# Conjugacy of Self-Adjoint Difference Equations of Even Order

Authors: Petr Hasil
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study oscillation properties of2n-order Sturm-Liouville difference equations. For these equations, we show a conjugacy criterion using the p-criticality (the existence of linear dependent recessive solutions at ∞ and
-∞). We also show the equivalent condition of
p-criticality for one term 2n-order equations.


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## Body

## 1. Introduction

In this paper, we deal with2n-order Sturm-Liouville difference equations and operators
(1.1)L(y)k=∑ν=0n(-Δ)ν(rk[ν]Δνyk+n-ν)=0,rk[n]>0,k∈Z,
where Δ is the forward difference operator, that is, Δyk=yk+1-yk, and r[ν],ν=0,…,n, are real-valued sequences. The main result is the conjugacy criterion which we formulate for the equation L(y)k+qkyk+n=0, that is viewed as a perturbation of (1.1), and we suppose that (1.1) is at least p-critical for some p∈{1,…,n}. The concept of p-criticality (a disconjugate equation is said to be p-critical if and only if it possesses p solutions that are recessive both at ∞ and -∞, see Section 3) was introduced for second-order difference equations in [1], and later in [2] for (1.1). For the continuous counterpart of the used techniques, see [3–5] from where we get an inspiration for our research.The paper is organized as follows. In Section2, we recall necessary preliminaries. In Section 3, we recall the concept of p-criticality, as introduced in [2], and show the first result—the equivalent condition of p-criticality for the one term difference equation
(1.2)Δn(rkΔnyk)=0
(Theorem 3.4). In Section 4 we show the conjugacy criterion for equation
(1.3)(-Δ)n(rkΔnyk)+qkyk+n=0,
and Section 5 is devoted to the generalization of this criterion to the equation with the middle terms
(1.4)∑ν=0n(-Δ)ν(rk[ν]Δνyk+n-ν)+qkyk+n=0.


## 2. Preliminaries

The proof of our main result is based on equivalency of (1.1) and the linear Hamiltonian difference systems
(2.1)Δxk=Axk+1+Bkuk,Δuk=Ckxk+1-ATuk,
where A,Bk, and Ck are n×n matrices of which Bk and Ck are symmetric. Therefore, we start this section recalling the properties of (2.1), which we will need later. For more details, see the papers [6–11] and the books [12, 13].The substitution(2.2)xk[y]=(yk+n-1Δyk+n-2⋮Δn-1yk),uk[y]=(∑ν=1n(-Δ)ν-1(rk[ν]Δνyk+n-ν)⋮-Δ(rk[n]Δnyk)+rk[n-1]Δn-1yk+1rk[n]Δnyk)
transforms  (1.1) to linear Hamiltonian system (2.1) with the n×n matrices A,Bk, and Ck given by
(2.3)A=(aij)i,j=1n,aij={1,ifj=i+1,i=1,…,n-1,0,elsewhere,Bk=diag⁡{0,…,0,1rk[n]},Ck=diag⁡{rk[0],…,rk[n-1]}.
Then, we say that the solution (x,u) of (2.1) is generated by the solution y of (1.1).Let us consider, together with system (2.1), the matrix linear Hamiltonian system
(2.4)ΔXk=AXk+1+BkUk,ΔUk=CkXk+1-ATUk,
where the matrices A,Bk, and Ck are also given by (2.3). We say that the matrix solution (X,U) of (2.4) is generated by the solutions y[1],…,y[n] of (1.1) if and only if its columns are generated by y[1],…,y[n], respectively, that is, (X,U)=(x[y1],…,x[yn],u[y1],…,u[yn]). Reversely, if we have the solution (X,U) of (2.4), the elements from the first line of the matrix X are exactly the solutions y[1],…,y[n] of (1.1). Now, we can define the oscillatory properties of (1.1) via the corresponding properties of the associated Hamiltonian system (2.1) with matrices A,Bk, and Ck given by (2.3), for example, (1.1) is disconjugate if and only if the associated system (2.1) is disconjugate, the system of solutions y[1],…,y[n] is said to be recessive if and only if it generates the recessive solution X of (2.4), and so forth. Therefore, we define the following properties just for linear Hamiltonian systems.For system (2.4), we have an analog of the continuous Wronskian identity. Let (X,U) and (X̃,Ũ) be two solutions of (2.4). Then,
(2.5)XkTŨk-UkTX̃k≡W
holds with a constant matrix W. We say that the solution (X,U) of (2.4) is a conjoined basis, if (2.6)XkTUk≡UkTXk,rank⁡(XU)=n.
Two conjoined bases (X,U),(X̃,Ũ) of (2.4) are called normalized conjoined bases of (2.4) if W=I in (2.5) (where I denotes the identity operator).System (2.1) is said to be right disconjugate in a discrete interval [l,m], l,m∈ℤ, if the solution (XU) of (2.4) given by the initial condition Xl=0, Ul=I satisfies
(2.7)Ker⁡Xk+1⊆Ker⁡Xk,XkXk+1†(I-A)-1Bk≥0,
for k=l,…,m-1, see [6]. Here ker, †, and ≥ stand for kernel, Moore-Penrose generalized inverse, and nonnegative definiteness of the matrix indicated, respectively. Similarly, (2.1) is said to be left disconjugate on [l,m], if the solution given by the initial condition Xm=0, Um=-I satisfies
(2.8)Ker⁡Xk⊆Ker⁡Xk+1,Xk+1Xk†Bk(I-A)T-1≥0,k=l,…,m-1.
System (2.1) is disconjugate on ℤ, if it is right disconjugate, which is the same as left disconjugate, see [14, Theorem 1], on [l,m] for every l,m∈ℤ, l<m. System (2.1) is said to be nonoscillatory at ∞ (nonoscillatory at -∞), if there exists l∈ℤ such that it is right disconjugate on [l,m] for every m>l (there exists m∈ℤ such that (2.1) is left disconjugate on [l,m] for every l<m).We call a conjoined basis(X̃Ũ) of (2.4) the recessive solution at ∞, if the matrices X̃k are nonsingular, X̃kX̃k+1-1(I-Ak)-1Bk≥0 (both for large k), and for any other conjoined basis (XU), for which the (constant) matrix XTŨ-UTX̃ is nonsingular, we have (2.9)lim⁡k→∞Xk-1X̃k=0.
The solution (X,U) is called the dominant solution at ∞. The recessive solution at ∞ is determined uniquely up to a right multiple by a nonsingular constant matrix and exists whenever (2.4) is nonoscillatory and eventually controllable. (System is said to be eventually controllable if there exist N,κ∈ℕ such that for any m≥N the trivial solution (xu)=(00) of (2.1) is the only solution for which xm=xm+1=⋯=xm+κ=0.) The equivalent characterization of the recessive solution (X̃Ũ) of eventually controllable Hamiltonian difference systems (2.1) is
(2.10)lim⁡k→∞(∑kX̃j+1-1(I-A)-1BjX̃jT-1)-1=0,
see [12]. Similarly, we can introduce the recessive and the dominant solutions at -∞. For related notions and results for second-order dynamic equations, see, for example, [15, 16].We say that a pair(x,u) is admissible for system (2.1) if and only if the first equation in (2.1) holds.The energy functional of (1.1) is given by
(2.11)F(y):=∑k=-∞∞∑ν=0nrk[ν](Δνyk+n-ν)2.
Then, for admissible (x,u), we have
(2.12)F(y)=∑k=-∞∞∑ν=0nrk[ν](Δνyk+n-ν)2=∑k=-∞∞[∑ν=0n-1rk[ν](Δνyk+n-ν)2+1rk[n](rk[n]Δnyk)2]=∑k=-∞∞[xk+1TCkxk+1+ukTBkuk]=:F(x,u).
To prove our main result, we use a variational approach, that is, the equivalency of disconjugacy of (1.1) and positivity of ℱ(x,u); see [6].Now, we formulate some auxiliary results, which are used in the proofs of Theorems3.4 and 4.1. The following Lemma describes the structure of the solution space of
(2.13)Δn(rkΔnyk)=0,rk>0.Lemma 2.1 (see [17, Section 2]).
Equation (2.13) is disconjugate on ℤ and possesses a system of solutions y[j],ỹ[j], j=1,…,n, such that
(2.14)y[1]≺⋯≺y[n]≺ỹ[1]≺⋯≺ỹ[n]
as k→∞, where f≺g as k→∞ for a pair of sequences f,g means that limk→∞(fk/gk)=0. If (2.14) holds, the solutions y[j] form the recessive system of solutions at ∞, while ỹ[j] form the dominant system, j=1,…,n. The analogous statement holds for the ordered system of solutions as k→-∞.Now, we recall the transformation lemma.Lemma 2.2 (see [14, Theorem 4]).
Lethk>0,L(y)=∑ν=0n(-Δ)ν(rk[ν]Δνyk+n-ν) and consider the transformation yk=hkzk. Then, one has
(2.15)hk+nL(y)=∑ν=0n(-Δ)ν(Rk[ν]Δνzk+n-ν),
where
(2.16)Rk[n]=hk+nhkrk[n],Rk[0]=hk+nL(h),
that is, y solves L(y)=0 if and only if z solves the equation
(2.17)∑ν=0n(-Δ)ν(Rk[ν]Δνzk+n-ν)=0.The next lemma is usually called the second mean value theorem of summation calculus.Lemma 2.3 (see [17, Lemma 3.2]).
Letn∈ℕ and the sequence ak be monotonic for k∈[K+n-1,L+n-1] (i.e., Δak does not change its sign for k∈[K+n-1,L+n-2]). Then, for any sequence bk there exist n1,n2∈[K,L-1] such that
(2.18)∑j=KL-1an+jbj≤aK+n-1∑i=Kn1-1bi+aL+n-1∑i=n1L-1bi,∑j=KL-1an+jbj≥aK+n-1∑i=Kn2-1bi+aL+n-1∑i=n2L-1bi.Now, let us consider the linear difference equation(2.19)yk+n+ak[n-1]yk+n-1+⋯+ak[0]yk=0,
where k≥n0 for some n0∈ℕ and ak[0]≠0, and let us recall the main ideas of [18] and [19, Chapter IX].An integerm>n0 is said to be a generalized zero of multiplicity k of a nontrivial solution y of (2.19) if ym-1≠0,ym=ym+1=⋯=ym+k-2=0, and (-1)kym-1ym+k-1≥0.  Equation (2.19) is said to be eventually disconjugate if there exists N∈ℕ such that no non-trivial solution of this equation has n or more generalized zeros (counting multiplicity) on [N,∞).A system of sequencesuk[1],…,uk[n] is said to form the D-Markov system of sequences for k∈[N,∞) if Casoratians
(2.20)C(u[1],…,u[j])k=|uk[1]⋯uk[j]uk+1[1]⋯uk+1[j]⋮⋮uk+j-1[1]⋯uk+j-1[j]|,j=1,…,n
are positive on (N+j,∞).Lemma 2.4 (see [19, Theorem 9.4.1]).
Equation (2.19) is eventually disconjugate if and only if there exist N∈ℕ and solutions y[1],…,y[n] of (2.19) which form a D-Markov system of solutions on (N,∞). Moreover, this system can be chosen in such a way that it satisfies the additional condition
(2.21)lim⁡k→∞yk[i]yk[i+1]=0,i=1,…,n-1.


## 3. Criticality of One-Term Equation

Suppose that (1.1) is disconjugate on ℤ and let ŷ[i] and ỹ[i], i=1,…,n, be the recessive systems of solutions of L(y)=0 at -∞ and ∞, respectively. We introduce the linear space
(3.1)H=Lin{ŷ[1],…,ŷ[n]}∩Lin{ỹ[1],…,ỹ[n]}.Definition 3.1 (see [2]).
Let (1.1) be disconjugate on ℤ and let dimℋ=p∈{1,…,n}. Then, we say that the operator L (or (1.1)) is p-critical on ℤ. If dimℋ=0, we say that L is subcritical on ℤ. If (1.1) is not disconjugate on ℤ, that is, L≱0, we say that L is supercritical on ℤ.To prove the result in this section, we need the following statements, where we use the generalized power function(3.2)k(0)=1,k(i)=k(k-1)⋯(k-i+1),i∈N.
For reader's convenience, the first statement in the following lemma is slightly more general than the corresponding one used in [2] (it can be verified directly or by induction).Lemma 3.2 (see [2]).
The following statements hold.
(i) Letzk be any sequence, m∈{0,…,n},  and
(3.3)yk:=∑j=0k-1(k-j-1)(n-1)zj,
then(3.4)Δmyk={(n-1)(m)∑j=0k-1(k-j-1)(n-1-m)zj,m≤n-1,(n-1)!zk,m=n.
(ii) The generalized power function has the binomial expansion(3.5)(k-j)(n)=∑i=0n(-1)i(ni)k(n-i)(j+i-1)(i).We distinguish two types of solutions of (2.13). The polynomial solutions k(i),i=0,…,n-1, for which Δnyk=0, and nonpolynomial solutions
(3.6)∑j=0k-1(k-j-1)(n-1)j(i)rj-1,i=0,…,n-1,
for which Δnyk≠0. (Using Lemma 3.2(i) we obtain Δnyk=(n-1)!k(i)rk-1.)Now, we formulate one of the results of [20].Proposition 3.3 (see [20, Theorem 4]).
If for somem∈{0,…,n-1}(3.7)∑k=-∞0[k(n-m-1)]2rk-1=∞=∑k=0∞[k(n-m-1)]2rk-1,
then
(3.8)Lin{1,…,k(m)}⊆H,
that is, (2.13) is at least (m+1)-critical on ℤ.Now, we show that (3.7) is also sufficient for (2.13) to be at least (m+1)-critical.Theorem 3.4.
Letm∈{0,…,n-1}. Equation (2.13) is at least (m+1)-critical if and only if (3.7) holds.Proof.
Let𝒱+ and 𝒱- denote the subspaces of the solution space of (2.13) generated by the recessive system of solutions at ∞ and -∞, respectively. Necessity of (3.7) follows directly from Proposition 3.3. To prove sufficiency, it suffices to show that if one of the sums in (3.7) is convergent, then {1,…,k(m)}⊈𝒱+∩𝒱-. We show this statement for the sum ∑∞. The other case is proved similarly, so it will be omitted. Particularly, we show
(3.9)∑k=0∞[k(n-m-1)]2rk-1<∞⟹k(m)∉V+.
Let us denote p:=n-m-1, and let us consider the following nonpolynomial solutions of (2.13):
(3.10)yk[l]=∑j=0k-1(k-j-1)(n-1)j(p+l-1)rj-1-∑i=0p[(-1)i(n-1i)(k-1)(n-1-i)∑j=0∞j(p+l-1)(j+i-1)(i)rj-1],
where ℓ=1-p,…,m+1. By Stolz-Cesàro theorem, since (using Lemma 3.2(i)) Δnyk[ℓ]=(n-1)!k(p+ℓ-1)rk-1, these solutions are ordered, that is, y[i]≺y[i+1],i=1-p,…,m, as well as the polynomial solutions, that is, k(i)≺k(i+1),i=0,…,n-2.
By some simple calculation and by Lemma3.2 (at first, we use (i), and at the end, we use (ii)), we have
(3.11)Δmyk[1]=(n-1)!(n-m-1)!∑j=0k-1(k-j-1)(n-m-1)j(p)rj-1-∑i=0p[(-1)i(n-1i)(n-1-i)!(n-m-1-i)!(k-1)(n-m-1-i)∑j=0∞j(p)(j+i-1)(i)rj-1]=(n-1)!p!∑j=0k-1(k-j-1)(p)j(p)rj-1-∑i=0p[(-1)i(n-1)!(n-1-i)!(n-1-i)!i!(p-i)!(k-1)(p-i)∑j=0∞j(p)(j+i-1)(i)rj-1]=(n-1)!p!{∑j=0k-1(k-j-1)(p)j(p)rj-1-∑i=0p[(-1)i(pi)(k-1)(p-i)∑j=0∞j(p)(j+i-1)(i)rj-1]}=(n-1)!p![∑j=0k-1(k-j-1)(p)j(p)rj-1-∑j=0∞(k-j-1)(p)j(p)rj-1]=-(n-1)!p!∑j=k∞(k-j-1)(p)j(p)rj-1=(-1)p+1(n-1)!p!∑j=k∞(j+1-k)(p)j(p)rj-1,∑j=k∞(j+1-k)(p)j(p)rj-1≤∑j=k∞[j(p)]2rj-1.
Hence, from this and by Stolz-Cesàro theorem, we get
(3.12)lim⁡k→∞yk[1]k(m)=1m!lim⁡k→∞Δmyk[1]=0,
thus yk[1]≺k(m). We obtained that {1,k,…,k(m-1),y[1-p],…,y[1]}≺k(m), which means that we have n solutions less than k(m), therefore k(m)∉𝒱+ and (2.13) is at most m-critical.


## 4. Conjugacy of Two-Term Equation

In this section, we show the conjugacy criterion for two-term equation.Theorem 4.1.
Letn>1, qk be a real-valued sequence, and let there exist an integer m∈{0,…,n-1} and real constants c0,…,cm such that (2.13) is at least (m+1)-critical and the sequence hk:=c0+c1k+⋯+cmk(m) satisfies
(4.1)limsup⁡K↓-∞,L↑∞∑k=KLqkhk+n2≤0.
If q≢0, then
(4.2)(-Δ)n(rkΔnyk)+qkyk+n=0
is conjugate on ℤ.Proof.
We prove this theorem using the variational principle; that is, we find a sequencey∈ℓ02(ℤ) such that the energy functional F(y)=∑k=-∞∞[rk(Δnyk)2+qkyk+n2]<0.
At first, we estimate the first term ofF(y). To do this, we use the fact that this term is an energy functional of  (2.13). Let us denote it by F̃ that is,
(4.3)F̃(y)=∑k=-∞∞rk(Δnyk)2.
Using the substitution (2.2), we find out that (2.13) is equivalent to the linear Hamiltonian system (2.1) with the matrix Ck≡0; that is,
(4.4)Δxk=Akxk+1+Bkuk,Δuk=-ATuk,
and to the matrix system
(4.5)ΔXk=AkXk+1+BkUk,ΔUk=-ATUk.
Now, let us denote the recessive solutions of (4.5) at -∞ and ∞ by (X-,U-) and (X+,U+), respectively, such that the first m+1 columns of X+ and X- are generated by the sequences 1,k,…,k(m). Let K,L,M, and N be arbitrary integers such that N-M>2n,M-L>2n, and L-K>2n (some additional assumptions on the choice of K,L,M,N will be specified later), and let (x[f],u[f]) and (x[g],u[g]) be the solutions of (4.4) given by the formulas
(4.6)xk[f]=Xk-(∑j=Kk-1Bj-)(∑j=KL-1Bj-)-1(XL-)-1xL[h],uk[f]=Uk-(∑j=Kk-1Bj-)(∑j=KL-1Bj-)-1(XL-)-1xL[h]+(Xk-)T-1(∑j=KL-1Bj-)-1(XL-)-1xL[h],xk[g]=Xk+(∑j=kN-1Bj+)(∑j=MN-1Bj+)-1(XM+)-1xM[h],uk[g]=Uk+(∑j=kN-1Bj+)(∑j=MN-1Bj+)-1(XM+)-1xM[h]-(Xk+)T-1(∑j=MN-1Bj+)-1(XM+)-1xM[h],
where
(4.7)Bk-=(Xk+1-)-1(I-A)-1Bk(Xk-)T-1,Bk+=(Xk+1+)-1(I-A)-1Bk(Xk+)T-1,
and (x[h],u[h]) is the solution of (4.4) generated by h. By a direct substitution, and using the convention that ∑kk-1=0, we obtain
(4.8)xK[f]=0,xL[f]=xL[h],xM[g]=xM[h],xN[g]=0.
Now, from (4.1), together with the assumption q≢0, we have that there exist k̃∈ℤ and ɛ>0 such that qk̃≤-ɛ. Because the numbers K,L,M, and N have been “almost free” so far, we may choose them such that L<k̃<M-n-1.
Let us introduce the test sequence(4.9)yk:={0,k∈(-∞,K-1],fk,k∈[K,L-1],hk(1+Dk),k∈[L,M-1],gk,k∈[M,N-1],0,k∈[N,∞),
where
(4.10)Dk={δ>0,k=k̃+n,0,otherwise.
To finish the first part of the proof, we use (4.4) to estimate the contribution of the term
(4.11)F̃(y)=∑k=-∞∞rk(Δnyk)2=∑k=-∞∞uk[y]TBkuk[y]=∑k=KN-1uk[y]TBkuk[y].
Using the definition of the test sequence y, we can split F̃ into three terms. Now, we estimate two of them as follows. Using (4.4), we obtain
(4.12)∑k=KL-1uk[f]TBkuk[f]=∑k=KL-1[uk[f]T(Δxk[f]-Axk+1[f])]=∑k=KL-1[uk[f]TΔxk[f]-uk[f]TAxk+1[f]]=∑k=KL-1[Δ(uk[f]Txk[f])-Δuk[f]Txk+1[f]-uk[f]TAxk+1[f]]=∑k=KL-1[Δ(uk[f]Txk[f])-xk+1[f]T(Δuk[f]+ATuk[f])]=uk[f]Txk[f]|KL=xL[f]TuL[f]=xL[h]T[UL-(XL-)-1xL[h]+(XL-)T-1(∑j=KL-1Bj-)-1(XL-)-1xL[h]]=xL[h]T(XL-)T-1(∑j=KL-1Bj-)-1(XL-)-1xL[h]=:G,
where we used the fact that xL[h]TUL-(XL-)-1xL[h]≡0 (recall that the last n-m-1 entries of xL[h] are zeros and that the first m+1 columns of X- and U- are generated by the solutions 1,…,k(m)). Similarly,
(4.13)∑k=MN-1uk[g]TBkuk[g]=-xM[g]TuM[g]=xM[h]T(XM+)T-1(∑j=MN-1Bj+)-1(XM+)-1xM[h]=:H.
Using property (2.10) of recessive solutions of the linear Hamiltonian difference systems, we can see that 𝒢→0 as K→-∞ and ℋ→0 as N→∞. We postpone the estimation of the middle term of F̃ to the end of the proof.
To estimate the second term ofF(y), we estimate at first its terms
(4.14)∑k=KL-1qkfk+n2,∑k=MN-1qkgk+n2.
For this estimation, we use Lemma 2.3. To do this, we have to show the monotonicity of the sequences
(4.15)fkhkfork∈[K+n-1,L+n-1],gkhkfork∈[M+n-1,N+n-1].
Let x[1],…,x[2n] be the ordered system of solutions of (2.13) in the sense of Lemma 2.1. Then, again by Lemma 2.1, there exist real numbers d1,…,dn such that h=d1x[1]+⋯+dnx[n]. Because h≢0, at least one coefficient di is nonzero. Therefore, we can denote p:=max{i∈[1,n]:di≠0}, and we replace the solution x[p] by h. Let us denote this new system again x[1],…,x[2n] and note that this new system has the same properties as the original one.
Following Lemma2.2, we transform (2.13) via the transformation yk=hkzk, into
(4.16)∑ν=0n(-Δ)ν(Rk[ν]Δνzk+n-ν)=0,
that is,
(4.17)(-Δ)n(rkhkhk+nΔn-1wk)+⋯-Δ(Rk[1]wk+n-1)=0
possesses the fundamental system of solutions
(4.18)w[1]=-Δ(x[1]h),…,w[p-1]=-Δ(x[p-1]h),w[p]=Δ(x[p+1]h),…,w[2n-1]=Δ(x[2n]h).
Now, let us compute the Casoratians
(4.19)C(w[1])=w[1]=-Δ(x[1]h)=C(x[1],h)hkhk+1>0,C(w[1],w[2])=C(x[1],x[2],h)hkhk+1hk+2>0,⋮C(w[1],…,w[2n-1])=C(x[1],…,x[p-1],x[p+1],…,x[2n],h)hk⋯hk+2n-1>0.
Hence, w[1],…,w[2n-1] form the D-Markov system of sequences on [M,∞), for M sufficiently large. Therefore, by Lemma 2.4, (4.17) is eventually disconjugate; that is, it has at most 2n-2 generalized zeros (counting multiplicity) on [M,∞). The sequence Δ(g/h) is a solution of (4.17), and we have that this sequence has generalized zeros of multiplicity n-1 both at M and at N; that is,
(4.20)Δ(gM+ihM+i)=0=Δ(gN+ihN+i),i=0,…,n-2.
Moreover,gM/hM=1 and gN/hN=0. Hence,Δ(gk/hk)≤0,k∈[M,N+n-1]. We can proceed similarly for the sequencef/h.
Using Lemma2.3, we have that there exist integers ξ1∈[K,L-1]andξ2∈[M,N-1] such that
(4.21)∑k=KL-1qkfk+n2=∑k=KL-1[qkhk+n2(fk+nhk+n)2]≤∑k=ξ1L-1qkhk+n2,∑k=MN-1qkgk+n2=∑k=MN-1[qkhk+n2(gk+nhk+n)2]≤∑k=Mξ2-1qkhk+n2.
Finally, we estimate the remaining term of F(y). By (4.9), we have
(4.22)∑k=LM-1[rk(Δnyk)2+qkyk+n2]=∑k=LM-1{rk[Δnhk+Δn(hkDk)]2+qk(hk+n+hk+nDk+n)2}=∑k=LM-1{rk[Δn(hkDk)]2+qkhk+n2+2qkhk+n2Dk+n+qkhk+n2Dk+n2}=∑k=k̃k̃+n{rk[Δn(hkDk)]2}+∑k=LM-1[qkhk+n2]+2qk̃hk̃+n2Dk̃+n+qk̃hk̃+n2Dk̃+n2=∑k=k̃k̃+n{rk[(-1)k-k̃(nk-k̃)hk̃+nδ]2}+∑k=LM-1[qkhk+n2]+2δqk̃hk̃+n2+δ2qk̃hk̃+n2≤δ2hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]+∑k=LM-1[qkhk+n2]-2δɛhk̃+n2-δ2ɛhk̃+n2<δ2hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]+∑k=LM-1[qkhk+n2]-2δɛhk̃+n2.
Altogether, we have
(4.23)F(y)<δ2hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]+∑k=LM-1[qkhk+n2]-2δɛhk̃+n2+G+H+∑k=ξ1L-1qkhk+n2+∑k=Mξ2-1qkhk+n2=δ2hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]-2δɛhk̃+n2+G+H+∑k=ξ1ξ2-1qkhk+n2,
where for K sufficiently small is 𝒢<δ2/3, for N sufficiently large is ℋ<δ2/3, and, from (4.1), ∑k=ξ1ξ2-1qkhk+n2<δ2/3 for ξ1<L and ξ2>M. Therefore,
(4.24)F(y)<δ2+δ2hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]-2δɛhk̃+n2=δ{δ[1+hk̃+n2∑k=k̃k̃+n[rk(nk-k̃)2]]-ɛhk̃+n2},
which means that F(y)<0 for δ sufficiently small, and (4.2) is conjugate on ℤ.


## 5. Equation with the Middle Terms

Under the additional conditionqk≤0 for large |k|, and by combining of the proof of Theorem 4.1 with the proof of [2, Lemma 1], we can establish the following criterion for the full 2n-order equation.Theorem 5.1.
Letn>1, qk be a real-valued sequence, and let there exist an integer m∈{0,…,n-1} and real constants c0,…,cm such that (1.1) is at least (m+1)-critical and the sequence hk:=c0+c1k+⋯+cmk(m) satisfies
(5.1)limsup⁡K↓-∞,L↑∞∑k=KLqkhk+n2≤0.
If qk≤0 for large |k| and q≢0, then
(5.2)L(y)k+qkyk+n=∑ν=0n(-Δ)ν(rk[ν]Δνyk+n-ν)+qkyk+n=0
is conjugate on ℤ.Remark 5.2.
Using Theorem3.4, we can see that the statement of Theorem 4.1 holds if and only if (3.7) holds. Finding a criterion similar to Theorem 3.4 for (1.1) is still an open question.Remark 5.3.
In the view of the matrix operator associated to (1.1) in the sense of [21], we can see that the perturbations in Theorem 4.1 affect the diagonal elements of the associated matrix operator. A description of behavior of (1.1), with regard to perturbations of limited part of the associated matrix operator (but not only of the diagonal elements), is given in [2].



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*Source: 814962-2011-07-13.xml*