# Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients

Authors: Petr Hasil; Michal Veselý
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate a type of the Sturm-Liouville difference equations with almost periodic coefficients. We prove that there exists a constant, which is the borderline between the oscillation and the nonoscillation of these equations. We compute this oscillation constant explicitly. If the almost periodic coefficients are replaced by constants, our result reduces to the well-known result about the discrete Euler equation.


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## Body

## 1. Introduction

In this paper, we analyse the second-order Sturm-Liouville equation(*)Δ(rkΔyk)+qkyk+1=0,rk>0,
whose oscillation properties are widely studied over the last few decades. We begin with a short literature overview concerning the (non)oscillation of (*) and of some direct generalizations (including half-linear equations and dynamic equations on time scales).Basic necessary and sufficient conditions for(*) in order to be oscillatory are derived in [1–3]. In [4] (see also [5]), the concept of a phase is established to obtain other oscillation criteria. For the matrix difference equations of the form of (*), we refer to [6]. Several oscillation criteria for slightly more general equations are presented in [7, 8]. The oscillation theory for the corresponding higher-order two-term Sturm-Liouville difference equations can be found in [9–11] (for differential case, see [12]).Fundamental aspects of Sturmian theory (and some oscillation criteria) for second-order Sturm-Liouville equations on arbitrary time scales are formulated in [13]. Oscillation criteria for second-order difference equations can be obtained from oscillation criteria for more general dynamic equations. The oscillation properties of second-order linear dynamic equations, which have the Sturm-Liouville difference equations as special cases, are considered, for example, in [14].As an illustration, we mention a particular Sturm-Liouville equation of which the complete oscillation classification is done as a consequence of general results on time scales. Using the comparison theorem for second-order linear dynamic equations, it is shown in [15] that the difference equation
(1.1)Δ2yk+b(-1)kkcyk+1=0
is oscillatory for any c<1 and b≠0. Further, it is obtained in [16] (based on results of [17, 18]) that the equation
(1.2)Δ2yk+[ak2+b(-1)kk]yk+1=0
is oscillatory if and only if 4a>1-b2. Finally in [16], applying the Willett-Wong-type theorems for second-order linear dynamic equations, there is given the full oscillation analysis of (*) for
(1.3)rk=1,qk=akc+1+b(-1)kkc
with regard to arbitrary a,b,c∈ℝ.The importance of the oscillation results about second-order equations lies among others in the fact that such results can be used to study the oscillation and nonoscillation properties of solutions of different equations. For example (see [19] and also [20]), all solutions of the delay equation Δyk+pkyk-n=0 oscillate if and only if all solutions of a certain type of (*) with rk=1 oscillate.The main aim of this paper is to present a sharp oscillation constant for the Euler-type difference equation(**)Δ(rkΔyk)+γsk(k+1)kyk+1=0,
where γ∈ℝ, inf{rk}>0, and {rk},{sk} are positive almost periodic sequences. More precisely, we show that (**) is the so-called conditionally oscillatory; that is, we prove that there exists a positive constant K (the oscillation constant) such that (**) is oscillatory for γ>K and non-oscillatory for γ<K.Our research is motivated by the continuous case. It is a famous result due to Kneser [21] that the differential Euler equation
(1.4)y′′(t)+γt2y(t)=0
is conditionally oscillatory with the oscillation constant K=1/4. It is known (see [22]) that the equation
(1.5)[r(t)y′(t)]′+γs(t)t2y(t)=0,
where r, s are positive periodic continuous functions, is conditionally oscillatory as well. We also refer to [23] and [24–29] which generalize [23] (for the discrete case, see [30]). Since the Euler difference equation
(1.6)Δ2yk+γ(k+1)kyk+1=0
is conditionally oscillatory with the oscillation constant K=1/4 (see [31]), it is natural to analyse the conditional oscillation of (**). Note that the announced result is more general than the results known in the continuous case, because (**) has almost periodic coefficients. The conditional oscillation of discrete equations with constant coefficients can be generalized in other ways. Point out [32], where an oscillation constant is characterized. The constant from [32] coincides with our oscillation constant if the considered coefficients are asymptotically constant.Solutions of the second-order Sturm-Liouville difference equations with periodic coefficients are studied in [33] (see also [34, 35]). In [36], the half-linear differential equations of the second order with the Besicovitch almost periodic coefficients are considered and an oscillation theorem for these equations is obtained.In the last years, many results dealing with the conditional oscillation of second-order equations and two-term equations of even order appeared. The two-term difference equation of even order(1.7)(-1)n+1Δn(Γ(k+1)Γ(k-α+1)Δnyk)+qkyk+n=0,
where Γ denotes the gamma function, is studied in [9, 10]. Results concerning the half-linear difference equation
(1.8)Δ[rkΦ(Δyk)]+qkΦ(yk+1)=0,
where
(1.9)rk>0,Φ(y)=|y|α-1sgny,α>1,
can be found in [37] for rk=1, qk=γ(k+1)-α and also in [38, 39] (for dynamic half-linear equations on time scales, see [40–42]).The paper is organized as follows. In Section2, we mention only necessary preliminaries and an auxiliary result. Our main result is proved in Section 3, where the particular case concerning the equation with periodic coefficients is formulated as well. The paper is finished by concluding remarks and simple examples.


## 2. Preliminaries

We begin this section recalling some elements of the oscillation theory of the Sturm-Liouville difference equation(2.1)Δ(rkΔyk)+qkyk+1=0,rk>0,k∈ℕ.
For more details, we can refer to books [43, 44] and references cited therein.We recall that an interval(a,a+1], a∈ℕ, contains the generalized zero of a solution {yk} of (2.1) if ya≠0 and yaya+1≤0. Equation (2.1) is said to be conjugate on {a,…,a+n}, n∈ℕ, if there exists a solution which has at least two generalized zeros on (a,…,a+n+1] or if the solution {y~k} satisfying y~a=0 has at least one generalized zero on (a,…,a+n+1]. Otherwise, (2.1) is said to be disconjugate on {a,…,a+n}. Since Sturmian theory is valid for difference equations, all solutions of (2.1) have either a finite or an infinite number of generalized zeros on ℕ. Hence, we can categorize these equations as oscillatory and non-oscillatory.Definition 2.1.
Equation (2.1) is called non-oscillatory provided a solution of (2.1) is disconjugate at infinity, that is, there exists N∈ℕ such that (2.1) is disconjugate on any set [N,N+m]∩ℕ, m∈ℕ. Otherwise, we say that (2.1) is oscillatory.Since we study a special case of (2.1), when the coefficients are almost periodic, we also mention the basics of the theory of almost periodic sequences. Here, we refer to each one of books [45, 46].Definition 2.2.
A real sequence{fk}k∈ℤ is called almost periodic if, for any ε>0, there exists a positive integer p(ε) such that any set consisting of p(ε) consecutive integers contains at least one integer l with the property that
(2.2)|fk+l-fk|<ε,k∈ℤ.
We say that a sequence {gk}k=1∞ is almost periodic if there exists an almost periodic sequence {fk}k∈ℤ for which fk=gk, k∈ℕ.The above definition of almost periodicity is based on the Bohr concept. Now we formulate a necessary and sufficient condition for a sequence to be almost periodic. The following theorem is often used as an equivalent definition (the Bochner one) of almost periodicity fork∈ℤ.Theorem 2.3.
A sequence{fk}k∈ℤ⊂ℝ is almost periodic if and only if any sequence of the form {fk+h(n)}, where h(n)∈ℤ, n∈ℕ, has a uniformly convergent subsequence with respect to k.Proof.
See [45, Theorem 1.26].Corollary 2.4.
Let{fk} be almost periodic. The sequence {1/fk} is almost periodic if and only if
(2.3)inf{|fk|;k∈ℕ}>0.Proof.
The corollary follows from [45, Theorem 1.27] and [47, Theorem 1.9] (or directly from Theorem 2.3). It suffices to use that (2.3) implies inf{|fk|}>0 for any almost periodic sequence {fk} if k∈ℤ.Note that there exist nonzero almost periodic sequences{fk} for which (2.3) is not satisfied (see, e.g., [48, Theorem 3]).Theorem 2.5.
If{fk} is an almost periodic sequence, then the limit
(2.4)M({fk}):=limn→∞fk+fk+1+⋯+fk+nn+1
exists uniformly with respect to k.Proof.
See [45, Theorem 1.28].Definition 2.6.
Let{fk} be almost periodic. The number M({fk}) introduced in (2.4) is called the mean value of {fk}.Remark 2.7.
For any positive almost periodic sequence{fk}, we have M({fk})>0. Indeed, if we put ε=f1/2 and find a corresponding p(ε) in Definition 2.2, then we obtain
(2.5)M({fk})≥f12p(ε)>0.In the proof of our main result, we use an adapted Riccati technique. The classical Riccati technique deals with the so-called Riccati difference equation, which we obtain from (2.1) using the substitution wk=rk(Δyk/yk), that is, we obtain the equation
(2.6)Δwk+qk+wk2wk+rk=0,k∈ℕ.
Putting ζk=-kwk, we adapt (2.6) to our purposes. A direct calculation leads to the equation
(2.7)Δζk=1k[k(k+1)qk+ζk+(k+1)ζk2krk-ζk],k∈ℕ.We also mention two lemmas which we use to prove the main result.Lemma 2.8.
Let the equation(2.8)Δ(rkΔyk)+qkyk+1=0,k∈ℕ,
where sup{rk;k∈ℕ}<∞ and rk, qk>0, k∈ℕ, be non-oscillatory. For any solution {ζk} of the associated equation (2.7), there exists k0∈ℕ such that, if ζk0+m<0 for some m∈ℕ, then ζk+m<0, k≥k0.Proof.
Let{yk} be a solution of the non-oscillatory equation (2.8) for which ykyk+1>0, k≥k0. From [43, Lemma  6.6.1] it follows that the sequence {wk}k=k0∞, where wk=rk(Δyk/yk), is decreasing. Further, [43, Theorem  6.6.2] implies that limk→∞wk=0. Thus, the sequence {wk}k=k0∞ is positive, that is, ζk=-kwk<0, k≥k0.Lemma 2.9.
If there exists a solution{ζk} of the associated equation (2.7) satisfying ζk<0 for all k≥k0, then (2.8) is non-oscillatory.Proof.
The statement of the lemma follows from [44, Theorem 6.16].


## 3. Oscillation Constant

This section is devoted to the main result of our paper. After its proof, within the concluding remarks, we formulate as a corollary the result which deals with periodic equations. This corollary is the discrete counterpart of the main result of [49].Theorem 3.1.
Let the equation(3.1)Δ(rkΔxk)+γsk(k+1)kxk+1=0,k∈ℕ,
where γ∈ℝ and {rk} and {sk} are positive almost periodic sequences satisfying
(3.2)inf{rk;k∈ℕ}>0,
be arbitrarily given. Let
(3.3)K:=[4M({rk-1})M({sk})]-1.
Then, (3.1) is oscillatory for γ>K and non-oscillatory for γ<K.Proof.
At first, let us prepare several estimates which we will use to prove the theorem. Henceforth, for givenγ≠K, we will consider α∈ℕ and ϑ>0 such that
(3.4)1α∑i=kk+α-1‍si>2ϑ,k∈ℕ,(3.5)8|1α∑i=kk+α-1‍1ri-1α∑i=ll+α-1‍1ri|<(1α∑i=1α‍1ri)2|γ-K|ϑ,k,l∈ℕ,(3.6)2|K-(4α2∑i=kk+α-1‍1ri∑i=ll+α-1‍si)-1|<|γ-K|,k,l∈ℕ.
The fact that such numbers α, ϑ exist follows from Theorem 2.5 and Remark 2.7 (consider also Corollary 2.4 with (3.2)). We put
(3.7)r-:=inf{rk;k∈ℕ},r+:=sup{rk;k∈ℕ},s+:=sup{sk;k∈ℕ}.
The adapted Riccati equation associated to (3.1) has the form (see (2.7))
(3.8)Δζk=1k[γsk+ζk+(k+1)ζk2krk-ζk].
Since one can express
(3.9)ζk+(k+1)ζk2krk-ζk=kζk(rk+ζk)krk+|ζk|ifζk<0forsomek∈ℕ,
it is valid that
(3.10)Δζk>γskk>0ifζk<-r+forsomek∈ℕ,
and that (C>0 is arbitrarily given)
(3.11)|Δζk|≤1k[γs++k|ζk|(rk+|ζk|)krk+|ζk|]<γs+r-+C(r++C)kr-ifζk∈(-C,0)forsomek∈ℕ.
Particularly, ifζk<0 for all sufficiently large k, then there exists δ<0 such that
(3.12)ζk>δforconsideredk.
Indeed, it follows directly from (3.10) and (3.11).
Similarly, applying (3.10) and (3.11), it is seen that there exists k¯=k¯(δ,α)∈ℕ for any δ<0 and α∈ℕ with the property that the solution {ζk}k=k¯+m∞ of the Cauchy problem
(3.13)Δζk=1k[γsk+ζk+(k+1)ζk2krk-ζk],ζk¯+m=ζ0∈(2δ,δ],
where m∈ℕ, satisfies
(3.14){ζk}k=k¯+m∞⊆(min{2ζ0,-2r+},∞),{ζk}k=k¯+mk¯+m+α-1⊆(min{2ζ0,-2r+},0),
and hence there exists Θ>0 (consider again (3.11)) for which
(3.15)|ζk¯+m-ζk¯+m+j|<Θk¯,j∈{1,…,α-1},m∈ℕ.
Now we can proceed to the oscillatory part of the theorem. By contradiction, we suppose thatγ>K and that (3.1) is non-oscillatory. According to Lemma 2.8, any solution {ζk}k=k0∞ of the associated adapted Riccati equation (3.8) for which ζk0<0 satisfies ζk<0, k≥k0, if k0 is enough large. Further, (3.12) gives the existence of δ<0 with the property that ζk∈(δ,0) for all k≥k0. Using (3.11), we obtain
(3.16)|Δζk|<γs+r--δ(r+-δ)kr-,k≥k0.
Our goal is to achieve a contradiction withζk∈(δ,0) by estimating the arithmetic mean of α subsequent values of ζk. We denote
(3.17)ξk:=1α∑i=kk+α-1‍ζi∈(δ,0),k≥k0,
and compute (for Δξk>0)
(3.18)Δξk=1α∑i=kk+α-1‍Δζi=1α∑i=kk+α-1‍1i[γsi+ζi+(i+1)ζi2iri-ζi]≥1k+α-1[γα∑i=kk+α-1‍si+ξk+1α∑i=kk+α-1‍(i+1)ζi2iri-ζi]=1k+α-1{γα∑i=kk+α-1‍si-Ak22+1α∑i=kk+α-1‍[(i+1)ζi2iri-ζi-ζi2ri]+ξk+Ak22+Bk22+1α∑i=kk+α-1‍ζi2ri-Bk22}
or (for Δξk<0)
(3.19)Δξk=1α∑i=kk+α-1‍Δζi≥1k{γα∑i=kk+α-1‍si-Ak22+1α∑i=kk+α-1‍[(i+1)ζi2iri-ζi-ζi2ri]+ξk+Ak22+Bk22+1α∑i=kk+α-1‍ζi2ri-Bk22},
where
(3.20)Ak:=(2α∑i=kk+α-1‍1ri)-1/2,Bk:=|ξk|(2α∑i=kk+α-1‍1ri)1/2,k≥k0.
Note that we can choose k0≥α. For reader’s convenience, we will estimate Δξk stepwise.
Step1. We show that there exist k1≥k0 and Γ>0 such that
(3.21)γα∑i=kk+α-1‍si-Ak22+1α∑i=kk+α-1‍[(i+1)ζi2iri-ζi-ζi2ri]≥Γ,k≥k1.
Applying ζk∈(δ,0) for k≥k0, we have (see (3.4) and (3.6))
(3.22)γα∑i=kk+α-1‍si-Ak22+1α∑i=kk+α-1‍[(i+1)ζi2iri-ζi-ζi2ri]=γα∑i=kk+α-1‍si-12(2α∑i=kk+α-1‍1ri)-1+1α∑i=kk+α-1‍[ζi2ri+ζi3ri(iri-ζi)]≥γα∑i=kk+α-1‍si-14(1α∑i=kk+α-1‍1ri)-1-1α∑i=kk+α-1‍δ2r+-δr-(ir-)≥(1α∑i=kk+α-1‍si)[γ-14(1α∑i=kk+α-1‍1ri)-1(1α∑i=kk+α-1‍si)-1]-1αk∑i=kk+α-1‍δ2(r+-δ)(r-)2>ϑ(γ-K)-1k(δ2(r+-δ)(r-)2),k≥k0.
Thus, there exist Γ>0 and k1 with the property that (3.21) is satisfied for all k≥k1.
Step2. It holds (see (3.17) and (3.20))
(3.23)ξk+Ak22+Bk22=Ak22-|ξk|+Bk22=12(Ak-Bk)2≥0,k≥k0.
Step3. We prove that there exists k2≥k1 satisfying
(3.24)1α∑i=kk+α-1‍ζi2ri-Bk22≥-Γ2,k≥k2,
where Γ is taken from Step 1. Considering (3.16), we obtain
(3.25)|ζm-ζn|≤∑i=kk+α-1‍|Δζi|<∑i=kk+α-1‍γs+r--δ(r+-δ)ir-≤1k∑i=kk+α-1‍γs+r--δ(r+-δ)r-
for each m,n∈{k,…,k+α-1}, k≥k0. Thus, it is true
(3.26)|ζm-ζn|<Dk,m,n∈{k,…,k+α-1},k≥k0,
where
(3.27)D:=αγs+r--δ(r+-δ)r-.
Now we can calculate (see (3.17))
(3.28)1α∑i=kk+α-1‍ζi2ri-Bk22=1α∑i=kk+α-1‍ζi2ri-12(2α∑i=kk+α-1‍1ri)ξk2=1α∑i=kk+α-1‍ζi2-ξk2ri=-1α∑i=kk+α-1‍(|ξk|+|ζi|)(|ξk|-|ζi|)ri>2δα∑i=kk+α-1‍|ξk-ζi|ri>2δDαk∑i=kk+α-1‍1ri≥2δDkr-,k≥k0.
Let us discuss the inequality before the final one in more detail. If we denote
(3.29)ζ^-k:=max{ζj;ζj≤ξk,j∈{k,…,k+α-1}},ζ^+k:=min{ζj;ζj≥ξk,j∈{k,…,k+α-1}},
then we easily get (applying (3.26))
(3.30)|ζi-ξk|≤max{|ζi-ζ^-k|,|ζi-ζ^+k|}<Dk,i∈{k,…,k+α-1},k≥k0.
Of course, (3.28) implies the existence of k2≥k1 such that (3.24) is satisfied.
Using the previous steps, it is possible to prove the following result. Ifl tends to infinity, then so do ξl. Combining (3.21), (3.23), and (3.24), we obtain
(3.31)Δξk≥1k+α-1(Γ+0-Γ2)=Γ2(k+α-1),k≥k2.
We use the estimate (3.18) because Δξk>0. Summing inequality (3.31) from k2 to an integer (l-1)≥k2, we have
(3.32)ξl≥ξk2+Γ2∑i=k2l-1‍1i+α-1.
This estimate implies that
(3.33)liminfl→∞ξl≥ξk2+Γ2∑i=k2∞‍1i+α-1=∞.
Particularly, ξk>0 for sufficiently large k which means that ζk>0 for infinitely many k. This contradiction gives that (3.1) is oscillatory for γ>K.
To prove the non-oscillatory part of Theorem3.1, we will consider the initial value problem
(3.34)Δζk=1k[γsk+ζk+(k+1)ζk2krk-ζk],ζk~=-(2α∑i=1α‍1ri)-1
for some integer
(3.35)k~>k¯(-(2α∑i=1α‍1ri)-1,α),
where k¯ satisfies (3.14) and (3.15). Let γ<K. Analogously as in the first part of the proof, we put
(3.36)ξk~:=1α∑i=k~k~+α-1‍ζi<0
and we express (for Δξk~>0)
(3.37)Δξk~=1α∑i=k~k~+α-1‍Δζi=1α∑i=k~k~+α-1‍1i[γsi+ζi+(i+1)ζi2iri-ζi]≤1k~[γα∑i=k~k~+α-1‍si+ξk~+1α∑i=k~k~+α-1‍(i+1)ζi2iri-ζi]=1k~[γα∑i=k~k~+α-1‍si-14(1α∑i=1α‍1ri)-1+14(1α∑i=1α‍1ri)-1+ξk~+1α∑i=k~k~+α-1‍ζi2ri-1α∑i=k~k~+α-1‍ζi2ri+1α∑i=k~k~+α-1‍(i+1)ζi2iri-ζi]
or (for Δξk~<0)
(3.38)Δξk~=1α∑i=k~k~+α-1‍Δζi≤1k~+α-1[γα∑i=k~k~+α-1‍si-14(1α∑i=1α‍1ri)-1+14(1α∑i=1α‍1ri)-11α∑i=k~k~+α-1‍Δζi≤≤1k~+α-1+ξk~+1α∑i=k~k~+α-1‍ζi2ri-1α∑i=k~k~+α-1‍ζi2ri+1α∑i=k~k~+α-1‍(i+1)ζi2iri-ζi].
Again, we estimateΔξk~ stepwise. Using (3.14) and
(3.39)-r+<-r+2≤-(2α∑i=1α‍1ri)-1,
we have
(3.40)ζk>-2r+,k≥k~,ζk~,…,ζk~+α-1<0.
Similarly to the first part of the proof, we can show that
(3.41)γα∑i=k~k~+α-1‍si-14(1α∑i=1α‍1ri)-1+1α∑i=k~k~+α-1‍(i+1)ζi2iri-ζi-1α∑i=k~k~+α-1‍ζi2ri=(1α∑i=k~k~+α-1‍si)[γ-14(1α∑i=1α‍1ri)-1(1α∑i=k~k~+α-1‍si)-1]+1α∑i=k~k~+α-1‍riζi2+ζi3iri2-ζiri≤(1α∑i=k~k~+α-1‍si)γ-K2+1α∑i=k~k~+α-1‍r+(2r+)2+(2r+)3i(r-)2≤(1α∑i=k~k~+α-1‍si)γ-K2+12(r+)3k~(r-)2.
Thus (consider (3.4)), there exist k^∈ℕ and
(3.42)Γ^>K-γ2ϑ
such that
(3.43)γα∑i=k~k~+α-1‍si-14(1α∑i=1α‍1ri)-1+1α∑i=k~k~+α-1‍(i+1)ζi2iri-ζi-1α∑i=k~k~+α-1‍ζi2ri≤-Γ^
for k~≥k^. Henceforth, let k~≥k^.
Now we want to estimate(3.44)Y:=14(1α∑i=1α‍1ri)-1+ξk~+1α∑i=k~k~+α-1‍ζi2ri.
Firstly, consider that, for ζi=ζk~, i∈{k~,…,k~+α-1}, it is valid
(3.45)Y=14(1α∑i=1α‍1ri)-1-(2α∑i=1α‍1ri)-1+(2α∑i=1α‍1ri)-21α∑i=k~k~+α-1‍1ri,
and hence (see (3.5) and (3.42))
(3.46)|Y-0|=|(2α∑i=1α‍1ri)-2[1α∑i=k~k~+α-1‍1ri-1α∑i=1α‍1ri]|<18(K-γ)ϑ<Γ^4
because
(3.47)14(1α∑i=1α‍1ri)-1-(2α∑i=1α‍1ri)-1+(2α∑i=1α‍1ri)-21α∑i=1α‍1ri=0.
We repeat that (see (3.15))
(3.48)|ζk~-ζk~+j|<Θk~,j∈{1,…,α-1},
which gives
(3.49)|ζk~-ξk~|<Θk~.
Considering (3.40), (3.46), (3.48), and (3.49) together for general Y, we have
(3.50)|Y|≤Γ^4+|ξk~-ζk~|+1α∑i=0α-1‍|ζk~+i2-ζk~2|rk~+i=Γ^4+|ξk~-ζk~|+1α∑i=0α-1‍|ζk~+i+ζk~|·|ζk~+i-ζk~|rk~+i<Γ^4+(1+4r+r-)Θk~.
Since it suffices to consider very large k~, we can assume that |Y|<2Γ^/3.
Altogether, we obtain(3.51)Δξk~≤1k~+α-1(-Γ^+2Γ^3)<0.
The resulting inequality (3.51) implies
(3.52)Δξk~=1α∑i=k~k~+α-1‍Δζi=ζk~+α-ζk~α<0,i.e.,ζk~+α<ζk~.
Partially, if
(3.53)ζk=-(2α∑i=1α‍1ri)-1forsomek≥k~,
then from (3.14) it follows
(3.54)ζk,ζk+1,…,ζk+α<0,ζk+α<ζk.
In fact (see the below given), this result remains true also if(3.55)ζk∈(-η-(2α∑i=1α‍1ri)-1,-(2α∑i=1α‍1ri)-1)
for a number η>0 which depends only on γ and K. Considering (3.15) for large k¯, it is seen that the solution of the Cauchy problem (3.34) satisfies
(3.56)ζk~,ζk~+α,…,ζk~+nα,…<-(2α∑i=1α‍1ri)-1,
and hence (see (3.54))
(3.57)ζk~+n<0,n∈ℕ.
Therefore, (3.57) and Lemma 2.9 say that (3.1) is non-oscillatory for γ<K.
It means that, to complete the proof, it suffices to findη>0 which guaranties the above-mentioned generalization, that is, we need to prove (3.51) for (3.55) with k=k~. The concrete initial value was not used in the proof of (3.43). Thus, Γ^ depends only on γ and K. Let
(3.58)η:=min{(2α∑i=1α‍1ri)-1,(1+3r+r-)-1Γ^4}.
In the estimate of Y, since (3.14) and (3.15) remain true, we have (consider also (3.39), (3.48), and (3.49))
(3.59)ζk~+j∈(-2r+,0),|ζk~+j+(2α∑i=1α‍1ri)-1|<Θk~+η,|ξk~+(2α∑i=1α‍1ri)-1|<Θk~+η,
where j∈{0,1,…,α-1}, and (see again (3.39))
(3.60)|Y|≤Γ^4+|ξk~+(2α∑i=1α‍1ri)-1|+1α∑j=0α-1‍[1rk~+j|ζk~+j2-(2α∑i=1α‍1ri)-2|]<Γ^4+Θk~+η+1α∑j=0α-1‍[1rk~+j|ζk~+j-(2α∑i=1α‍1ri)-1|·|ζk~+j+(2α∑i=1α‍1ri)-1|]<Γ^4+Θk~+η+1r-(Θk~+η)(2r++r+)=Γ^4+[1+3r+r-](Θk~+η)≤[1+3r+r-]Θk~+Γ^2,
which confirms |Y|<2Γ^/3 and then the validity of (3.51).Remark 3.2.
Let us point out that the constantK arises from the calculations in Step 1.Remark 3.3.
Ifrk=sk=1, k∈ℕ, then K=1/4 (see (3.3)); that is, Theorem 3.1 reduces to the result about the discrete Euler equation.Example 3.4.
For arbitrarily given continuous functionf:[-1,1]→ℝ+ and a>1, b,c∈ℝ, let us consider
(3.61)Δ(Δxka+sin(bk)cos(ck))+γf(sink)k(k+1)xk+1=0,k∈ℕ.
The almost periodicity of {rk}={[a+sin(bk)cos(ck)]-1}k∈ℕ and {sk}={f(sink)}k∈ℕ follows from Corollary 2.4 and from, for example, [45, Theorem 1.27] and [47, Theorem 1.9]. It is seen that
(3.62)M({rk-1})=a,M({sk})=12π∫-ππf(sinx)dx.
Thus, (3.61) is oscillatory if
(3.63)γ>K=π2a∫-ππ‍f(sinx)dx,
and non-oscillatory if γ<K.
Analogously, under the additional conditionb≠0, the oscillation constant for the equation
(3.64)Δ([a+sin(bk)cos(bk)]Δxk)+γf(cosk)k(k+1)xk+1=0,k∈ℕ,
is
(3.65)K=π2b∫-ππ‍f(cosx)dx∫-π/bπ/b‍dy/(a+sin(by)cos(by))=π4a2-14∫-ππ‍f(cosx)dx.Evidently, any periodic sequence is almost periodic. Thus, we also obtain this new result.Corollary 3.5.
The equation(3.66)Δ(rkΔxk)+γsk(k+1)kxk+1=0,k∈ℕ,
where γ∈ℝ and {rk} and {sk} are positive sequences with period n∈ℕ, is oscillatory if
(3.67)γ>K:=[(2n∑i=1n‍1ri)(2n∑i=1n‍si)]-1,
and non-oscillatory if γ<K.Remark 3.6.
The border case given byγ=K remains open. Nevertheless, based on the corresponding continuous case (see [23]) and other cases which generalize the discrete equation with constant coefficients (see, e.g., [10] with references cited therein), we conjecture that (3.66) (with periodic coefficients) is non-oscillatory even for γ=K.Example 3.7.
Let an odd integerm≥3 be given. We can use Corollary 3.5 for the equation
(3.68)Δ(|sin(2k-1)π12|Δxk)+γk(k+1)(1+cos2kπm)xk+1=0,k∈ℕ.
Since we can choose n=6m, we obtain
(3.69)M({rk-1})=16m∑i=16m‍1ri=16∑i=16‍|sin(2i-1)π12|-1=23(1+23),M({sk})=16m∑i=16m‍si=1m∑i=1m‍(1+cos2iπm)=1.
Hence, the oscillation constant for (3.68) is
(3.70)K=342(1+23)≈217.We add that we can use Theorem3.1 also in the case when one of the sequences {rk} and {sk} in (3.1) changes its sign. If the sequence {rk} in (3.71) changes its sign, then we have to generalize the definition of the generalized zeros as follows. An interval (a,a+1], a∈ℕ, contains the generalized zero of a solution {xk} of (3.71) if xa≠0 and raxaxa+1≤0.Corollary 3.8.
Let the equation(3.71)Δ(rkΔxk)+γsk(k+1)kxk+1=0,k∈ℕ,
where γ∈ℝ and {rk} and {sk} are nonzero almost periodic sequences, be given. (i)
Ifinf{rk;k∈ℕ}>0 and γ<[4M({rk-1})M({|sk|})]-1, then (3.71) is non-oscillatory.(ii)
Ifinf{|rk|;k∈ℕ}>0, {sk} is positive and γ>[4M({|rk|-1})M({sk})]-1, then (3.71) is oscillatory.Proof.
Since the almost periodicity of{fk} implies the almost periodicity of {|fk|}, it suffices to apply the discrete Sturm comparison theorem and Theorem 3.1.At the end we remark that it is possible to find several definitions of almost periodicity fork∈ℕ in the literature. For example, concerning almost periodic sequences with indices k∈ℕ, we refer to [50]. There is proved that, for any precompact sequence {xk}k∈ℕ, there exists a permutation P of the set of positive integers such that the sequence {xP(k)}k∈ℕ is almost periodic. In fact, the so-called asymptotically almost periodic sequences are considered in [50] (based on the Bochner concept), where a bounded sequence {xk}k∈ℕ is called asymptotically almost periodic if the set of sequences {xk+p}k∈ℕ, p∈ℕ, is precompact in the space of all bounded sequences. We add that a sequence {xk}k∈ℕ is asymptotically almost periodic if and only if it is the sum of an almost periodic sequence and a sequence which approaches zero as k→∞. One finds that this representation is unique. See, for example, [51, 52].We consider difference equations with almost periodic coefficients given by the limitation of almost periodic sequences onℤ because this approach is the standard one. But we conjecture that the main result can be similarly proved for almost periodic coefficients defined in other ways (e.g., for the above-mentioned asymptotically almost periodic sequences).



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