INVOLVING QUASI-DIFFERENCES
ZUZANA DOˇSL ´A AND ALEˇS KOBZAReceived 30 June 2004; Revised 20 September 2004; Accepted 12 October 2004
We study the third-order linear difference equation with quasi-differences and its adjoint equation. The main results of the paper describe relationships between the oscillatory and nonoscillatory solutions of both equations.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Consider the third-order linear difference equation
ΔpnΔrnΔxn+qnxn+1=0 (E)
and its adjoint equation
Δrn+1Δ
pnΔun−qn+1un+2=0, (EA)
whereΔis the forward difference operator defined byΔxn=xn+1−xn, (pn), (rn), and (qn) are sequences of positive real numbers forn∈N.
This paper has been motivated by the paper [9], where third-order difference equa-tions
Δ3v
n−pn+1Δvn+1+qn+1vn+1=0, ΔΔ2u
n−pn+1un+1
−qn+2un+2=0
(1.1)
had been investigated. As it is noted here, these equations are not adjoint equations and are referred to asquasi-adjointequations.
Equation (E) is a special case of linearnth-order difference equations with quasi-diff er-ences. Such equations have been widely studied in the literature, see, for example, [6,11] and the references therein. The natural question which arises is to find the adjoint equa-tion to (E) and to examine the connection between solutions of (E) and its adjoint one.
Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 65652, Pages1–13
In the continuous case, it holds (see, e.g., [5, Theorem 8.33]) that
1 p(t)
1 r(t)x
(t)
+q(t)x(t)=0 (1.2)
is oscillatory if and only if the adjoint equation
1 r(t)
1
p(t)x(t)
−q(t)x(t)=0 (1.3)
has the same property. In addition, nonoscillatory solutions of these equations satisfy some interesting relationships, see, for example, [2,5].
The aim of this paper is to investigate oscillatory and asymptotic properties of solu-tions of (E) and (EA). We will prove that (EA) is the adjoint equation to (E) and we will give discrete analogues of the above-quoted results for third-order differential equations. Moreover, the oscillation of (E) and (EA) is characterized by means of second-order linear difference equations and the problem of the number of oscillatory solutions in a given ba-sis for the solution space of (E) and (EA) is investigated. Our results extend and complete results of [7–10] stated for the various forms of third-order difference equations.
A solutionxof (E) is a real sequence (xn) defined for alln∈Nand satisfying (E) for alln∈N. A solution of (E) is callednontrivialif for anyn0≥1, there existsn > n0such thatxn=0. Otherwise, the solution is calledtrivial. A nontrivial solutionxof (E) is said to be oscillatory if for anyn0≥1, there exists n > n0 such thatxn+1xn≤0. Otherwise, the nontrivial solution is said to benonoscillatory. Equation (E) isoscillatoryif it has an oscillatory solution. The same terminology is used for (EA).
Denote quasi-differencesx[i],i=0, 1, 2, of a solutionxof (E) as follows:
x[0]
n =xn, xn[1]=rnΔxn, xn[2]=pnΔx[1]n , x[3]n =Δxn[2]. (1.4)
Similarly, denote quasi-differencesu[i],i=0, 1, 2, of a solutionuof (EA) as follows:
u[0]
n =un, u[1]n =pnΔun, u[2]n =rn+1Δu[1]n , u[3]n =Δu[2]n . (1.5)
All nonoscillatory solutionsxof (E) can be a priori classified to the following classes:
N0=x:∃nxs.t.xnx[1]n <0,xnxn[2]>0∀n≥nx , N1=x:∃nxs.t.xnx[1]n >0,xnxn[2]<0∀n≥nx , N2=x:∃nxs.t.xnx[1]n >0,xnxn[2]>0∀n≥nx , N3=x:∃nxs.t.xnx[1]n <0,xnxn[2]<0∀n≥nx ,
(1.6)
2. Relationship between (E) and (EA)
Solutions of (E) and (EA) are related by the following properties.
Theorem2.1. (a)Letx,ybe solutions of (E). Then the sequenceC=(Cn) (n≥2)such that
Cn−1=C
xn−1,yn−1
≡
xxn[1]−1 yn−1 n−1 y[1]n−1
(2.1)
is a solution of (EA).
(b)Letu,vbe solutions of (EA). Then the sequenceD=(Dn) (n≥2)such that
Dn=Dun−1,vn−1≡
un−
1 vn−1 u[1]
n−1 vn[1]−1
(2.2)
is a solution of (E).
Proof. Claim (a). For any two solutionsx,yof (E), we have
xnΔy[2]
n−1−ynΔx [2]
n−1= −xnqn−1yn+ynqn−1xn=0. (2.3)
Therefore,
ΔCn−1=xnΔy[1]n−1+yn[1]−1Δxn−1−ynΔx[1]n−1−x[1]n−1Δyn−1
=xnΔy[1]
n−1+rn−1Δyn−1Δxn−1−ynΔxn[1]−1−rn−1Δxn−1Δyn−1 =xnΔy[1]
n−1−ynΔx[1]n−1.
(2.4)
Using the factx[2]n−1=x[2]n −Δxn[2]−1and (2.3), we obtain
C[1]
n−1=pn−1ΔCn−1=xny[2]n−1−ynxn[2]−1
=xn
y[2]
n −Δyn[2]−1
−yn
x[2]
n −Δxn[2]−1
=xny[2]n −ynxn[2].
(2.5)
By a direct computation in view of (2.3), we get
ΔC[1]
n−1=xn+1Δy[2]n +yn[2]Δxn−yn+1Δx[2]n −x[2]n Δyn=yn[2]Δxn−x[2]n Δyn, (2.6)
hence
C[2]
Finally
Claim (b). By the similar argument as in (a), we get
ΔDn=unΔvn[1]−1−vnΔu[1]n−1. (2.9)
Using the same argument as before, we get
ΔD[1]
Remark 2.2. According to [1, page 60], ifX= {Xn}is a nontrivial solution of the system
Xn+1=AnXn, (2.14)
thenU= {Un}, whereUn=(XnT)−1is a solution of the system
Un=ATnUn+1. (2.15)
System (2.15) is called theadjoint systemof (2.14).
Equation (E) can be written as a first-order difference system
Δx[0]
Using the usual convention that no index actually means the indexn, otherwise the index is explicitly specified, we obtain
⎛
Hence (E) can be interpreted as the system of the form (2.14). Its adjoint system is
and the last equation givesΔu[1]n+1= −Δ(pnΔu [2]
n ). Replacing the shiftnbyn+ 1 and sub-stituting into the second equation, we have
−ΔpnΔu[2]
Multiplying this equation by−rn+1and differentiating it, we obtain
Δrn+1Δ
Substituting from the first equation in (2.20), we get
Δrn+1Δ
Notation 2.3. LetSdenote the solution space of (E) and letSdenote the solution space of (EA). For (x,u)∈S×S, defineᏸ=(ᏸn), where
ᏸn=ᏸxn,un=xn+1un[2]−x[1]n+1u[1]n +xn[2]+1un+1. (2.24)
The functionalᏸhas the following properties.
Lemma2.4. The sequenceᏸ:S×S→Ris a constant which depends only on the choice of
solutionsxandu, and not onn. Proof. By a direct computation we get
Δᏸn=Δ
which completes the proof.
Proof. ExpandingRnalong its third column, we obtain
Rn=zn
x
[1] n yn[1] x[2]
n yn[2]
−z[1]n
xx[2]n yn n y[2]n
+z[2]
n
xx[1]n yn n yn[1]
. (2.28)
Using (2.24), we have
ᏸzn−1,Cn−1
=znC[2]n−1−z[1]n Cn[1]−1+z[2]n Cn. (2.29)
From here, (2.1), (2.5), and (2.7) show that (2.27) holds.
3. Nonoscillatory solutions of adjoint equations
In this section, we study nonoscillatory solutions. We start with the following auxiliary results.
Lemma3.1. There always exists nonoscillatory solutionuof (EA) with the property
un>0, u[1]n >0, u[2]n >0 forn∈N, (3.1)
that is, (EA) has a strongly monotone solution. For the proof, see [4, Theorem 3.2].
Lemma3.2. If a solutionyof (E) satisfies for some integerm >1that
ym≥0, ym[1]≤0, y[2]m >0, (3.2)
then
yk>0, yk[1]<0, yk[2]>0 (3.3)
for eachk∈Nsuch that1≤k < m.
The proof follows from the proof of [3, Proposition 2].
The existence of Kneser solutions of (E) is ensured by the following result.
Theorem3.3. There always exists nonoscillatory solutionxof (E) with the property
xn>0, x[1]n <0, xn[2]>0 forn∈N, (3.4)
that is, (E) has a Kneser solution.
Proof. Letx=(x(n)),y=(y(n)),z=(z(n)) be a basis of the solution spaceSof (E). For k∈N, define
ωk(n)=akx(n) +bky(n) +ckz(n), (3.5)
whereak,bk,ckare chosen such that
Thenω[1]k (k)=0. By [3, Lemma 1],ωk(k+ 2)=0. Without loss of generality, assume that ωk(k+ 2)>0. Then
ω[1]
k (k+ 1)=rk+1Δωk(k+ 1)=rk+1
ωk(k+ 2)−ωk(k+ 1)>0, (3.7)
hence
ω[2]
k (k)=pkΔω[1]k (k)=pk
ω[1]
k (k+ 1)−ω[1]k (k)
>0. (3.8)
Since
ωk(k)=0, ω[1]k (k)=0, ωk[2](k)>0, (3.9)
byLemma 3.2
ωk(n)>0, ωk[1](n)<0, ω[2]k (n)>0, for 1≤n < k. (3.10)
PutAk=(ak,bk,ck). ThenAk =1 for eachk. The unit ball is compact inR3, so (Ak) has a convergent subsequence (Aki). Denote
A=lim
i→∞Aki=(a,b,c). (3.11)
Thena2+b2+c2=1 and
ω(n)=lim
i→∞ωki=lim
akix(n) +bkiy(n) +ckiz(n) (3.12)
is a nontrivial solution of (E). Then in view of (3.10) and the fact thatk is arbitrary integer, we get
ω(n)≥0, ω[1](n)≤0, ω[2](n)≥0 forn≥1. (3.13)
Ifω(n0)=0 for somen=n0, thenω(n)=0 for alln≥n0which is a contradiction with the fact thatωis a nontrivial solution. Thusω(n)>0 for everyn≥1, and so
Δω[2](n)= −q(n)ω(n+ 1)<0 forn≥1. (3.14)
Hence,ω[2]is decreasing and soω[2](n)>0 forn∈N. From hereΔω[1](n)>0 forn∈
N, which implies thatω[1]is increasing andω[1](n)<0 forn∈N.
Theorem3.4. Every nonoscillatory solution of (EA) is strongly monotone if and only if every
nonoscillatory solution of (E) is a Kneser solution.
Theorem 2.1solution of (EA) and in view of (2.5) and (2.7) it satisfies, for largen,
Cn−1>0, C[1]n−1<0 (ifi=1) Cn−1>0, C[2]n−1<0 (ifi=2) C[1]
n−1<0, Cn[2]−1>0 (ifi=3).
(3.15)
This is a contradiction with the fact thatCis strongly monotone solution.
Now suppose that every solution of (E) is a Kneser solution. Assume by contradiction that there exists solutionvof (EA) which belongs to the classNi, wherei∈ {0, 1, 3}. Let ube a strongly monotone solution of (EA). Without loss of generality, we may suppose thatun>0 andvn>0 for largen. Then the sequenceDdefined by (2.2) is according to Theorem 2.1solution of (E) and it satisfies, for largen,
Dn<0, D[2]n >0 (ifi=0) D[1]
n <0, D[2]n <0 (ifi=1) Dn<0, Dn[1]<0 (ifi=3).
(3.16)
This is a contradiction with the fact thatDis a Kneser solution.
4. Oscillatory properties of adjoint equations
Lemma4.1. Letube a strongly monotone solution andvan oscillatory solution of (EA).
Then their CasoratianDdefined by (2.2) is an oscillatory solution of (E).
Proof. ByTheorem 2.1,Dis a solution of (E). We will show thatDis an oscillatory so-lution. Without loss of generality, we may suppose thatusatisfies (3.1). Sincevis an os-cillatory solution, there exist increasing sequences of positive integers (in) and (jn), with properties
vin≤0, v[1]in >0 forn∈N, vjn≥0, v[1]jn <0 forn∈N.
(4.1)
From the above inequalities, (2.2), and (3.1), we have
Din+1=uinv[1]in −vinu[1]in >0 forn∈N, (4.2)
and similarlyDjn+1<0 for n∈N. Hence the sequenceD is an oscillatory solution of
(E).
Lemma 4.2. Let x be a Kneser solution and y an oscillatory solution of (E). Then their
CasoratianCdefined by (2.1) is an oscillatory solution of (EA).
with properties
i1> M, yin≤0, y[1]in >0 forn∈N,
j1> M, yjn≥0, y[1]jn <0 forn∈N, (4.3)
whereM=min{n∈N:ynyn+1≤0}.
Assume thaty[2]jn >0 for somen∈N. Then byLemma 3.2, we get
yk>0, yk[1]<0 for 1≤k < jn, (4.4)
which is a contradiction withj1> M. Hencey[2]jn ≤0 forn∈N. From here and using (2.7) follows
C[2]
jn−1=x[1]jn y[2]jn −y[1]jn x[2]jn >0 forn∈N. (4.5)
By similar argument as before, we obtainyin[2]≥0 forn∈N, which implies that
C[2]
in−1=x[1]in yin[2]−yin[1]x[2]in <0 forn∈N. (4.6)
By [3, Lemma 2], it follows from inequalities (4.5) and (4.6) thatCis an oscillatory
solu-tion of (EA). The proof is now complete.
Our next result characterizes the existence of oscillatory solutions of the adjoint equa-tions.
Theorem4.3. Equation (EA) is oscillatory if and only if (E) is oscillatory. The proof follows fromTheorem 3.3and Lemmas3.1,4.1, and4.2.
In the sequel, we study the existence of an oscillatory solution in terms of second-order equations.
Theorem4.4. (a)Ifuis a nonoscillatory solution of (EA), then two linearly independent
solutions of (E) satisfy the second-order difference equation
pn+1Δ
r
n+1Δxn+1 un+1
+ u
[2] n
un+1un+2xn+2=
0. (4.7)
(b)Ifxis a nonoscillatory solution of (E), then two linearly independent solutions of (EA) satisfy the second-order difference equation
rn+1Δ
pnΔu
n xn+1
+ x
[2] n+1
xn+1xn+2un+1=0. (4.8)
Proof. Claim (a). Letu be a fixed nonoscillatory solution of (EA) such thatun>0 for n≥N. LetL:S→Rbe the functional onSdefined byL(x)=ᏸ(xn,un). The set
is the kernel of linear functionalLdefined onS. Thenx∈Ksatisfies
In view of the identity x[1]
is the kernel of linear functional defined onS. Thenu∈Ksatisfies
Corollary 4.5. (a)If (E) is oscillatory, then there exists a basis forS consisting of one nonoscillatory solution and two oscillatory solutions.
(b)If (EA) is oscillatory, then there exists a basis forSconsisting of one nonoscillatory solution and two oscillatory solutions.
Proof. ByTheorem 3.3andLemma 3.1, there exist Kneser solution of (E) and strongly monotone solution of (EA). Assume that (E) and (EA) are oscillatory. In view ofTheorem 4.4and its proof, (E) has two independent solutions which must be oscillatory. Similarly, there exist two independent oscillatory solutions of (EA). Because dimS=dimS=3,
the proof is complete.
Theorem4.6. (a)If (4.7), whereuis a nonoscillatory solution of (EA), is oscillatory, then
(E) and (EA) are oscillatory.
(b)If (4.8), wherexis a nonoscillatory solution of (E), is oscillatory, then (E) and (EA) are oscillatory.
Proof. Assume that (4.7) is oscillatory, that is, there exists an oscillatory solution x of (4.7). Using the same argument as in the proof ofTheorem 4.4, we obtain thatx∈K, whereK is defined by (4.9). Hencexis an oscillatory solution of (E). ByTheorem 4.3, (EA) is oscillatory, too.
Similarly, ifuis an oscillatory solution of (4.8), thenu∈K. Henceuis oscillatory solution of (EA) and this implies that (E) is oscillatory too.
Theorem4.7. Equation (E) is oscillatory if and only if (4.8) is oscillatory.
Proof. Assume that (E) is oscillatory, that is, there exists an oscillatory solutionyof (E). ByTheorem 3.3, there exists a Kneser solutionxof (E). According to (2.27),
0=
xn yn xn x[1]
n y[1]n xn[1] x[2]
n y[2]n xn[2]
=ᏸ
xn−1,Cn−1, (4.19)
whereC is defined by (2.1). ByTheorem 2.1, Cis solution of (EA). From Lemma 4.2
follows thatCis an oscillatory solution of (EA). In view ofLemma 2.4, L(C)=ᏸx
n,Cn=0, (4.20)
henceC∈K, whereKis defined by (4.15). From here and the proof ofTheorem 4.4, we get the fact thatCis an oscillatory solution of (4.8). The opposite statement follows
fromTheorem 4.6.
Open problems.
(1) It is an open problem whether the existence of an oscillatory solution of (EA) im-plies the oscillation of (4.7). To solve this problem, it would be useful to find the functionalᏸdefined onS×Swith similar properties to those ofᏸdescribed in Lemmas2.4and2.5.
Acknowledgment
This work was supported by the Czech Grant Agency, Grant 201/04/0586.
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Zuzana Doˇsl´a: Department of Mathematics, Masaryk University, Jan´aˇckovo n´am. 2a, 602 00 Brno, Czech Republic
E-mail address:[email protected]
Aleˇs Kobza: Department of Mathematics, Masaryk University, Jan´aˇckovo n´am. 2a, 602 00 Brno, Czech Republic
