Chatzarakis and Li Advances in Difference Equations (2017) 2017:292 DOI 10.1186/s13662-017-1353-5

RESEARCH

Open Access

Oscillations of differential equations generated by several deviating arguments George E Chatzarakis1 and Tongxing Li2* *

Correspondence: [email protected] 2 School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P.R. China Full list of author information is available at the end of the article

Abstract Sufficient conditions, involving lim sup and lim inf, for the oscillation of all solutions of differential equations with several not necessarily monotone deviating arguments and nonnegative coefficients are established. Corresponding differential equations of both delayed and advanced type are studied. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in MATLAB. MSC: 34K06; 34K11 Keywords: differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution

1 Introduction Consider the differential equations with several variable deviating arguments of either delayed x (t) +

m 

  pi (t)x τi (t) =  for all t ≥ t ,

(E)

i=

or advanced type 

x (t) –

m 

  qi (t)x σi (t) =  for all t ≥ t ,

(E )

i=

where pi , qi ,  ≤ i ≤ m, are functions of nonnegative real numbers, and τi , σi ,  ≤ i ≤ m, are functions of positive real numbers such that τi (t) < t,

t ≥ t

σi (t) > t,

t ≥ t ,

and

lim τi (t) = ∞,

t→∞

 ≤ i ≤ m,

(.)

and  ≤ i ≤ m,

(. )

respectively. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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In addition, we consider the initial condition for (E) t ≤ t ,

x(t) = ϕ(t),

(.)

where ϕ : (–∞, t ] → R is a bounded Borel measurable function. A solution of (E), (.) is an absolutely continuous on [t , ∞) function satisfying (E) for almost all t ≥ t and (.) for all t ≤ t . By a solution of (E ) we mean an absolutely continuous on [t , ∞) function satisfying (E ) for almost all t ≥ t . A solution of (E) or (E ) is oscillatory if it is neither eventually positive nor eventually negative. If there exists an eventually positive or an eventually negative solution, the equation is nonoscillatory. An equation is oscillatory if all its solutions oscillate. The problem of establishing sufficient conditions for the oscillation of all solutions of equations (E) or (E ) has been the subject of many investigations. The reader is referred to [–] and the references cited therein. Most of these papers concern the special case where the arguments are nondecreasing, while a small number of these papers are concerned with the general case where the arguments are not necessarily monotone. See, for example, [–, ] and the references cited therein. In the present paper, we establish new oscillation criteria for the oscillation of all solutions of (E) and (E ) when the arguments are not necessarily monotone. Our results essentially improve several known criteria existing in the literature. Throughout this paper, we are going to use the following notation: 

t

α := lim inf t→∞

pi (s) ds,

(.)

τ (t) i=

 β := lim inf t→∞

D(ω) :=

m 

m σ (t) 

t

⎧ ⎨, ⎩ –ω–

if ω > /e, √ –ω–ω , 

 MD := lim sup t→∞

(.)

t

m 

if ω ∈ [, /e],

pi (s) ds,

(.)

(.)

τ (t) i=

 MA := lim sup t→∞

qi (s) ds,

i=

t

m σ (t) 

qi (s) ds,

(.)

i=

where τ (t) = max≤i≤m τi (t), σ (t) = min≤i≤m σi (t) and τi (t), σi (t) (in (.) and (.)) are nondecreasing, i = , , . . . , m.

1.1 DDEs By Remark .. in [], it is clear that if τi (t),  ≤ i ≤ m, are nondecreasing and MD > ,

(.)

then all solutions of (E) are oscillatory. This result is similar to Theorem .. [] which is a special case of Ladas, Lakshmikantham and Papadakis’s result [].

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In  Ladde [] and in  Ladas and Stavroulakis [] proved that if  α> , e

(.)

then all solutions of (E) are oscillatory. In , Hunt and Yorke [] proved that if τi (t) are nondecreasing, t – τi (t) ≤ τ ,  ≤ i ≤ m, and lim inf

m 

t→∞

i=

   pi (t) t – τi (t) > , e

(.)

then all solutions of (E) are oscillatory. Assume that τi (t),  ≤ i ≤ m, are not necessarily monotone. Set hi (t) := sup τi (s) and t ≤s≤t

h(t) := max hi (t), ≤i≤m

i = , , . . . , m,

(.)

for t ≥ t , and  m

t pi (ζ ) dζ , a (t, s) := exp s

i=

 m

t   ar+ (t, s) := exp pi (ζ )ar ζ , τi (ζ ) dζ , s

(.) r ∈ N.

i=

Clearly, hi (t), h(t) are nondecreasing and τi (t) ≤ hi (t) ≤ h(t) < t for all t ≥ t . In , Braverman et al. [] proved that if, for some r ∈ N, 

t

lim sup t→∞

m 

  pi (ζ )ar h(t), τi (ζ ) dζ > ,

(.)

  pi (ζ )ar h(t), τi (ζ ) dζ >  – D(α),

(.)

   pi (ζ )ar h(t), τi (ζ ) dζ > , e

(.)

h(t) i=

or 

t

lim sup t→∞

m 

h(t) i=

or 

m 

t

lim inf t→∞

h(t) i=

then all solutions of (E) oscillate. In , Chatzarakis and Péics [] proved that if  lim sup t→∞

t

m 

h(t) i=

   + ln λ pi (ζ )ar h(ζ ), τi (ζ ) dζ > – D(α), λ

(.)

where λ is the smaller root of the transcendental equation eαλ = λ, then all solutions of (E) are oscillatory.

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Very recently, Chatzarakis [] proved that if, for some j ∈ N, 

 P(s) exp

t

lim sup t→∞

h(t)

Pj (u) du ds > ,

(.)

Pj (u) du ds >  – D(α),

(.)

τ (s)

h(t)

or 

 P(s) exp

t

lim sup t→∞

h(t)

τ (s)

h(t)

or 

 P(s) exp

t

lim sup t→∞

Pj (u) du ds >

t

τ (s)

h(t)

 , D(α)

(.)

or 

 P(s) exp

t

lim sup t→∞

τ (s)

h(t)

 + ln λ Pj (u) du ds > – D(α), λ

(.)

 Pj (u) du ds > , e

(.)

h(s)

or 

 P(s) exp

t

lim inf t→∞

h(s)

τ (s)

h(t)

where

 Pj (t) = P(t)  +

t

 P(s) exp

τ (t)

with P (t) = P(t) =

m

i= pi (t),

t

 P(u) exp

τ (s)

u

 Pj– (ξ ) dξ du ds ,

(.)

τ (u)

then all solutions of (E) are oscillatory.

1.2 ADEs For equation (E ), the dual condition of (.) is MA > 

(.)

(see [], paragraph .). In  Ladde [] and in  Ladas and Stavroulakis [] proved that if  β> , e

(.)

then all solutions of (E ) are oscillatory. In , Zhou [] proved that if σi (t) are nondecreasing, σi (t) – t ≤ σ ,  ≤ i ≤ m, and lim inf t→∞

m  i=

   qi (t) σi (t) – t > , e

then all solutions of (E ) are oscillatory. (See also [], Corollary ...)

(.)

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Assume that σi (t),  ≤ i ≤ m, are not necessarily monotone. Set t ≥ t

ρi (t) := inf σi (s), s≥t

and

ρ(t) := min ρi (t), ≤i≤m

t ≥ t

(.)

and  m

s qi (ζ ) dζ , b (t, s) := exp t

 br+ (t, s) := exp t

i= m s

  qi (ζ )br t, σi (ζ ) dζ ,

(.) r ∈ N.

i=

Clearly, ρi (t), ρ(t) are nondecreasing and σi (t) ≥ ρi (t) ≥ ρ(t) > t for all t ≥ t . In , Braverman et al. [] proved that if, for some r ∈ N, 

m ρ(t) 

lim sup t→∞

t

  qi (ζ )br ρ(t), σi (ζ ) dζ > ,

(.)

  qi (ζ )br ρ(t), σi (ζ ) dζ >  – D(β),

(.)

   qi (ζ )br ρ(t), σi (ζ ) dζ > , e

(.)

i=

or 

m ρ(t) 

lim sup t→∞

t

i=

or 

m ρ(t) 

lim inf t→∞

t

i=

then all solutions of (E ) are oscillatory. Very recently, Chatzarakis [] proved that if, for some j ∈ N, 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s)

Qj (u) du ds > ,

(.)

Qj (u) du ds >  – D(β),

(.)

Qj (u) du ds >

(.)

ρ(t)

or 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s) ρ(t)

or 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s) t

 , D(β)

or 

ρ(t)

lim sup t→∞

 Q(s) exp

t

 + ln λ Qj (u) du ds > – D(β), λ

(.)

 Qj (u) du ds > , e

(.)

σ (s) ρ(s)

or  lim inf t→∞

t

ρ(t)

 Q(s) exp

σ (s) ρ(s)

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where

 Qj (t) = Q(t)  +

σ (t)

 Q(s) exp

t

with Q (t) = Q(t) =



σ (s)

t

m

i= qi (t),

σ (u)

Q(u) exp

 Qj– (ξ ) dξ du ds ,

(.)

u

then all solutions of (E ) are oscillatory.

2 Main results 2.1 DDEs We further study (E) and derive new sufficient oscillation conditions, involving lim sup and lim inf, which essentially improve all known results in the literature. For this purpose, we will use the following three lemmas. The proofs of them are similar to the proofs of Lemmas .., .. and .. in [], respectively. Lemma  Assume that h(t) is defined by (.). Then 

m 

t

lim inf t→∞



m 

t

pi (s) ds = lim inf t→∞

τ (t) i=

pi (s) ds.

(.)

h(t) i=

Lemma  Assume that x is an eventually positive solution of (E), h(t) is defined by (.) and α by (.) with  < α ≤ /e. Then lim inf t→∞

x(t) ≥ D(α). x(h(t))

(.)

Lemma  Assume that x is an eventually positive solution of (E), h(t) is defined by (.) and α by (.) with  < α ≤ /e. Then lim inf t→∞

x(h(t)) ≥ λ , x(t)

(.)

where λ is the smaller root of the transcendental equation λ = eαλ . Based on the above lemmas, we establish the following theorems. Theorem  Assume that h(t) is defined by (.) and, for some j ∈ N, 

t

lim sup t→∞

 P(s) exp



h(t)

P(u) exp

τ (s)

h(t)

u



Rj (ξ ) dξ du ds > ,

(.)

τ (u)

where

 Rj (t) = P(t)  +



t

τ (t)



t

P(s) exp

u

P(u) exp τ (s)

 Rj– (ξ ) dξ du ds ,

(.)

τ (u)

 with P(t) = m i= pi (t), R (t) = λ P(t), and λ is the smaller root of the transcendental equaαλ tion λ = e . Then all solutions of (E) are oscillatory. Proof Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (E). Since –x(t) is also a solution of (E), we can confine our discussion only to the case

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where the solution x(t) is eventually positive. Then there exists a t > t such that x(t) >  and x(τi (t)) > ,  ≤ i ≤ m, for all t ≥ t . Thus, from (E) we have x (t) = –

m 

  pi (t)x τi (t) ≤ 

for all t ≥ t ,

i=

which means that x(t) is an eventually nonincreasing function of positive numbers. Taking into account that τi (t) ≤ h(t), (E) implies that  

x (t) +

m 



m      pi (t) x h(t) ≤ x (t) + pi (t)x τi (t) =  for all t ≥ t ,

i=

i=

or   x (t) + P(t)x h(t) ≤  for all t ≥ t .

(.)

Observe that (.) implies that, for each > , there exists a t such that x(h(t)) > λ – x(t)

for all t ≥ t ≥ t .

(.)

Combining inequalities (.) and (.), we obtain x (t) + (λ – )P(t)x(t) ≤ ,

t ≥ t ,

or x (t) + R (t, )x(t) ≤ ,

t ≥ t ,

(.)

where R (t, ) = (λ – )P(t).

(.)

Applying the Grönwall inequality in (.), we conclude that  t x(s) ≥ x(t) exp R (ξ , ) dξ ,

t ≥ s ≥ t .

s

Now we divide (E) by x(t) >  and integrate on [s, t], so 

t

– s

x (u) du = x(u)

 t m

x(τi (u)) du x(u) s i=   t  m x(τ (u)) du pi (u) ≥ x(u) s i=  t x(τ (u)) du P(u) = x(u) s pi (u)

(.)

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or ln

x(s) ≥ x(t)



t

P(u) s

x(τ (u)) du, x(u)

t ≥ s ≥ t .

(.)

Since τ (u) < u, setting u = t, s = τ (u) in (.), we take    x τ (u) ≥ x(u) exp

R (ξ , ) dξ .

u

(.)

τ (u)

Combining (.) and (.), we obtain, for sufficiently large t, ln

x(s) ≥ x(t)





t

u

P(u) exp

R (ξ , ) dξ du

τ (u)

s

or  t  x(s) ≥ x(t) exp P(u) exp

u



R (ξ , ) dξ du .

(.)

τ (u)

s

Hence,    x τ (s) ≥ x(t) exp



t

u

P(u) exp

τ (s)



R (ξ , ) dξ du .

(.)

τ (u)

Integrating (E) from τ (t) to t, we have   x(t) – x τ (t) +



t

m 

  pi (s)x τi (s) ds = ,

τ (t) i=

or   x(t) – x τ (t) +



t



τ (t)

m 



  pi (s) x τ (s) ds ≤ ,

i=

i.e.,   x(t) – x τ (t) +



t

  P(s)x τ (s) ds ≤ .

(.)

τ (t)

It follows from (.) and (.) that   x(t) – x τ (t) + x(t)





t

P(s) exp τ (t)



t

u

P(u) exp τ (s)



R (ξ , ) dξ du ds ≤ .

τ (u)

Multiplying the last inequality by P(t), we find   P(t)x(t) – P(t)x τ (t)   t P(s) exp + P(t)x(t) τ (t)



t

τ (s)

u

P(u) exp τ (u)

R (ξ , ) dξ du ds ≤ .

(.)

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Furthermore, x (t) = –

m 

m       pi (t)x τi (t) ≤ –x τ (t) pi (t) = –P(t)x τ (t) .

i=

(.)

i=

Combining inequalities (.) and (.), we have x (t) + P(t)x(t)  + P(t)x(t)



t



t

P(s) exp

τ (t)

R (ξ , ) dξ du ds ≤ .

u

P(u) exp τ (s)

τ (u)

Hence, 





t

x (t) + P(t)  +



t

P(s) exp



 R (ξ , ) dξ du ds x(t) ≤ ,

u

P(u) exp

τ (t)

τ (s)

τ (u)

or x (t) + R (t, )x(t) ≤ ,

(.)

where

 R (t, ) = P(t)  +



t



t

P(s) exp

 R (ξ , ) dξ du ds .

u

P(u) exp

τ (t)

τ (s)

τ (u)

Clearly, (.) resembles (.) with R replaced by R , so an integration of (.) on [s, t] leads to  t x(s) ≥ x(t) exp R (ξ , ) dξ .

(.)

s

Taking the steps starting from (.) to (.), we may see that x satisfies the inequality    x τ (u) ≥ x(u) exp

R (ξ , ) dξ .

u

(.)

τ (u)

Combining now (.) and (.), we obtain  t  x(s) ≥ x(t) exp P(u) exp

u

R (ξ , ) dξ du ,

τ (u)

s

from which we take    x τ (s) ≥ x(t) exp



t

u

P(u) exp

τ (s)

R (ξ , ) dξ du .

τ (u)

By (.) and (.) we have   x(t) – x τ (t) + x(t)





t

τ (t)



t

P(s) exp

u

P(u) exp τ (s)

τ (u)

R (ξ , ) dξ du ds ≤ .

(.)

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Multiplying the last inequality by P(t), as before, we find

 x (t) + P(t)  +



t



t

P(s) exp

 R (ξ , ) dξ du ds x(t) ≤ .

u

P(u) exp

τ (t)

τ (s)

τ (u)

Therefore, for sufficiently large t, x (t) + R (t, )x(t) ≤ ,

(.)

where





t

R (t, ) = P(t)  +



t

P(s) exp

u

P(u) exp

τ (t)

τ (s)



 R (ξ , ) dξ du ds .

τ (u)

Repeating the above procedure, it follows by induction that for sufficiently large t x (t) + Rj (t, )x(t) ≤ ,

j ∈ N,

where

 Rj (t) = P(t)  +



t



t

P(s) exp

u

P(u) exp

τ (t)

τ (s)

 Rj– (ξ , ) dξ du ds .

τ (u)

Moreover, since τ (s) ≤ h(s) ≤ h(t), we have      x τ (s) ≥ x h(t) exp



h(t)



Rj (ξ , ) dξ du .

u

P(u) exp

τ (s)

(.)

τ (u)

Integrating (E) from h(t) to t and using (.), we obtain    = x(t) – x h(t) +



m 

t

  pi (s)x τi (s) ds

h(t) i=

  ≥ x(t) – x h(t) +





t

m 

h(t)

  = x(t) – x h(t) +





  pi (s) x τ (s) ds

i=

  P(s)x τ (s) ds

t

h(t)

    ≥ x(t) – x h(t) + x h(t)





t



h(t)

P(s) exp

u

P(u) exp τ (s)

h(t)



Rj (ξ , ) dξ du ds,

τ (u)

i.e.,   x(t) – x h(t)    + x h(t)

t

 P(s) exp



h(t) τ (s)

h(t)

u

P(u) exp

Rj (ξ , ) dξ du ds ≤ .

τ (u)

The strict inequality is valid if we omit x(t) >  on the left-hand side. Therefore,

   x h(t)



t

h(t)



h(t)

P(s) exp

u

P(u) exp τ (s)

τ (u)

 Rj (ξ , ) dξ du ds –  < ,

(.)

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or 



t



h(t)

P(s) exp

u

P(u) exp τ (s)

h(t)

Rj (ξ , ) dξ du ds –  < .

τ (u)

Taking the limit as t → ∞, we have 

 P(s) exp

t

lim sup t→∞



h(t)

τ (s)

h(t)

u

P(u) exp

Rj (ξ , ) dξ du ds ≤ .

τ (u)

Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.



Theorem  Assume that α is defined by (.) with  < α ≤ /e and h(t) by (.). If for some j∈N 

 P(s) exp

t

lim sup t→∞



h(t)

P(u) exp

τ (s)

h(t)

u



Rj (ξ ) dξ du ds >  – D(α),

(.)

τ (u)

where Rj is defined by (.), then all solutions of (E) are oscillatory. Proof Let x be an eventually positive solution of (E). Then, as in the proof of Theorem , (.) is satisfied, i.e., 

    x(t) – x h(t) + x h(t)



t



h(t)

P(s) exp

u

P(u) exp τ (s)

h(t)

Rj (ξ , ) dξ du ds ≤ .

τ (u)

That is, 



t



h(t)

P(s) exp

P(u) exp τ (s)

h(t)



x(t) , Rj (ξ , ) dξ du ds ≤  – x(h(t)) τ (u) u

which gives 



t

P(s) exp

lim sup t→∞

t→∞



Rj (ξ , ) dξ du ds

u

P(u) exp τ (s)

h(t)

≤  – lim inf



h(t)

τ (u)

x(t) . x(h(t))

(.)

By combining Lemmas  and , it becomes obvious that inequality (.) is fulfilled. So, (.) leads to 

t

lim sup t→∞

h(t)

 P(s) exp



h(t)

τ (s)

u

P(u) exp

Rj (ξ , ) dξ du ds ≤  – D(α).

τ (u)

Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.



Remark  It is clear that the left-hand sides of both conditions (.) and (.) are identical, also the right-hand side of condition (.) reduces to (.) in case that α = . So it

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seems that Theorem  is the same as Theorem  when α = . However, one may notice that the condition  < α ≤ /e is required in Theorem  but not in Theorem . Theorem  Assume that α is defined by (.) with  < α ≤ /e and h(t) by (.). If for some j∈N 

t

lim sup t→∞

 P(s) exp



t

P(u) exp

τ (s)

h(t)

u



Rj (ξ ) dξ du ds >

τ (u)

 – , D(α)

(.)

where Rj is defined by (.), then all solutions of (E) are oscillatory. Proof Assume, for the sake of contradiction, that there exists a nonoscillatory solution x of (E) and that x is eventually positive. Then, as in the proof of Theorem , (.) is satisfied, which yields    x τ (s) ≥ x(t) exp



t

Rj (ξ , ) dξ du .

u

P(u) exp

τ (s)

τ (u)

Integrating (E) from h(t) to t, we have 

  x(t) – x h(t) +

t

m 

  pi (s)x τi (s) ds = ,

h(t) i=

or 

  x(t) – x h(t) +

t



h(t)

m 



  pi (s) x τ (s) ds ≤ .

i=

Thus 

  x(t) – x h(t) +

t

  P(s)x τ (s) ds ≤ .

h(t)

By virtue of (.), the last inequality gives 

  x(t) – x h(t) +

t

 P(s)x(t) exp



t

P(u) exp

τ (s)

h(t)

u



Rj (ξ , ) dξ du ds ≤ ,

τ (u)

or   x(t) – x h(t) + x(t)





t



t

P(s) exp τ (s)

h(t)

u

P(u) exp

Rj (ξ , ) dξ du ds ≤ .

τ (u)

Thus, for all sufficiently large t, it holds 



t



t

P(s) exp

P(u) exp τ (s)

h(t)



x(h(t)) – . Rj (ξ , ) dξ du ds ≤ x(t) τ (u) u

Letting t → ∞, we take 

t

lim sup t→∞

h(t)

 P(s) exp



t

τ (s)

P(u) exp

x(h(t)) – , Rj (ξ , ) dξ du ds ≤ lim sup x(t) t→∞ τ (u) u

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which, in view of (.), gives 

t

lim sup t→∞

 P(s) exp



t

τ (s)

h(t)

Rj (ξ , ) dξ du ds ≤

u

P(u) exp τ (u)

 – . D(α)

Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.



Theorem  Assume that α is defined by (.) with  < α ≤ /e and h(t) by (.). If for some j∈N 

t

lim sup t→∞

 P(s) exp



h(s)

P(u) exp

τ (s)

h(t)



 + ln λ Rj (ξ ) dξ du ds > – D(α), (.) λ τ (u) u

where Rj is defined by (.) and λ is the smaller root of the transcendental equation λ = eαλ , then all solutions of (E) are oscillatory. Proof Assume, for the sake of contradiction, that there exists a nonoscillatory solution x of (E) and that x is eventually positive. Then, as in Theorem , (.) holds. Observe that (.) implies that, for each > , there exists a t such that λ – <

x(h(t)) x(t)

for all t ≥ t .

(.)

Noting that by nonincreasingness of the function x(h(t))/x(s) in s it holds =

x(h(t)) x(h(t)) x(h(t)) ≤ ≤ , x(h(t)) x(s) x(t)

t ≤ h(t) ≤ s ≤ t,

in particular for ∈ (, λ – ), by continuity we see that there exists a t ∗ ∈ (h(t), t] such that  < λ – =

x(h(t)) . x(t ∗ )

(.)

By (.), it is obvious that      x τ (s) ≥ x h(s) exp



h(s)

u

P(u) exp

τ (s)

Rj (ξ , ) dξ du .

τ (u)

Integrating (E) from t ∗ to t, we have   x(t) – x t ∗ +

 t m t∗

  pi (s)x τi (s) ds = ,

i=

or   x(t) – x t ∗ +

 t  m t∗

i=



  pi (s) x τ (s) ds ≤ ,



(.)

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i.e.,   x(t) – x t ∗ +



t t∗

  P(s)x τ (s) ds ≤ .

By using (.) along with h(s) ≤ h(t) in combination with the nonincreasingness of x, we have     x(t) – x t ∗ + x h(t)



t

t∗

 P(s) exp



h(s)

u

P(u) exp τ (s)

Rj (ξ , ) dξ du ds ≤ ,

τ (u)

or 

t

t∗

 P(s) exp



h(s)

P(u) exp τ (s)



x(t) x(t ∗ ) – . Rj (ξ , ) dξ du ds ≤ x(h(t)) x(h(t)) τ (u) u

In view of (.) and Lemma , for the considered, there exists a t  ≥ t such that 

t

t∗

 P(s) exp



h(s)



Rj (ξ , ) dξ du ds <

u

P(u) exp τ (s)

τ (u)

 – D(α) + λ –

(.)

for t ≥ t  . Dividing (E) by x(t) and integrating from h(t) to t ∗ , we find 

m t∗ 

h(t) i=

x(τi (s)) ds = – pi (s) x(s)



t∗ h(t)

x (s) ds, x(s)

or 

t∗

h(t)





m 

x(τ (s)) pi (s) ds ≤ – x(s)

i=



t∗ h(t)

x (s) ds, x(s)

i.e., 

t∗

P(s) h(t)

x(τ (s)) ds ≤ – x(s)



t∗ h(t)

x (s) ds, x(s)

and using (.), we find  h(s)  u  t∗  x (s) x(h(s)) exp ds. (.) P(s) P(u) exp Rj (ξ , ) dξ du ds ≤ – x(s) h(t) h(t) x(s) τ (s) τ (u)



t∗

By (.), for s ≥ h(t) ≥ t  , we have x(h(s))/x(s) > λ – , so from (.) we get 

t∗

(λ – ) h(t)

 P(s) exp



h(s)

u

P(u) exp τ (s)

τ (u)

 Rj (ξ , ) dξ du ds < –

t∗ h(t)

x (s) ds. x(s)

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Hence, for all sufficiently large t, we have 

t∗

 P(s) exp

h(t)

<–

 λ –



h(s) τ (s)



u

P(u) exp t∗

h(t)

Rj (ξ , ) dξ du ds

τ (u)

 x(h(t)) ln(λ – ) x (s) ds = ln = , x(s) λ – x(t ∗ ) λ –

i.e., 



t∗



h(s)

P(s) exp

P(u) exp τ (s)

h(t)

ln(λ – ) Rj (ξ , ) dξ du ds < . λ – τ (u) u

(.)

Adding (.) and (.), and then taking the limit as t → ∞, we have 

 P(s) exp

t

lim sup t→∞



h(s)

τ (s)

h(t)

u

P(u) exp

Rj (ξ , ) dξ du ds

τ (u)

 + ln(λ – ) – D(α) + . ≤ λ – Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.



Theorem  Assume that h(t) is defined by (.) and for some j ∈ N  t→∞



t

h(t)



h(s)

P(s) exp

lim inf

P(u) exp τ (s)

 Rj (ξ ) dξ du ds > , e τ (u) u

(.)

where Rj is defined by (.). Then all solutions of (E) are oscillatory. Proof Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (E). Since –x(t) is also a solution of (E), we can confine our discussion only to the case where the solution x(t) is eventually positive. Then there exists a t > t such that x(t) >  and x(τi (t)) > ,  ≤ i ≤ m for all t ≥ t . Thus, from (E) we have x (t) = –

m 

  pi (t)x τi (t) ≤ 

for all t ≥ t ,

i=

which means that x(t) is an eventually nonincreasing function of positive numbers. Furthermore, as in previous theorem, (.) is satisfied. Dividing (E) by x(t) and integrating from h(t) to t, for some t ≥ t , we get  t   m x(τi (s)) x(h(t)) = ds pi (s) ln x(t) x(s) h(t) i=   t  m x(τ (s)) ds pi (s) ≥ x(s) h(t) i=  t x(τ (s)) ds. P(s) = x(s) h(t)

(.)

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Combining inequalities (.) and (.), we obtain   u  t  h(s) x(h(t)) x(h(s)) ln P(s) P(u) exp Rj (ξ , ) dξ du ds. ≥ exp x(t) x(s) h(t) τ (s) τ (u) Taking into account that x is nonincreasing and h(s) < s, the last inequality becomes   t  h(s)  u x(h(t)) ln ≥ P(s) exp P(u) exp Rj (ξ , ) dξ du ds. x(t) h(t) τ (s) τ (u)

(.)

From (.), it follows that there exists a constant c >  such that for sufficiently large t 



t



h(s)

P(s) exp

 Rj (ξ ) dξ du ds ≥ c > . e τ (u)

P(u) exp τ (s)

h(t)

u



Choose c such that c > c > /e. For every >  such that c – > c , we have 



t



h(s)

P(s) exp

 Rj (ξ , ) dξ du ds > c – > c > . e τ (u)

P(u) exp τ (s)

h(t)

u



(.)

Combining inequalities (.) and (.), we obtain  x(h(t)) ln ≥ c , x(t)

t ≥ t .

Thus x(h(t))  ≥ ec ≥ ec > , x(t) which yields, for some t ≥ t ≥ t ,     x h(t) ≥ ec x(t). Repeating the above procedure, it follows by induction that for any positive integer k, x(h(t))   k ≥ ec x(t)

for sufficiently large t.

Since ec > , there is a k ∈ N satisfying k > (ln() – ln(c ))/( + ln(c )) such that for t sufficiently large x(h(t))   k ≥ ec > x(t)

   . c

(.)

Next we split the integral in (.) into two integrals, each integral being no less than c /: c Rj (ξ , ) dξ du ds ≥ ,  h(t) τ (s) τ (u)  h(s)  u  t c P(s) exp P(u) exp Rj (ξ , ) dξ du ds ≥ .  tm τ (s) τ (u)





tm

P(s) exp



h(s)

u

P(u) exp

(.)

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Integrating (E) from tm to t, we deduce that x(t) – x(tm ) +

 t m

  pi (s)x τi (s) ds = ,

tm i=

or x(t) – x(tm ) +



 t  m tm

  pi (s) x τ (s) ds ≤ .

i=

Thus 

t

x(t) – x(tm ) +

  P(s)x τ (s) ds ≤ ,

tm

which, in view of (.), gives 

  x(t) – x(tm ) + x h(t)



t



h(s)

P(s) exp τ (s)

tm

Rj (ξ , ) dξ du ds ≤ .

u

P(u) exp τ (u)

The strict inequality is valid if we omit x(t) >  on the left-hand side:   –x(tm ) + x h(t)





t



h(s)

P(s) exp

u

P(u) exp τ (s)

tm



Rj (ξ , ) dξ du ds < .

τ (u)

Using the second inequality in (.), we conclude that x(tm ) >

 c  x h(t) . 

(.)

Similarly, integration of (E) from h(t) to tm with a later application of (.) leads to     x(tm ) – x h(t) + x h(tm )





tm



h(s)

P(s) exp τ (s)

h(t)

u

P(u) exp

Rj (ξ , ) dξ du ds ≤ .

τ (u)

The strict inequality is valid if we omit x(tm ) >  on the left-hand side:     –x h(t) + x h(tm )



tm

h(t)

 exp



h(s)

u

P(u) exp

τ (s)



Rj (ξ , ) dξ du ds < .

τ (u)

Using the first inequality in (.) implies that   c   x h(t) > x h(tm ) . 

(.)

Combining inequalities (.) and (.), we obtain      x h(tm ) <  x h(t) < c

   x(tm ), c

which contradicts (.). The proof of the theorem is complete.



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2.2 ADEs Similar oscillation conditions for the (dual) advanced differential equation (E ) can be derived easily. The proofs are omitted since they are quite similar to the delay equation. Theorem  Assume that ρ(t) is defined by (.), and for some j ∈ N 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s)

 Q(u) exp

σ (u)



Lj (ξ ) dξ du ds > ,

(.)

u

ρ(t)

where

 Lj (t) = Q(t)  +

σ (t)

 Q(s) exp

t

σ (s)

 Q(u) exp

t

σ (u)



 Lj– (ξ ) dξ du ds ,

(.)

u

 with Q(t) = m i= qi (t), L (t) = λ Q(t) and λ is the smaller root of the transcendental equation λ = eβλ . Then all solutions of (E ) are oscillatory. Theorem  Assume that β is defined by (.) with  < β ≤ /e and ρ(t) by (.). If for some j ∈ N 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s)

 Q(u) exp

σ (u)



Lj (ξ ) dξ du ds >  – D(β),

(.)

u

ρ(t)

where Lj is defined by (.), then all solutions of (E ) are oscillatory. Remark  It is clear that the left-hand sides of both conditions (.) and (.) are identical, also the right-hand side of condition (.) reduces to (.) in case that β = . So it seems that Theorem  is the same as Theorem  when β = . However, one may notice that the condition  < β ≤ /e is required in Theorem  but not in Theorem . Theorem  Assume that β is defined by (.) with  < β ≤ /e and ρ(t) by (.). If for some j ∈ N 

ρ(t)

lim sup t→∞

 Q(s) exp

t

σ (s)

 Q(u) exp

t

σ (u)

Lj (ξ ) dξ du ds >

u

 – , D(β)

(.)

where Lj is defined by (.), then all solutions of (E ) are oscillatory. Theorem  Assume that β is defined by (.) with  < β ≤ /e and ρ(t) by (.). If for some j ∈ N 

ρ(t)

lim sup t→∞

>

 Q(s) exp

t

 + ln λ – D(β), λ

σ (s)

ρ(s)

 Q(u) exp

σ (u)

Lj (ξ ) dξ du ds

u

(.)

where Lj is defined by (.) and λ is the smaller root of the transcendental equation λ = eβλ , then all solutions of (E ) are oscillatory.

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Theorem  Assume that ρ(t) is defined by (.) and for some j ∈ N  lim inf t→∞

ρ(t)

 Q(s) exp

t



σ (s)

σ (u)

Q(u) exp u

ρ(s)

 Lj (ξ ) dξ du ds > , e

(.)

where Qj is defined by (.). Then all solutions of (E ) are oscillatory.

2.3 Differential inequalities A slight modification in the proofs of Theorems - leads to the following results about differential inequalities. Theorem  Assume that all the conditions of Theorem  [] or  [] or  [] or  [] or  [] hold. Then (i) the delay [advanced] differential inequality x (t) +

m 

  pi (t)x τi (t) ≤ 

 x (t) –

m 

i=

   qi (t)x σi (t) ≥  ,

∀t ≥ t ,

   qi (t)x σi (t) ≤  ,

∀t ≥ t ,

i=

has no eventually positive solutions; (ii) the delay [advanced] differential inequality 

x (t) +

m 

  pi (t)x τi (t) ≥ 

 

x (t) –

m 

i=

i=

has no eventually negative solutions.

2.4 An example We give an example that illustrates a case when Theorem  of the present paper yields oscillation, while previously known results fail. The calculations were made by the use of MATLAB software. Example  Consider the delay differential equation x (t) +

         x τ (t) + x τ (t) + x τ (t) = ,   

t ≥ ,

with (see Figure , (a)) ⎧ ⎪ –t + k – , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪t – k – , ⎪ ⎪ ⎪ ⎪ ⎨–t + k + , τ (t) = ⎪t – , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ –t + k + , ⎪ ⎪ ⎪ ⎪ ⎩t – k – ,

if t ∈ [k, k + ], if t ∈ [k + , k + ], if t ∈ [k + , k + ], if t ∈ [k + , k + ],

and

if t ∈ [k + , k + ], if t ∈ [k + , k + ],

where k ∈ N and N is the set of nonnegative integers.

τ (t) = τ (t) – ., τ (t) = τ (t) – .,

(.)

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Figure 1 The graphs of τ1 (t) and h1 (t).

By (.), we see (Figure , (b)) that ⎧ ⎪ k – , if t ∈ [k, k + .], ⎪ ⎪ ⎪ ⎪ ⎨t – k – , if t ∈ [k + ., k + ], h (t) = ⎪ ⎪k + , if t ∈ [k + , k + .], ⎪ ⎪ ⎪ ⎩ t – k – , if t ∈ [k + ., k + ],

and

h (t) = h (t) – ., h (t) = h (t) – .,

and consequently,     h(t) = max hi (t) = h (t) and τ (t) = max τi (t) = τ (t). ≤i≤

≤i≤

It is easy to verify that  α = lim inf t→∞

t

 



k+

pi (s) ds = . · lim inf k→∞

τ (t) i=

ds = ., k+

and therefore, the smaller root of e.λ = λ is λ = .. Observe that the function Fj : [, ∞) → R+ defined as 

t

Fj (t) =

 P(s) exp



h(t)

P(u) exp

τ (s)

h(t)

u



Rj (ξ ) dξ du ds

τ (u)

attains its maximum at t = k + ., k ∈ N , for every j ≥ . Specifically, 

k+.

F (t = k + .) =

 P(s) exp

k+



k+

u

P(u) exp τ (s)



R (ξ ) dξ du ds

τ (u)

with

 R (ξ ) = P(ξ )  +

ξ τ (ξ )

 P(v) exp



ξ

τ (v)

w

P(w) exp τ (w)

 λ P(z) dz dw dv .

Chatzarakis and Li Advances in Difference Equations (2017) 2017:292

Page 21 of 24

By using an algorithm on MATLAB software, we obtain F (t = k + .)  ., and so lim sup F (t)  . > . t→∞

That is, condition (.) of Theorem  is satisfied for j = , and therefore all solutions of (.) are oscillatory. Observe, however, that 

 k+. 

MD = lim sup k→∞

k+

pi (s) ds = . < ,

i=

 α = . < , e and lim inf

 

t→∞

  pi (t) t – τi (t)

i=

           t – τ (t) + t – τ (t) – . + t – τ (t) – . = lim inf t→∞           = lim inf . t – τ (t) + . = lim inf . t – τ (t) + . t→∞



t→∞

   = . · lim inf t – τ (t) + . = . ·  + . = . < . t→∞ e Also, observe that the function Gr : [, ∞) → R+ defined as  Gr (t) =

t

m 

  pi (ζ )ar h(t), τi (ζ ) dζ

h(t) i=

attains its maximum at t = k + . and its minimum at t = k + , k ∈ N , for every r ∈ N. Specifically,  G (t = k + .) =

 k+. 

k+

 =

k+ 

  pi (ζ )a k + , τi (ζ ) dζ

i=

    p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ )

k+

  + p (ζ )a k + , τ (ζ ) dζ  k+      + p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ ) k+

  + p (ζ )a k + , τ (ζ ) dζ  k+      + p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ ) k+

Chatzarakis and Li Advances in Difference Equations (2017) 2017:292

  + p (ζ )a k + , τ (ζ ) dζ  k+      p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ ) + k+

  + p (ζ )a k + , τ (ζ ) dζ  k+.      p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ ) + k+

  + p (ζ )a k + , τ (ζ ) dζ  . and  G (t = k + ) =

 k+ 

k+

 =

  pi (ζ )a k + , τi (ζ ) dζ

i=

k+ 

    p (ζ )a k + , τ (ζ ) + p (ζ )a k + , τ (ζ )

k+

  + p (ζ )a k + , τ (ζ ) dζ  .. Thus lim sup G (t)  . < , t→∞

lim inf G (t)  . < /e, t→∞

and . <  – D(α)  .. Also 

 k+. 

k+

  pi (ζ )a h(ζ ), τi (ζ ) dζ ≤ G (t = k + .)  ..

i=

Thus  lim sup k→∞

 k+. 

k+

i=

   + ln λ pi (ζ )a h(ζ ), τi (ζ ) dζ ≤ . < – D(α)  .. λ

Also 

t

lim sup t→∞

h(t)

 P(s) exp

h(t)

P (u) du ds  . < ,

τ (s)

. <  – D(α)  .,

Page 22 of 24

Chatzarakis and Li Advances in Difference Equations (2017) 2017:292



t

lim sup t→∞

h(t)

 P(s) exp

P (u) du ds

τ (s)



k+.

= lim sup k→∞

t

Page 23 of 24

 P(s) exp

k+

k+.

P (u) du ds  .

τ (s)

  ., D(α)   t lim sup P(s) exp <

t→∞

h(t)



≤ lim sup t→∞

h(s)

P (u) du ds

τ (s)

t

h(t)

 P(s) exp

h(t)

P (u) du ds

τ (s)

 + ln λ – D(α)  .,  . < λ  h(t)  t  lim inf P(s) exp P (u) du ds  . < . t→∞ e h(t) τ (s) That is, none of the conditions (.)-(.), (.)-(.) (for r = ) and (.)-(.) (for j = ) is satisfied. Comments It is worth noting that the improvement of condition (.) to the corresponding condition (.) is significant, approximately .%, if we compare the values on the left-hand side of these conditions. Also, the improvement compared to conditions (.) and (.) is very satisfactory, around .% and .%, respectively. Finally, observe that conditions (.)-(.) do not lead to oscillation for the first iteration. On the contrary, condition (.) is satisfied from the first iteration. This means that our condition is better and much faster than (.)-(.). Remark  Similarly, one can construct examples to illustrate the other main results.

Acknowledgements The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China. Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors contributed equally to this work. They both read and approved the final version of the manuscript. Author details 1 Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklio, Athens, 14121, Greece. 2 School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P.R. China.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 29 May 2017 Accepted: 7 September 2017 References 1. Braverman, E, Chatzarakis, GE, Stavroulakis, IP: Iterative oscillation tests for differential equations with several non-monotone arguments. Adv. Differ. Equ. (2016). doi:10.1186/s13662-016-0817-3, 18 pp.

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