doi:10.1155/2010/973432
Research Article
Positive Decreasing Solutions of Higher-Order
Nonlinear Difference Equations
B. Krasznai, I. Gy ˝ori, and M. Pituk
Department of Mathematics, University of Pannonia, P.O. Box 158, 8201 Veszpr´em, Hungary
Correspondence should be addressed to M. Pituk,[email protected]
Received 23 December 2009; Accepted 13 May 2010
Academic Editor: A ˘gacik Zafer
Copyrightq2010 B. Krasznai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is shown that the decay rates of the positive, monotone decreasing solutions approaching the zero equilibrium of higher-order nonlinear difference equations are related to the positive characteristic values of the corresponding linearized equation. If the nonlinearity is sufficiently smooth, this result yields an asymptotic formula for the positive, monotone decreasing solutions.
1. Introduction and the Main Results
LetR,C, andZbe the set of real and complex numbers and the set of integers, respectively. The symbolZdenotes the set of nonnegative integers.
Recently, Aprahamian et al.1 have studied the second-order nonlinear difference equation
Δ2xn−1 cΔxn fxn 0, n∈Z, 1.1
where c > 0, Δxn xn1 −xn so that Δ2xn−1 xn1 −2xn xn−1
and f : R → Ris a Lipschitz continuous function such that f > 0 in0,1 and f 0 is identically in−∞,0∪1,∞ . As noted in1,1.1 is related to the discretization of traveling wave solutions of the Fisher-Kolmogorov partial differential equation. The main result of1
Theorem 1.1. In addition to the above hypotheses onf, suppose that there exist∈0,1 andk >0
such that
y
0
fs ds≤ k
2y
2, y∈0, , 1.2
kk24k < c. 1.3
Then1.1 has a strictly decreasing solutionx:Z → 0, such that
lim
n→ ∞xn 0, 1.4
∞
n1
1c n|Δxn−1 |2 <∞. 1.5
Note that in1, solutions of1.1 satisfying conditions1.4 and1.5 ofTheorem 1.1
are calledfast solutions.
In this paper, we give an asymptotic description of the decreasing fast solutions of
1.1 described inTheorem 1.1. We will also consider a similar problem for the more general higher-order equation
xn1 gxn , xn−1 , . . . , xn−k , n∈Z, 1.6
where k ∈ Z and g : Ω → R, withΩ being a convex open neighborhood of 0 in Rk1.
Throughout the paper, we will assume that the partial derivativesDjg, 1 ≤ j ≤ k 1, are continuous on Ω and g0 0 so that 1.6 has the zero equilibrium; moreover, the mild technical assumption
Dk1g0 /0 1.7
holds. We are interested in the asymptotic properties of those positive, monotone decreasing solutions of1.6 which tend to the zero equilibrium, that is,
lim
n→ ∞xn 0. 1.8
We will show that under appropriate assumptions the decay rates of these solutions are equal to the characteristic values of the corresponding linearized equation belonging to the interval
0,1. Our result, combined with asymptotic theorems from 2 or 3, yields asymptotic formulas for the positive, monotone decreasing solutions of1.6 .
Before we formulate our main theorems, we introduce some notations and definitions. Associated with 1.6 is the linearization about the zero equilibrium, namely, the linear homogeneous equation
yn1 k
j0 ajy
with coefficients
ajDj1g0 , 0≤j≤k. 1.10
By acharacteristic valueof1.9, we mean a complex root of thecharacteristic polynomial
Pz zk1− k
j0
ajzk−j. 1.11
Thus, the set of all characteristic values of1.9 is given by
ZP {ζ∈C|Pζ 0}. 1.12
To eachζ∈ZP , there corresponds solutions of1.9 of the form
yn qn ζn, n∈Z, 1.13
whereqis a polynomial of degree less thanmζ , the multiplicity ofζas a root ofP. Such solutions are calledcharacteristic solutionscorresponding toζ. IfS ⊂ ZP is a nonempty set of characteristic values, then by acharacteristic solution corresponding to the setS, we mean a finite sum of characteristic solutions corresponding to valuesζ∈S.
Now we can formulate our main results. The first theorem applies to the positive, monotone decreasing solutions of1.6 provided that zero is a hyperbolic equilibrium of1.6 . Recall that the zero equilibrium of1.6 ishyperbolicif the linearized equation 1.9 has no characteristic values on the unit circle{z∈C| |z|1}.
Theorem 1.2. In addition to the above hypotheses ong, suppose that the partial derivativesDjg, 1≤j≤k1, are Lipschitz continuous on compact subsets ofΩ. Assume also that zero is a hyperbolic equilibrium of 1.6 . Letxbe a positive, monotone decreasing solution of 1.6 satisfying1.8 . Then
the limit
λ lim n→ ∞
n
xn 1.14
exists andλis a characteristic value of the linearized equation1.9 belonging to the interval0,1 .
Moreover, there exists∈0, λ such that the asymptotic representation
xn yn Oλ− n, n−→ ∞, 1.15
holds, whereyis a positive characteristic solution of the linearized equation1.9 corresponding to the set
In contrast toTheorem 1.2, the next result applies also in some cases when the zero equilibrium of1.6 is not hyperbolic.
Theorem 1.3. In addition to the hypotheses on g, suppose that the linearized equation1.9 has exactly one characteristic valueλin the interval0,1. Assume also thatλis the only characteristic value of 1.9 on the circle{z∈C| |z|λ}and
mλ ≤2, 1.17
wheremλ is the multiplicity ofλas a root of the characteristic polynomialP. Letxbe a positive, monotone decreasing solution of 1.6 satisfying1.8 . Then
lim n→ ∞
xn1
xn λ. 1.18
Note that conclusion1.18 ofTheorem 1.3is stronger than1.14 .
For the second-order equation1.1 , we have the following theorem which provides new information about the decreasing fast solutions obtained by Aprahamian et al.1.
Theorem 1.4. Adopt the hypotheses ofTheorem 1.1. Letx:Z → 0, be a monotone decreasing solution of1.1 satisfying conditions1.4 and1.5 . If the (finite) right-hand derivativef0 exists, then
lim n→ ∞
xn1
xn λ, 1.19
whereλis the unique root of the equation
z2−2c−f
0
1c z
1
1c 0, 1.20
in the interval0,1/√1c .
Iffis Lipschitz continuous on0,1, then1.19 can be replaced with the stronger conclusion
lim n→ ∞
xn
λn L for someL∈0,∞ . 1.21
The proofs of the above theorems are given inSection 3.
2. Preliminary Results
Consider the linear homogeneous difference equation
xn1 k
j0 bjn x
n−j, n∈Z, 2.1
where the coefficientsbj :Z → R, 0≤j ≤k, areasymptotically constant, that is, thefinite limits
aj lim
n→ ∞bjn , 0≤j≤k, 2.2
exist. We will assume that
ak/0. 2.3
If we replace the coefficients in2.1 with their limits, we obtain thelimiting equation
un1 k
j0 aju
n−j, n∈Z. 2.4
Theorem 1.2will be deduced from the following proposition.
Proposition 2.1. Suppose2.2 and2.3 hold. Assume that the convergence in2.2 is exponential, that is, there exists a constantη∈0,1 such that
bjn ajO
ηn, n−→ ∞. 2.5
Letxbe a positive, monotone decreasing solution of 2.1 . Then the limit
λ lim n→ ∞
n
xn 2.6
exists andλis a characteristic value of the limiting equation2.4 belonging to the interval0,1.
Moreover, there exists∈0, λ such that the asymptotic representation
xn un Oλ− n, n−→ ∞, 2.7
holds, whereuis a positive characteristic solution of the limiting equation2.4 corresponding to the set of characteristic values
S{ζ∈ZP | |ζ|λ}, 2.8
Proof. Equation2.1 can be written in the form
whereris the radius of convergence given by
rlim sup n→ ∞
n
xn . 2.12
The boundedness ofximplies thatr ≤1. By the application of a Perron-type theoremsee4, Theorem 2or5, Theorem 8.47 , we conclude thatr |ζ|for some characteristic valueζof
it follows by easy calculations that
Pz xz Fz forz∈Cwith|z|> r, 2.16
wherePis the polynomial given by1.11 and
Fz Qz zkhz 2.17
Since the coefficients of thez-transformxare positive, according to Prinsheim’s theoremsee
6, Theorem 18.3or7, Theorem 2.1, page 262 x has a singularity atz r. Sincehand henceF is holomorphic in the region|z|> ηr, this implies thatPr 0. Otherwise, 2.16
would imply thatxcan be extended as a holomorphic function to a neighborhood ofzrby
Sincexis monotone decreasing, it follows forn∈Z,
xn xn1 · · ·xnk ≤k1 xn 2.20
Lettingn → ∞in the last inequality, and using2.19 , we find that
lim inf
This proves the existence of the limit2.6 withλr. Finally, conclusion2.7 is an immediate consequence of2, Theorem 2.3 and Remark 2.5.
Proposition 2.2. Suppose2.2 and2.3 hold. Assume that the limiting equation2.4 has exactly one characteristic valueλin the interval0,1. Assume also thatλis the only characteristic value of 2.4 on the circle{z∈C| |z|λ}and
mλ ≤2, 2.23
wheremλ is the multiplicity ofλas a root of the characteristic polynomial corresponding to2.4 .
Letxbe a positive, monotone decreasing solution of 2.1 . Then
lim n→ ∞
xn1
xn λ. 2.24
Before we give a proof ofProposition 2.2, we establish two lemmas.
Lemma 2.3. Suppose2.2 and2.3 hold. Letxbe a positive, monotone decreasing solution of2.1 .
Then there existsα∈0,1 such that
α≤ xn1
xn ≤1, n∈Z
. 2.25
Proof. The second inequality in2.25 follows from the monotonicity ofx. In order to prove the first inequality, suppose by the way of contradiction that there exists a strictly increasing sequence{ni}∞i0inZsuch that
From this and from the fact thatxis monotone decreasing, we find forn∈Z,
|bkn | ≤
Lemma 2.4. Suppose that2.4 has exactly one characteristic valueλin the interval0,1. Assume
also thatλis the only characteristic value of 2.4 on the circle{z∈C| |z|λ}and2.23 holds. Let
u:Z → 0,∞ be a positive, monotone decreasing biinfinite sequence satisfying2.4 onZ, that is,
un1 k
j0 aju
n−j, n∈Z. 2.30
Then
un u0 λn, n∈Z. 2.31
Proof. We will prove the lemma by using a similar method as in the proof of8, Lemma 3.2. The radius of convergencerof thez-transform
uz
∞
n0
unz−n 2.32
ofuis given by
rlim sup n→ ∞
n
un . 2.33
Sinceuis bounded onZ,r ≤1, and4, Theorem 2orLemma 2.3implies thatr >0. Taking thez-transform of2.4, we obtain
Pz uz Qz for z∈Cwith|z|> r, 2.34
wherePis given by1.11 and
Qz u0 zk1 k
j1 aj
−1
n−j
un zk−n−j. 2.35
Relation2.34 , combined with Pringsheim’s theorem6, Theorem 18.3, implies thatPr
0.Otherwise,ucan be extended as a holomorphic function to a neighborhood ofz r by
uQ/P. Sincer∈0,1and the only root ofPin0,1isλ, we have thatrλ. Define
Uz
∞
n1
u−n zn forz∈Cwith|z|< R, 2.36
whereRis the radius of convergence of the above power series given by
R 1
Sinceuis monotone decreasing, we have
n
u−n ≥n
u0 , n∈Z. 2.38
Therefore,
lim sup n→ ∞
n
u−n ≥1 2.39
and henceR≤1. As a solution of a constant coefficient equation,uis a sum of characteristic solutions. Therefore,
lim sup n→ ∞
n
u−n <∞ 2.40
and henceR >0. From2.30 , we find forn∈Zandz∈C,
u−n−1 zn k
j0 aju
−njzn. 2.41
Summation fromn1 to infinity and the definition ofUyield
Pz Uz −Qz forz∈Cwith|z|< R, 2.42
withQas in2.35 . This, combined with Prinsheim’s theorem, implies thatPR 0. Since
R ∈ 0,1and the only root of P in0,1isλ, we have thatR λ. Thus,r R λ. This, together with2.34 and2.42 , implies that the holomorphic functionHdefined on the open setW{z∈C| |z|/λ}by
Hz ⎧ ⎨ ⎩
uz if|z|> λ,
−Uz if|z|< λ.
2.43
satisfies
Pz Hz Qz , z∈W. 2.44
By hypotheses,z λis the only root ofP on the circle|z| λ. Therefore,2.44 implies that
Hcan be extended as a holomorphic function toC\ {λ}by
Hz Qz
Moreover, sincemλ ≤2, the functionE:C\ {λ} → Cdefined by
Ez z−λ 2Hz , z∈C\ {λ}, 2.46
has a removable singularity atz λ. Thus, it can be regarded as an entire function. Using
1.11 ,2.35 , and2.44 , we obtain forz∈C\ZP ,
Hz Qz
Pz O1 , |z| −→ ∞. 2.47
Hence
Ez O|z|2, |z| −→ ∞. 2.48
By the application of the Extended Liouville Theorem6, Theorem 5.11, we conclude thatE
is a polynomial of degree at most 2, that is,
Ez az2bzc, z∈C, 2.49
for somea,b,c∈C. SinceH0 U0 0, we have
cE0 λ2H0 0. 2.50
Therefore,
Hz Ez
z−λ 2
az2bz
z−λ 2
azz−λ λz bz
z−λ 2 , z /λ. 2.51
Hence
Hz d λz
z−λ 2 a z
z−λ, z /λ, 2.52
wheredabλ−1. This, together with2.43 , implies
uz d λz
z−λ 2 a z
z−λ, |z|> λ. 2.53
From this, in view of the uniqueness of thez-transform, we obtain
From2.43 and2.52 , we obtain
Uz −Hz −d λz
z−λ 2 −a z
z−λ, |z|< λ. 2.55
From this and 2.36 , in view of the uniqueness of the coefficients of the power series, we obtain
u−n −dnλ−naλ−n, n∈Z. 2.56
This, together with2.54 , yields
un dnλnaλn, n∈Z. 2.57
From this, taking into account thatun >0 for alln∈Z, we see thatd0. Therefore,2.57
reduces to2.31 .
Now we are in a position to give a proof ofProposition 2.2.
Proof ofProposition 2.2. Letabe an arbitrary accumulation point of{xn1 /xn}∞n0so that for some strictly increasing sequence{ni}∞i0inZ,
lim i→ ∞
xni1 xni
a. 2.58
In order to prove2.24 , it is enough to show thataλ. From conclusion2.25 ofLemma 2.3
and the relation
xnm xn
xn1
xn
xn2
xn1 · · ·
xnm
xnm−1 , n, m∈Z
, 2.59
we obtain
αm≤ xnm
xn ≤1, n, m∈Z
,
1≤ xn−m
xn ≤α
−m, n≥m, n, m∈Z.
2.60
From2.60 , it follows by the standard diagonal choicesee the proof of8, Theorem 1.3 that there exists a subsequence{di}∞i0of{ni}∞i0such that the limits
um lim i→ ∞
xdim xdi
exist and are finite. Letm∈Zbe fixed. Writingndimin2.1 and dividing the resulting equation byxdi , we obtain
xdim1 xdi
k
j0
bjdim
xdim−j
xdi
, i≥i0, 2.62
wherei0 ∈ Z is so large that di ≥ −mfori ≥ i0. From this, lettingi → ∞and using2.2 and2.61 , we see that the biinfinite sequenceusatisfies2.30 . Further, it is easily seen that
uinherits the monotone decreasing property ofx. Finally, from estimates2.60 , we see that
u >0 onZ. By the application ofLemma 2.4, we conclude thatuhas the form2.31 . Since
u0 1see2.61 ,2.31 yields
un λn, n∈Z. 2.63
From this and2.61 , taking into account that{di}∞i0is a subsequence of{ni}∞i0, we obtain
a lim i→ ∞
xni1 xni
lim i→ ∞
xdi1 xdi
u1 λ. 2.64
3. Proofs of the Main Theorems
Proof ofTheorem 1.2. Sinceg0 0, from1.6 , we find forn∈Z,
xn1 1
0 d
dsgsxn , sxn−1 , . . . , sxn−k ds
k1
i1 1
0
Digsxn , sxn−1 , . . . , sxn−k ds xn−i1 .
3.1
Thus,xis a solution of2.1 with coefficients
bjn 1
0
Dj1gsxn , sxn−1 , . . . , sxn−k ds, 0≤j ≤k. 3.2
By virtue of1.8 , the limits in2.2 exist and are given by1.10 , that is, the limiting equation
2.4 coincides with the linearized equation 1.9. By virtue of 1.7 , hypothesis 2.3 of
Proposition 2.1is satisfied. Let
rlim sup n→ ∞
n
By virtue of the boundedness ofx,r ≤1. As noted in the proof ofProposition 2.1, according to4, Theorem 2, 1 ≥ r |ζ|for someζ ∈ ZP . Since zero is a hyperbolic equilibrium of
1.6 ,|ζ|/1 for allζ∈ZP . Therefore,r <1. Chooseη∈r,1 . By virtue of3.3 ,
xn Oηn, n−→ ∞. 3.4
From this,1.10 ,3.2 and the Lipschitz continuity of the partial derivatives Djg, 1 ≤ j ≤ k1, it is easily shown that hypothesis2.5 ofProposition 2.1also holds. The conclusions of
Theorem 1.2follow fromProposition 2.1.
Proof ofTheorem 1.3. As noted in the proof ofTheorem 1.2,xis a solution of 2.1 with the asymptotically constant coefficients given by 3.2 and the limiting equation of 2.1 is the linearized equation 1.9 . Further, by virtue of 1.7, condition 2.3 holds. Thus, all hypotheses ofProposition 2.2are satisfied and the result follows fromProposition 2.2.
Proof ofTheorem 1.4. Equation1.1 can be written in the form
xn1 2c
Consequently,xis a solution of the equation
xn1
Therefore, the coefficients in3.6 are asymptotically constant and the corresponding limiting equation is
with the last but one equality being a consequence of L’Hospital’s rule. This, together with
1.3 , yields
Hence
c−α >2√α. 3.11
The characteristic polynomialPcorresponding to3.8 is
Pz z2− 2c−α
Further, it is easily shown that3.11 implies that
Hence
l lim n→ ∞
xn1
xn nlim→ ∞ n
xn ≤ √1
1c. 3.19
Sinceμ >1/√1c, we have thatlλand1.19 holds.
Now suppose thatfis Lipschitz continuous on0,1. For all largen, we have
fxn −αxn 1 0
fsxn −f0 dsxn ≤Kx2n , 3.20
whereKis the Lipschitz constant offon0,1. From this and3.18 , we see that
fxn
xn αO ⎛ ⎜ ⎝√ 1
1cn ⎞ ⎟
⎠, n−→ ∞. 3.21
This shows that the convergence of the coefficients of 3.6 to their limits is exponentially fast. Thus,Proposition 2.1applies and the limit relation1.21 follows from the asymptotic formula2.7.
Acknowledgment
This research was supported in part by the Hungarian National Foundation for Scientific ResearchOTKA Grant no. K 732724.
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