# Oscillations of Difference Equations with Several Oscillating Coefficients

Authors: L. Berezansky; G. E. Chatzarakis; A. Domoshnitsky; I. P. Stavroulakis
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the oscillatory behavior of the solutions of the difference equationΔx(n)+∑i=1mpi(n)x(τi(n))=0,n∈N0[∇xn-∑i=1mpinxσin=0,n∈N] where (pi(n)), 1≤i≤m are real sequences with oscillating terms, τi(n)[σi(n)], 1≤i≤m are general retarded (advanced) arguments, and Δ[∇] denotes the forward (backward) difference operator Δx(n)=x(n+1)-x(n)[∇x(n)=x(n)-x(n-1)]. Examples illustrating the results are also given.


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## Body

## 1. Introduction

In the present paper, we study the oscillatory behavior of the solutions of the difference equation(ER)Δx(n)+∑i=1mpi(n)x(τi(n))=0,n∈N0,
where N∋m≥2, pi, 1≤i≤m are real sequences with oscillating terms, and {τi(n)}n∈N0, 1≤i≤m are sequences of integers such that
(1)τi(n)≤n-1,n∈N0,lim⁡n→∞τi(n)=∞,1≤i≤m
and the (dual) advanced difference equation
(EA)∇x(n)-∑i=1m‍pi(n)x(σi(n))=0,n∈N,
where N∋m≥2, pi, 1≤i≤m are real sequences with oscillating terms and {σi(n)}n∈N, 1≤i≤m, are sequences of integers such that
(2)σi(n)≥n+1,n∈N,1≤i≤m.
Here, N0={0,1,2,…} and N={1,2,…}. Also, as usual, Δ denotes the forward difference operator Δx(n)=x(n+1)-x(n) and ∇ denotes the backward difference operator ∇x(n)=x(n)-x(n-1).Strong interest in(ER) is motivated by the fact that it represents a discrete analogue of the differential equation (see [1] and the references cited therein)
(3)x′(t)+∑i=1m‍pi(t)x(τi(t))=0,t≥0,
where, for every i∈{1,…,m}, pi is an oscillating continuous real-valued function in the interval [0,∞), and τi is a continuous real-valued function on [0,∞) such that
(4)τi(t)≤t,t≥0,lim⁡t→∞τi(t)=∞,
while, (EA) represents a discrete analogue of the advanced differential equation (see [1] and the references cited therein)
(5)x′(t)-∑i=1m‍pi(t)x(σi(t))=0,t≥1,
where, for every i∈{1,…,m}, pi is an oscillating continuous real-valued function in the interval [1,∞) and σi is a continuous real-valued function on [1,∞) such that
(6)σi(t)≥t,t≥1.By asolution of (ER), we mean a sequence of real numbers {x(n)}n≥-w which satisfies (ER) for all n∈N0. Here,
(7)w=-min⁡n≥0{τi(n):1≤i≤m}.
It is clear that, for each choice of real numbers c-w,c-w+1,…,c-1,c0, there exists a unique solution {x(n)}n≥-w of (ER) which satisfies the initial conditions x(-w)=c-w, x(-w+1)=c-w+1,…,x(-1)=c-1, and x(0)=c0.By asolution of the advanced difference equation (EA), we mean a sequence of real numbers {x(n)}n∈N0 which satisfies (EA) for all n∈N.A solution{x(n)}n≥-w[{x(n)}n∈N0] of (ER) [(EA)] is calledoscillatory, if the terms x(n) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to benonoscillatory.In the last few decades, the oscillatory behavior of all solutions of difference equations has been extensively studied when the coefficientspi(n) are nonnegative. See, for example, [2–20] and the references cited therein. However, for the general case when pi(n) are allowed to oscillate, it is difficult to study the oscillation of (ER) [(EA)], since the difference Δx(n)[∇x(n)] of any nonoscillatory solution of (ER) [(EA)] is always oscillatory. Thus, a small number of papers are dealing with this case. See, for example, [1, 21–32] and the references cited therein.For (3) and (5) with oscillating coefficients, Fukagai and Kusano [1] established the following theorems.Theorem 1 (see [1, Theorem 3′(i)]).
Assume (4) and that there is a continuous nondecreasing function τ*(t) such that τi(t)≤τ*(t)≤t for t≥0, 1≤i≤m. Suppose moreover that there is a sequence {tn}n∈N such that lim⁡n→∞tn=∞, the intervals ⋃n∈N[(τ*)n(tn),tn] are disjoint, and
(8)pi(t)≥0∀t∈⋃n∈N[(τ*)n(tn),tn],1≤i≤m.
If there is a constant c such that
(9)∫τ*(t)t‍∑i=1m‍pi(s)ds>c>1e∀t∈⋃n∈N[(τ*)n-1(tn),tn],
then all solutions of (3) oscillate.Theorem 2 (see [1, Theorem 3′(ii)]).
Assume (6) and that there is a continuous nondecreasing function σ*(t) such that t≤σ*(t)≤σi(t) for t≥0, 1≤i≤m. Suppose moreover that there is a sequence {tn}n∈N such that lim⁡n→∞tn=∞, the intervals ⋃n∈N[tn,(σ*)n(tn)] are disjoint, and
(10)pi(t)≥0∀t∈⋃n∈N[tn,(σ*)n(tn)],1≤i≤m.
If there is a constant c such that
(11)∫tσ*(t)‍∑i=1m‍pi(s)ds>c>1e∀t∈⋃n∈N[tn,(σ*)n-1(tn)],
then all solutions of (5) oscillate.For(ER) and (EA) with oscillating coefficients, recently, Bohner et al. [21, 23] established the following theorems.Theorem 3 (see [23, Theorem 2.4]).
Assume (1) and that the sequences τi are increasing for all i∈{1,…,m}. Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞ and
(12)pk(n)≥0∀n∈⋂i=1m‍{⋃j∈N‍[τ(τ(ni(j))),ni(j)]∩N}≠∅,fsdfsdfsdfsdfsdfsdfsdfsdfsdfdsfsdfsddfghgh1≤k≤m,
where
(13)τ(n)=max⁡1≤i≤mτi(n),n∈N0.
If, moreover,
(14)limsup⁡j→∞∑i=1m‍∑q=τ(n(j))n(j)‍pi(q)>1,
where n(j)=min⁡{ni(j):1≤i≤m}, then all solutions of (ER) oscillate.Theorem 4 (see [23, Theorem 3.4]).
Assume (2) and that the sequences σi are increasing for all i∈{1,…,m}. Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞ and
(15)pk(n)≥0∀n∈⋂i=1m‍{⋃j∈N‍[ni(j),σ(σ(ni(j)))]∩N}≠∅,fdsfsdfsdfsdfsdfsdfdsfdsfsdfdsfsdfdsfdsfdfd1≤k≤m,
where
(16)σ(n)=min⁡1≤i≤mσi(n),n∈N.
If, moreover,
(17)limsup⁡j→∞∑i=1m‍∑q=n(j)σ(n(j))‍pi(q)>1,
where n(j)=max⁡{ni(j):1≤i≤m}, then all solutions of (EA) oscillate.Theorem 5 (see [21, Theorem 2.1]).
Assume (1) and that the sequences τi are increasing for all i∈{1,…,m}. Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞,
(18)pk(n)≥0∀n∈⋂i=1m‍{⋃j∈N‍[τi(τi(ni(j))),ni(j)]∩N}≠∅,fsdfsdfsdfdsfsdfssdfdsfsdfsdfsdfdfsdfsdfdsddddd1≤k≤m,(19)limsup⁡n→∞∑i=1m‍pi(n)>0∀n∈⋂i=1m‍{⋃j∈N‍[τi(τi(ni(j))),ni(j)]∩N}.
If, moreover,
(20)liminf⁡j→∞∑i=1m‍∑q=τi(ni(j))ni(j)-1pi(q)>1e,
then all solutions of (ER) oscillate.Theorem 6 (see [21, Theorem 3.1]).
Assume (2) and that the sequences σi are increasing for all i∈{1,…,m}. Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞,
(21)pk(n)≥0∀n∈⋂i=1m‍{⋃j∈N‍[ni(j),σi(σi(ni(j)))]∩N}≠∅,fsadfsdfadfdsfdsfsdfsdfdsfdfdfsdfsdfdsfsdfdddffd1≤k≤m,(22)limsup⁡n→∞∑i=1m‍pi(n)>0∀n∈⋂i=1m‍{⋃j∈N‍[ni(j),σi(σi(ni(j)))]∩N}.
If, moreover,
(23)liminf⁡j→∞∑i=1m‍∑q=ni(j)+1σi(ni(j))‍pi(q)>1e,
then all solutions of (EA) oscillate.In the present paper, the authors study further(ER) [(EA)] and derive new sufficient oscillation conditions when neither (14) [(17)] nor (20) [(23)] is satisfied (cf. [6–8] and the references cited therein in the case of the equations (ER) [(EA)] with nonnegative coefficients pi, 1≤i≤m). Examples illustrating the results are also given.


## 2. Retarded Equations

In this section, we present new sufficient conditions for the oscillation of all solutions of(ER) when the conditions (14) and (20) are not satisfied, under the assumption that the sequences τi are increasing for all i∈{1,…,m}. To that end, the following lemma provides a useful tool.Lemma 7.
Assume that (1) holds, the sequences τi are increasing for all i∈{1,…,m} and (x(n))n≥-w is a nonoscillatory solution of (ER). Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N, such that lim⁡j→∞ni(j)=∞, and (12) where τ is defined by (13). Set
(24)α:=liminf⁡j→∞∑i=1m‍∑q=τ(n(j))n(j)-1‍pi(q),wheren(j)=min⁡{ni(j):1≤i≤m}.
If0<α<1, then
(25)liminf⁡j→∞x(n(j)+1)x(τ(n(j)))≥α24(1-α).Proof.
Since the solution{x(n)}n≥-w of (ER) is nonoscillatory, it is either eventually positive or eventually negative. As {-x(n)}n≥-w is also a solution of (ER), we may restrict ourselves only to the case where x(n)>0 eventually.
By (12), it is obvious that there exists j0∈N such that
(26)pk(n)≥0∀n∈⋂i=1m‍[τ(τ(ni(j0))),ni(j0)]∩N,sdfsadfasdfasdfsdfasdfsdfsdfdsf1≤k≤m,(27)x(τk(n))>0∀n∈⋂i=1m‍[τ(τ(ni(j0))),ni(j0)]∩N,sdfasdafsdfsdfasdfasdfasdfsdfd1≤k≤m.
Also, by (24) we have
(28)∑i=1m‍∑q=τ(n(j0))n(j0)-1‍pi(q)≥α-ε,
where ε is an arbitrary real number with 0<ε<α.
In view of (26) and (27), (ER) gives
(29)x(n+1)-x(n)=-∑i=1m‍pi(n)x(τi(n))≤0,
for every n∈⋂i=1m[τ(τ(ni(j0))),ni(j0)]∩N. This guarantees that the sequence x is decreasing on ⋂i=1m[τ(τ(ni(j0))),ni(j0)]∩N.
Assume that0<α<1, where α is defined by (24). From inequality (28), it is clear that there exists n*(j0)≥n(j0) such that
(30)∑i=1m‍∑q=n(j0)n*(j0)-1‍pi(q)<α-ε2,∑i=1m‍∑q=n(j0)n*(j0)‍pi(q)≥α-ε2.
This is because in the case where pi(q)<(α-ε)/2, there exists n*(j0)>n(j0) such that (30) is satisfied, while in the case where pi(q)≥(α-ε)/2, then n*(j0)=n(j0), and, therefore,
(31)∑i=1m‍∑q=n(j0)n*(j0)-1‍pi(q)=∑i=1m‍∑q=n(j0)n(j0)-1‍pi(q)(bywhichwemean)=0<α-ε2,(32)∑i=1m‍∑q=n(j0)n*(j0)‍pi(q)=∑i=1m‍∑q=n(j0)n(j0)‍pi(q)≥pi(n(j0))≥α-ε2.
That is, in both cases (30) is satisfied.
Now, we will show thatτ(n*(j0))≤n(j0)-1. Indeed, in the case where pi(n(j0))≥(α-ε)/2, since n*(j0)=n(j0), it is obvious that τ(n*(j0))=τ(n(j0))≤n(j0)-1. In the case where pi(n(j0))<(α-ε)/2, then n*(j0)>n(j0). Assume, for the sake of contradiction, that τ(n*(j0))>n(j0)-1. Hence, n(j0)≤τ(n*(j0))≤n*(j0)-1 and then
(33)∑i=1m‍∑q=τ(n*(j0))n*(j0)-1‍pi(q)≤∑i=1m‍∑q=n(j0)n*(j0)-1‍pi(q)<α-ε2,
which contradicts (28). Thus, in both cases, we have τ(n*(j0))≤n(j0)-1. Therefore
(34)∑i=1m‍∑q=τ(n*(j0))n(j0)-1pi(q)=∑i=1m‍∑q=τ(n*(j0))n*(j0)-1pi(q)-∑i=1m‍∑q=n(j0)n*(j0)-1pi(q)‍>(α-ε)-α-ε2=α-ε2.
Summing up(ER) from n(j0) to n*(j0), and using the fact that the function x is decreasing and the function τ (as defined by (13)) is increasing, we have
(35)x(n(j0))=x(n*(j0)+1)+∑i=1m‍∑q=n(j0)n*(j0)‍pi(q)x(τi(q))≥x(n*(j0)+1)+∑i=1m‍∑q=n(j0)n*(j0)‍pi(q)x(τ(q)),
or
(36)x(n(j0))≥x(n*(j0)+1)+x(τ(n*(j0)))∑i=1m‍∑q=n(j0)n*(j0)‍pi(q),
which, in view of (30), gives
(37)x(n(j0))≥x(n*(j0)+1)+α-ε2x(τ(n*(j0))).
Summing up(ER) from τ(n*(j0)) to n(j0)-1, and using the same arguments, we have
(38)x(τ(n*(j0)))=x(n(j0))+∑i=1m‍∑q=τ(n*(j0))n(j0)-1‍pi(q)x(τi(q))≥x(n(j0))+∑i=1m‍∑q=τ(n*(j0))n(j0)-1‍pi(q)x(τ(q)),
or
(39)x(τ(n*(j0)))≥x(n(j0))+x(τ(n(j0)-1))∑i=1m‍∑q=τ(n*(j0))n(j0)-1‍pi(q),
which, in view of (34), gives
(40)x(τ(n*(j0)))>x(n(j0))+α-ε2x(τ(n(j0)-1)).
Combining inequalities (37) and (40), we obtain
(41)x(n(j0))>x(n*(j0)+1)+α-ε2×[x(n(j0))+α-ε2x(τ(n(j0)-1))],
or
(42)(1-α-ε2)x(n(j0))>x(n*(j0)+1)+(α-ε2)2.x(τ(n(j0)-1)).
Thus
(43)x(n(j0))>(α-ε)22[2-(α-ε)]x(τ(n(j0)-1)).
In view of (43), inequality (42) gives
(44)x(n(j0))>(α-ε)2/(2[2-(α-ε)])1-((α-ε)/2)x(τ(n*(j0)))+(α-ε)22[2-(α-ε)]x(τ(n(j0)-1)),
which, in view of (40) becomes
(45)x(n(j0))>(α-ε)2/(2[2-(α-ε)])1-((α-ε)/2)×[x(n(j0))+α-ε2x(τ(n(j0)-1))]+(α-ε)22[2-(α-ε)]x(τ(n(j0)-1)).
Thus
(46)x(n(j0))x(τ(n(j0)-1))>(α-ε)24[1-(α-ε)],
or
(47)x(n(j0)+1)x(τ(n(j0)))>(α-ε)24[1-(α-ε)].
Hence,
(48)liminf⁡j0→∞x(n(j0)+1)x(τ(n(j0)))≥(α-ε)24[1-(α-ε)],
which, for arbitrarily small values of ε, implies (25).
The proof of the lemma is complete.Theorem 8.
Assume that (1) holds, the sequences τi are increasing for all i∈{1,…,m} and τ is defined by (13). Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞, (12) and define α by (24), where n(j)=min⁡{ni(j):1≤i≤m}.
If0<α<1, and
(49)limsup⁡j→∞∑i=1m‍∑q=τ(n(j))n(j)‍pi(q)>1-α24(1-α),
then all solutions of (ER) oscillate.Proof.
Assume, for the sake of contradiction, that{x(n)}n≥-w is an eventually positive solution of (ER). Then there exists j0∈N such that
(50)pk(n)≥0∀n∈⋂k=1m‍[τ(τ(nk(j0))),nk(j0)]∩N,fdsfasdfafasdfdssfdafdsfsdfdfsdf1≤k≤m,x(τk(n))>0∀n∈⋂k=1m‍[τ(τ(nk(j0))),nk(j0)]∩N,dsafdsfdsafsdfsadfsdfdssdfsdfdf1≤k≤m.
Therefore, by (ER) we have
(51)x(n+1)-x(n)=-∑i=1m‍pi(n)x(τi(n))≤0,
for every n∈⋂k=1m[τ(τ(nk(j0))),nk(j0)]∩N. This guarantees that the sequence x is decreasing on ⋂k=1m[τ(τ(nk(j0))),nk(j0)]∩N.
Summing up(ER) from τ(n(j0)) to n(j0), and using the fact that the function x is decreasing and the function τ (as defined by (13)) is increasing, we obtain
(52)x(τ(n(j0)))=x(n(j0)+1)+∑i=1m‍∑q=τ(n(j0))n(j0)pi(q)x(τi(q))≥x(n(j0)+1)+x(τ(n(j0)))×∑i=1m‍∑q=τ(n(j0))n(j0)pi(q).
Consequently,
(53)∑i=1m‍∑q=τ(n(j0))n(j0)‍pi(q)≤1-x(n(j0)+1)x(τ(n(j0))),
which gives
(54)limsup⁡j0→∞∑i=1m‍∑q=τ(n(j0))n(j0)‍pi(q)≤1-liminf⁡j0→∞x(n(j0)+1)x(τ(n(j0))).
Assume that 0<α<1 and (49) holds. Then by Lemma 7, inequality (25) is fulfilled, and so (54) leads to
(55)limsup⁡j0→∞∑i=1m‍∑q=τ(n(j0))n(j0)‍pi(q)≤1-α24(1-α),
which contradicts condition (49).
The proof of the theorem is complete.


## 3. Advanced Equations

Oscillation of all solutions of(EA) is described by the theorem below. Note that the proof is an easy modification of the proof of Theorem 8 and hence is omitted.Theorem 9.
Assume (2) holds, the sequences σi are increasing for all i∈{1,…,m} and σ is defined by (16). Suppose also that for each i∈{1,…,m} there exists a sequence {ni(j)}j∈N such that lim⁡j→∞ni(j)=∞, (15) and
(56)α:=liminf⁡j→∞∑i=1m‍∑q=n(j)+1σ(n(j))‍pi(q),wheren(j)=max⁡{ni(j):1≤i≤m}.
If0<α<1 and
(57)limsup⁡j→∞∑i=1m‍∑q=n(j)σ(n(j))‍pi(q)>1-α24(1-α),
then all solutions of (EA) oscillate.Remark 10.
Whenα→0, then the conditions (49) and (57) reduce to the conditions (14) and (17), respectively. However the improvement is clear when α→1/e. The lower bound in (49) and (57) is 0.946475699. That is, when 0<α<1/e, our conditions (49) and (57) essentially improve (14) and (17).


## 4. Examples

The significance of the results is illustrated in the following examples.Example 1.
Consider the retarded difference equation(58)Δx(n)+p1(n)x(n-2)+p2(n)x(n-3)+p3(n)x(n-4)=0,n∈N0,
where p1(n), p2(n), and p3(n) are oscillating coefficients, as shown in Figure 1.
In view of (13), it is obvious that τ(n)=n-2. Observe that for
(59)n1(j)=20j+9,j∈N,
we have p1(n)>0 for every n∈A, where
(60)A=⋃j∈N‍[τ(τ(n1(j))),n1(j)]∩N=⋃j∈N‍[20j+5,20j+9]∩N.
For
(61)n2(j)=20j+8,j∈N,
we have p2(n)>0 for every n∈B, where
(62)B=⋃j∈N‍[τ(τ(n2(j))),n2(j)]∩N=⋃j∈N‍[20j+4,20j+8]∩N
and, for
(63)n3(j)=20j+9,j∈N,
we have p3(n)≥0 for every n∈C, where
(64)C=⋃j∈N‍[τ(τ(n3(j))),n3(j)]∩N=⋃j∈N‍[20j+5,20j+9]∩N.
Therefore,
(65)p1(n)>0,p2(n)>0,p3(n)≥0dsadsadsdasdsdss∀n∈A∩B∩C=⋃j∈N‍[20j+5,20j+8]∩N≠∅.
Observe that
(66)n(j)=min⁡{ni(j):1≤i≤3}=20j+8,j∈N.
Now,
(67)α=liminf⁡j→∞∑i=1m‍∑q=τ(n(j))n(j)-1‍pi(q)=liminf⁡j→∞[∑q=20j+620j+7‍p1(q)+∑q=20j+620j+7‍p2(q)+∑q=20j+620j+7‍p3(q)]=2·7100+2·6100+2·2100=0.3,limsup⁡j→∞∑i=12‍∑q=τ(n(j))n(j)‍pi(q)=limsup⁡j→∞[∑q=20j+620j+8‍p1(q)+∑q=20j+620j+8‍p2(q)+∑q=20j+620j+8‍p3(q)]=3·7100+2·6100+5981000+2·2100=0.968.
Observe that
(68)0.968>1-α24(1-α)≃0.967857142;
that is, condition (49) of Theorem 8 is satisfied and, therefore, all solutions of equation (58) oscillate.
On the other hand,(69)0.968<1.
Observe that p1(n)>0 for every n∈A′=A, p2(n)≥0 for every n∈B′, where
(70)B′=⋃j∈N‍[τ2(τ2(n2(j))),n2(j)]∩N=⋃j∈N‍[20j+2,20j+8]∩N,
and p3(n)≥0 for every n∈C′, where
(71)C′=⋃j∈N‍[τ3(τ3(n3(j))),n3(j)]∩N=⋃j∈N‍[20j+1,20j+9]∩N.
Therefore,
(72)p1(n)>0,p2(n)>0,p3(n)≥0fdsfsdfdfsdfdd∀n∈A′∩B′∩C′=⋃j∈N‍[20j+5,20j+8]∩N≠∅.
Also,
(73)liminf⁡j→∞∑i=12‍∑q=τi(ni(j))ni(j)-1‍pi(q)=liminf⁡j→∞[∑q=20j+720j+8‍p1(q)+∑q=20j+520j+7‍p2(q)+∑q=20j+520j+8‍p3(q)]=2·7100+3·6100+2·2100=0.36<1e.
Therefore none of the conditions (14) and (20) is satisfied.Figure 1Example 2.
Consider the advanced difference equation(74)∇x(n)-p1(n)x(n+1)-p2(n)x(n+2)=0,n∈N,
where p1(n) and p2(n) are oscillating coefficients, as shown in Figure 2.Figure 2In view of (16), it is obvious that σ(n)=n+1. Observe that for
(75)n1(j)=16j,j∈N,
we have p1(n)≥0 for every n∈A, where
(76)A=⋃j∈N‍[n1(j),σ(σ(n1(j)))]∩N=⋃j∈N‍[16j,16j+2]∩N.
Also, for
(77)n2(j)=16j+1,j∈N,
we have p2(n)>0 for every n∈B, where
(78)B=⋃j∈N‍[n2(j),σ(σ(n2(j)))]∩N=⋃j∈N‍[16j+1,16j+3]∩N.
Therefore,
(79)p1(n)>0,p2(n)>0∀n∈A∩B=⋃j∈N‍[16j+1,16j+2]∩N≠∅.
Observe that
(80)n(j)=max⁡{ni(j):1≤i≤2}=16j+1,j∈N.
Now
(81)α=liminf⁡j→∞∑i=12‍∑q=n(j)+1σ(n(j))‍pi(q)=liminf⁡j→∞[∑q=16j+216j+2‍p1(q)+∑q=16j+216j+2‍p2(q)]=15100+12100=0.27.
Also
(82)limsup⁡j→∞∑i=12‍∑q=n(j)σ(n(j))‍pi(q)=limsup⁡j→∞[∑q=16j+116j+2‍p1(q)+∑q=16j+116j+2‍p2(q)]=2·15100+12100+5561000=0.976.
Observe that
(83)0.976>1-α24(1-α)≃0.975034246;
that is, condition (57) of Theorem 9 is satisfied and, therefore, all solutions of equation (74) oscillate.On the other hand,(84)0.976<1.
Observe that p1(n)≥0 for every n∈A′=A and p2(n)≥0 for every n∈B′, where
(85)B′=⋃j∈N‍[n2(j),σ2(σ2(n2(j)))]∩N=⋃j∈N‍[16j+1,16j+5]∩N.
Therefore,
(86)p1(n)>0,p2(n)>0fsffdfd∀n∈A′∩B′=⋃j∈N‍[16j+1,16j+2]≠∅.
Also,
(87)liminf⁡j→∞∑i=1m‍∑q=ni(j)+1σi(ni(j))‍pi(q)=liminf⁡j→∞[∑q=16j+116j+1‍p1(q)+∑q=16j+216j+3‍p2(q)]=15100+12100+9100=0.36<1e.
Therefore none of the conditions (17) and (23) is satisfied.



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*Source: 392097-2014-06-01.xml*