∞. Hence, there existsn≥nsuch that
logx(n)≤ + ε l(n)≤
+ ε
l(n+ )
forn≥n. We therefore conclude that
w(n)≤–
n+w(n) w(n) –
+ λ
( + ε)l(n)l(n+ )
(.)
forn≥n.
Letv(n) be the function satisfyingv(n) =w(n) > andv(n+ ) =F(n,v(n)), where the
functionF:N×[,∞)→Rdefined by
F(n,v) =v–
n+v v–
+ λ
( + ε)l(n)l(n+ )
.
Using mathematical induction onn, we show that the functionv(n) is well defined and satisfiesv(n)≥w(n) > forn≥n. It is clear that the assertion is true forn=n. Assume
that the assertion is true forn=p. Thenv(p+ ) =F(p,v(p)) exists becausev(p) > . Since
d
dvF(p,v) =
(p+v) p+
+ λ
( + ε)l(p)l(p+ )
≥,
F(p,v) is nondecreasing with respect tov∈[,∞) for each fixedp. Hence, together with (.), we have
v(p+ ) =Fp,v(p)≥Fp,w(p)≥w(p+ ) > .
Thus, the assertion is also true forn=p+ . Letting
y(n) = n–
j=n
+v(j)
j
,
we can easily see thaty(n) is a positive solution of the difference equation
y(n) +
n(n+ )
+
λ
( + ε)l(n)l(n+ )
y(n) = .
Hence, from Proposition ., we have
λ
( + ε) ≤
,
4 Nonoscillation theorem
In this section, we give a sufficient condition for equation (.) to have a nonoscillatory solution. Letx(n) be a solution of equation (.) and puty(n) =nx(n) –x(n). Then we have
y(n) =x(n) + (n+ )x(n) –x(n) = –
nf
x(n),
and therefore, we can transform (.) into the system
nx(n) =y(n) +x(n), ny(n) = –fx(n). (.) To prove Theorem ., we need the following results.
Lemma . Let(x(n),y(n))be a nontrivial solution of system(.).If(x(n),y(n))∈Dk,then
(x(n+ ),y(n+ ))∈Dk∪Dk+for k= , , , ,where
D=
(x,y)∈R:x> ,y≥–x, D=
(x,y)∈R:x> ,y< –x,
D=
(x,y)∈R:x< ,y≤–x, D=
(x,y)∈R:x< ,y> –x
and D=D.
Proof We prove only the casek= , because the other cases are carried out in the same manner. Let (x(n),y(n))∈D. Then we havenx(n) =y(n) +x(n)≥, and therefore, we
obtainx(n+ )≥x(n). Hence, we conclude that (x(n+ ),y(n+ ))∈D∪D.
Lemma . Suppose thatθ(n)andϕ(n)satisfyθ(n) =ϕ(n)and
θ(n+ )≥Fn,θ(n), ϕ(n+ ) =Fn,ϕ(n)
for n≤n<n,where F(n,x)is nondecreasing with respect to x∈Rfor each fixed n.Then
θ(n)≥ϕ(n)for n≤n≤n.
Proof We use mathematical induction onn. It is clear that the assertion is true forn=n.
Assume thatθ(n)≥ϕ(n) forn=p<n. SinceF(p,x) is nondecreasing with respect tox
for each fixedp, we haveθ(p+ )≥F(p,θ(p))≥F(p,ϕ(p)) =ϕ(p+ ). Thus, the assertion is also true forn=p+ . This completes the proof.
We are now ready to prove Theorem ..
Proof of Theorem. We give only the proof of the case that
f(x)
x ≤
+
(logx)
forx> sufficiently large. The proof is by contradiction. Suppose that all nontrivial solu-tions of equation (.) are oscillatory. Let (x(n),y(n)) be the solution of system (.) satis-fying the initial condition
x(n),y(n)
= el(n+)/, – +
l(n)
el(n+)/
Thenx(n) is a nontrivial oscillatory solution of equation (.) and
Hence, we see that
We compare the functionθ(n) with a solution
together with (.) and (.), we have
–
<ϕ(n)≤θ(n) < – ,
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 equa-tion (.) is /, that is, we prove the following result.
Proposition A. Equation(.)can be classified into two types as follows. (i) Ifλ> /,then all nontrivial solutions of equation(.)are oscillatory. (ii) Ifλ≤/,then all nontrivial solutions of equation(.)are nonoscillatory.
Proof The general solution of equation (.) is given by
x(n) = (for the proof, see [–]). Hence, we consider only the case thatλ= / because we can
easily check that all nontrivial solutions of equation (.) are nonoscillatory ifλ= /. In caseλ> /, equation (A.) has the conjugate rootsz= (±iα)/, whereα=√λ– . Hence, by (A.) and Euler’s formula, the real solution of equation (.) can be written as
forn≤j≤n– . If (K,K)= (, ), then
x(n) =K n–
j=n
r(j)
sin
n–
j=n
θ(j) +K
,
whereK=
K+K,sinK=K/KandcosK=K/K. Since
tanθ(n) = α n+ →
asn→ ∞, we obtainθ(n)∼tanθ(n) asn→ ∞. Using Lemma ., we have
n–
j=n
θ(j)∼ n–
j=n
α
j+ ∼
α
logn
asn→ ∞. Hence, for any sufficiently largep∈N, there existsn∈Nsuch that
pπ≤
n–
j=n
θ(j) +K< (p+ )π,
and therefore,x(n) is oscillatory.
We next consider the case thatλ< /. Put
ϕ(n) = n–
j=n +z
j
and ψ(n) = n–
j=n
+ –z
j
,
wherezsatisfies (A.). Then, without loss of generality, we may assume thatz> /. From (A.), the solution of equation (.) can be represented as
x(n) =Kϕ(n) +Kψ(n) =ϕ(n)
K+K
ψ(n)
ϕ(n)
for someK∈RandK∈R. By Stirling’s formula for the gamma function, we see that
(t+ )∼√πt t e
t
ast→ ∞, whereis the gamma function (as to Stirling’s formula, for example, see []). Hence, we have
ϕ(n) = n–
j=n +z
j
=(n)(n+z)
(n+z)(n)∼
(n)
(n+z)
nz and ψ(n)∼ (n)
(n+ –z)
n–z
asn→ ∞, and therefore, we obtain
lim
n→∞
ψ(n)
ϕ(n) = .
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author thanks the referees for their valuable suggestions that helped to improve this manuscript.
Received: 18 August 2012 Accepted: 30 November 2012 Published: 19 December 2012 References
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doi:10.1186/1687-1847-2012-218
Cite this article as:Yamaoka:Oscillation criteria for second-order nonlinear difference equations of Euler type.
