# A survey on the oscillation of differential equations with several deviating arguments

## Abstract

Consider the first-order linear differential equation with several retarded arguments

$x ′ (t)+ ∑ i = 1 m p i (t)x ( τ i ( t ) ) =0,t≥ t 0 ,$

where the functions $p i , τ i ∈C([ t 0 , ∞), R + )$ for every $i=1,2,…,m$, $τ i (t)≤t$ for $t≥ t 0$ and $lim t → ∞ τ i (t)=∞$. A survey on the oscillation of all solutions to this equation is presented in the case of several non-monotone arguments and especially when well-known oscillation conditions are not satisfied. Examples illustrating the results are given.

MSC:34K11, 34K06.

## 1 Introduction

Consider the differential equation with several non-monotone retarded arguments

$x ′ (t)+ ∑ i = 1 m p i (t)x ( τ i ( t ) ) =0,t≥ t 0 ,$
(1.1)

where the functions $p i , τ i ∈C([ t 0 , ∞), R + )$ for every $i=1,2,…,m$ (here $R + =[0,∞)$), $τ i (t)≤t$ for $t≥ t 0$ and $lim t → ∞ τ i (t)=∞$.

Let $T 0 ∈[ t 0 ,+∞)$, $τ(t)=min{ τ i (t):i=1,…,m}$ and $τ ( − 1 ) (t)=inf{τ(s):s≥t}$. By a solution of equation (1.1), we understand a function $u∈C([ t 0 ,+∞);R)$ continuously differentiable on $[ τ ( − 1 ) ( T 0 ),+∞)$ that satisfies (1.1) for $t≥ τ ( − 1 ) ( T 0 )$. Such a solution is called oscillatory if it has arbitrarily large zeros, and otherwise it is called non-oscillatory.

In the special case where $m=1$, equation (1.1) reduces to the equation

$x ′ (t)+p(t)x ( τ ( t ) ) =0,t≥ t 0 ,$
(1.2)

where the functions $p,τ∈C([ t 0 , ∞), R + )$, $τ(t)≤t$ for $t≥ t 0$ and $lim t → ∞ τ(t)=∞$.

For the general theory of these equations, the reader is referred to [14].

In this paper we present a survey on the oscillation of all solutions to these equations in the case of several non-monotone arguments and especially when the well-known oscillation conditions

$lim sup t → ∞ ∫ τ ( t ) t p(s)ds>1and lim inf t → ∞ ∫ τ ( t ) t p(s)ds> 1 e$

for equation (1.2) are not satisfied.

## 2 Oscillation criteria for equation (1.2)

In this section we study the delay equation

$x ′ (t)+p(t)x ( τ ( t ) ) =0,t≥ t 0 ,$

where the functions $p,τ∈C([ t 0 , ∞), R + )$, $τ(t) for $t≥ t 0$ and $lim t → ∞ τ(t)=∞$.

The problem of establishing sufficient conditions for the oscillation of all solutions to the differential equation (1.2) has been the subject of many investigations. See, for example, [1, 526] and the references cited therein.

The first systematic study for the oscillation of all solutions to equation (1.2) was made by Myshkis. In 1950 [20] he proved that every solution of equation (1.2) oscillates if

$lim sup t → ∞ [ t − τ ( t ) ] <∞and lim inf t → ∞ [ t − τ ( t ) ] lim inf t → ∞ p(t)> 1 e .$

In 1972, Ladas et al. [27] proved that the same conclusion holds if in addition τ is a non-decreasing function and

($C 1$) $A:= lim sup t → ∞ ∫ τ ( t ) t p(s)ds>1$.

In 1979, Ladas [17] established integral conditions for the oscillation of equation (1.2) with constant delay, while in 1982, Koplatadze and Chanturija [14] established the following result. If

($C 2$) $a:= lim inf t → ∞ ∫ τ ( t ) t p(s)ds> 1 e$,

then all solutions of equation (1.2) oscillate; if

($N 1$) $lim sup t → ∞ ∫ τ ( t ) t p(s)ds< 1 e$,

then equation (1.2) has a non-oscillatory solution.

It is obvious that there is a gap between conditions ($C 1$) and ($C 2$) when the limit $lim t → ∞ ∫ τ ( t ) t p(s)ds$ does not exist. How to fill this gap is an interesting problem which has been recently investigated by several authors.

In 1988, Erbe and Zhang [10] developed new oscillation criteria by employing the upper bound of the ratio $x(τ(t))/x(t)$ for possible non-oscillatory solutions $x(t)$ of equation (1.2). Their result says that all the solutions of equation (1.2) are oscillatory if $0 and

($C 3$) $A>1− a 2 4$.

Since then several authors tried to obtain better results by improving the upper bound for $x(τ(t))/x(t)$.

In 1991, Jian [12] derived the condition

($C 4$) $A>1− a 2 2 ( 1 − a )$,

while in 1992, Yu et al. [25] obtained the condition

($C 5$) $A>1− 1 − a − 1 − 2 a − a 2 2$.

In 1990, Elbert and Stavroulakis [8] and in 1991 Kwong [16], using different techniques, improved ($C 3$), in the case where $0, to the conditions

($C 6$) $A>1− ( 1 − 1 λ 1 ) 2$

and

($C 7$) $A> ln λ 1 + 1 λ 1$,

respectively, where $λ 1$ is the smaller real root of the equation $λ= e a λ$.

In 1998, Philos and Sficas [21] and in 1999, Zhou and Yu [26] and Jaroš and Stavroulakis [11] derived the conditions

($C 8$) $A>1− a 2 2 ( 1 − a ) − a 2 2 λ 1$,

($C 9$) $A>1− 1 − a − 1 − 2 a − a 2 2 − ( 1 − 1 λ 1 ) 2$,

and

($C 10$) $A> ln λ 1 + 1 λ 1 − 1 − a − 1 − 2 a − a 2 2$,

respectively.

Consider equation (1.2) and assume that $τ(t)$ is continuously differentiable and that there exists $θ>0$ such that $p(τ(t)) τ ′ (t)≥θp(t)$ eventually for all t. Under this additional assumption, in 2000, Kon et al. [13] and in 2003, Sficas and Stavroulakis [22] established the conditions

($C 11$) $A>2a+ 2 λ 1 −1$,

and

($C 12$) $A> ln λ 1 − 1 + 5 − 2 λ 1 + 2 a λ 1 λ 1$,

respectively. In the case where $a= 1 e$, then $λ 1 =e$, and ($C 12$) leads to

$A> 7 − 2 e e ≈0.459987065.$

It is to be noted that for small values of $a$ ($a→0$), all previous conditions ($C 3$)-($C 11$) reduce to condition ($C 1$), i.e., $A>1$. However, condition ($C 12$) leads to

$A> 3 −1≈0.732,$

which is an essential improvement. Moreover, ($C 12$) improves all the above conditions for all values of $a∈(0, 1 e ]$. Note that the value of the lower bound on A cannot be less than $1 e ≈0.367879441$. Thus, the aim is to establish a condition which leads to a value as close as possible to $1 e$.

For illustrative purpose, we give the values of the lower bound on A under these conditions when (i) $a=1/1,000$ and (ii) $a=1/e$.

We see that condition ($C 12$) essentially improves all the known results in the literature.

Moreover, it should be pointed out that in 1994, Koplatadze and Kvinikadze [15] improved ($C 5$) as follows: Assume

$σ(t):= sup s ≤ t τ(s),t≥0.$
(2.1)

Clearly $σ(t)$ is non-decreasing and $τ(t)≤σ(t)$ for all $t≥0$. Define

(2.2)

Then the following theorem was established in [15].

Theorem 2.1 ([15])

Let $k∈{1,2,…}$ exist such that

$lim sup t → ∞ ∫ σ ( t ) t p(s)exp { ∫ σ ( s ) σ ( t ) p ( ξ ) ψ k ( ξ ) d ξ } ds>1−c(a),$
(2.3)

where σ, $ψ k$, $a$ are defined by (2.2), (2.4), ($C 2$), respectively, and

Then all solutions of equation (1.2) oscillate.

Concerning the constants 1 and $1 e$ which appear in conditions ($C 1$), ($C 2$) and ($N 1$), in 2011, Berezansky and Braverman [6] established the following theorem.

Theorem 2.2 ([6])

For any $α∈(1/e,1)$, there exists a non-oscillatory equation

$x ′ (t)+p(t)x(t−τ)=0,τ>0$

with $p(t)≥0$ such that

$lim sup t → ∞ ∫ t − τ t p(s)ds=α.$

Also in 2011, Braverman and Karpuz [7] investigated equation (1.2) in the case of a general argument (τ is not assumed monotone) and proved the following.

Theorem 2.3 ([7])

There is no constant $K>0$ such that

$lim sup t → ∞ ∫ τ ( t ) t p(s)ds>K$
(2.4)

implies oscillation of equation (1.2) for arbitrary (not necessarily non-decreasing) argument $τ(t)≤t$.

Remark 2.1 Observe that, because of condition ($N 1$), the constant K in (2.4) makes sense for $K>1/e$.

Moreover, in [7] the following result was established.

Theorem 2.4 ([7])

Assume that

$B:= lim sup t → ∞ ∫ σ ( t ) t p(s)exp { ∫ τ ( s ) σ ( t ) p ( ξ ) d ξ } ds>1,$
(2.5)

where $σ(t)$ is defined by (2.1). Then all solutions of equation (1.2) oscillate.

Observe that condition (2.5) improves ($C 1$).

Using the upper bound of the ratio $x ( τ ( t ) ) x ( t )$ for possible non-oscillatory solutions $x(t)$ of equation (1.2), presented in [8, 11, 13, 22], the above result was recently essentially improved in [23].

Theorem 2.5 ([23])

Assume that $0 and

$B:= lim sup t → ∞ ∫ σ ( t ) t p(s)exp { ∫ τ ( s ) σ ( t ) p ( ξ ) d ξ } ds>1− 1 2 ( 1 − a − 1 − 2 a − a 2 ) ,$
(2.6)

where $σ(t)$ is defined by (2.1). Then all solutions of equation (1.2) oscillate.

Remark 2.2 ([23])

Observe that as $a→0$, then condition (2.6) reduces to (2.5). However, the improvement is clear as $a→ 1 e$. Actually, when $a= 1 e$, the value of the lower bound on B is equal to ≈0.863457014. That is, (2.6) essentially improves (2.5).

Remark 2.3 ([23])

Note that, under the additional assumption that $τ(t)$ is continuously differentiable and that there exists $θ>0$ such that $p(τ(t)) τ ′ (t)≥θp(t)$ eventually for all t (see [13, 22]) condition (2.6) of Theorem 2.5 reduces to

(2.6′)

where M is given by

$M= e λ θ a − λ θ a − 1 ( λ θ ) 2$

and λ is the smaller root of the equation $λ= e λ a$. When $θ=1$, then from [22] it follows that

$1 2 ( 1 − a − ( 1 − a ) 2 − 4 M ) =1−a− 1 λ$

and in the case that $a= 1 e$, then $λ=e$ and (2.6′) leads to

$B>1− ( 1 − 2 e ) = 2 e ≈0.735758882.$

That is, condition (2.6′) essentially improves (2.6) but of course under the additional (stronger) assumptions on $τ(t)$ and $p(t)$.

The following example illustrates the significance of Theorem 2.5.

Example 2.1 (cf. [7, 23])

Consider the equation

$x ′ (t)+p(t)x ( τ ( t ) ) =0,t≥0,$
(2.7)

where $p(t)= 0.93 e$, and

$τ(t):= { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , − 3 t + ( 12 n + 3 ) , t ∈ [ 3 n + 1 , 3 n + 2 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2 , 3 n + 3 ] .$
(2.8)

We see that

$σ(t)= { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , 3 n , t ∈ [ 3 n + 1 , 3 n + 2.6 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2.6 , 3 n + 3 ] .$
(2.9)

Observe that

$a = lim inf t → ∞ ∫ τ ( t ) t 0.93 e d s = 0.93 e ≈ 0.34212788 < 1 e , lim sup t → ∞ ∫ σ ( t ) t 0.93 e d s = 2.6 0.93 e = 0.889532488 < 1 .$

Moreover, for $n≥0$, we have

$∫ σ ( 3 n + 3 ) 3 n + 3 0.93 e exp { ∫ τ ( s ) σ ( 3 n + 3 ) 0.93 e d ξ } d s = ∫ 3 n + 2 3 n + 3 0.93 e exp { ∫ 5 s − ( 12 n + 13 ) 3 n + 2 0.93 e d ξ } d s = ∫ 3 n + 2 3 n + 3 0.93 e exp { 4.65 e [ 3 n + 3 − s ] } d s = 1 5 [ exp { 4.65 e } − 1 ] ≈ 0.906499566 < 1 .$
(2.10)

That is, conditions ($C 1$), ($C 2$) and (2.5) are not satisfied. Observe, however, that for $a≈0.34212788$,

$1− 1 2 ( 1 − a − 1 − 2 a − a 2 ) ≈0.893938766$

and from (2.10) we see that

$lim sup t → ∞ ∫ σ ( t ) t p(s)exp { ∫ τ ( s ) σ ( t ) p ( ξ ) d ξ } ds>0.90>1− 1 2 ( 1 − a − 1 − 2 a − a 2 ) ≈0.89,$

that is, the conditions of Theorem 2.5 are satisfied and therefore all the solutions of equation (2.7) with the general argument (2.8) are oscillatory.

## 3 Oscillation criteria for equation (1.1)

For equation (1.1) with several arguments, the following results have been established.

In 1982, Ladas and Stavroulakis [28] (see also in 1984, Arino et al. [5]) studied the equation with several constant retarded arguments of the form

(1.1′)

under the assumption that $lim inf t → ∞ ∫ t − τ i / 2 t p(s)ds>0$, $i=1,2,…,m$, and proved that each one of the following conditions

(3.1)
(3.2)
$[ ∏ i = 1 m ( ∑ j = 1 m lim inf t → ∞ ∫ t − τ j t p i ( s ) d s ) ] 1 / m > 1 e ,$
(3.3)

or

$1 m ∑ i = 1 m ( lim inf t → ∞ ∫ t − τ i t p i ( s ) d s ) + 2 m ∑ i < j i , j = 1 m [ ( lim inf t → ∞ ∫ t − τ j t p i ( s ) d s ) ( lim inf t → ∞ ∫ t − τ i t p i ( s ) d s ) ] 1 2 > 1 e$
(3.4)

implies that all solutions of equation (1.1′) oscillate. Later in 1996, Li [19] proved that the same conclusion holds if

$lim inf t → ∞ ∑ i = 1 m ∫ t − τ i t p i (s)ds> 1 e .$
(3.5)

In 1984, Hunt and Yorke [29], considered the equation with variable coefficients of the form

(1.1′′)

under the assumption that there is a uniform upper bound $τ 0$ on the $τ i$’s and proved that if

$lim inf t → ∞ ∑ i = 1 m τ i (t) p i (t)> 1 e ,$

then all solutions of equation (1.1″) oscillate.

In 1984, Fukagai and Kusano [30], for equation (1.1), established the following theorem.

Theorem 3.1 ([[30], Theorem 1′(i)])

Consider equation (1.1) and assume that there is a continuous non-decreasing function $τ ∗ (t)$ such that $τ i (t)≤ τ ∗ (t)≤t$ for $t≥ t 0$, $1≤i≤m$. If

$lim inf t → ∞ ∫ τ ∗ ( t ) t ∑ i = 1 m p i (s)ds> 1 e ,$
(3.6)

then all solutions of equation (1.1) oscillate. If, on the other hand, there exists a continuous non-decreasing function $τ ∗ (t)$ such that $τ ∗ (t)≤ τ i (t)$ for $t≥ t 0$, $1≤i≤m$, $lim t → ∞ τ ∗ (t)=∞$ and

then equation (1.1) has a non-oscillatory solution.

In 2000, Grammatikopoulos et al. [31] improved the above results as follows.

Theorem 3.2 ([[31], Theorem 2.6])

Assume that the functions $τ i$ are non-decreasing for all $i∈{1,…,m}$,

$∫ 0 ∞ | p i (t)− p j (t)|dt<+∞,i,j=1,…,m$

and

$lim inf t → ∞ ∫ τ i ( t ) t p i (s)ds= β i >0,i=1,…,m.$

If

$∑ i = 1 m ( lim inf t → ∞ ∫ τ i ( t ) t p i ( s ) d s ) > 1 e ,$
(3.7)

then all solutions of equation (1.1) oscillate.

Observe that all the above mentioned oscillation conditions (3.1)-(3.7) involve lim inf only and coincide with condition ($C 2$) in the case of equation (1.2). It is obvious that there is a gap between conditions ($C 1$) and ($C 2$) when the limit $lim t → ∞ ∫ τ ( t ) t p(s)ds$ does not exist. Moreover, in view of Theorem 2.3, it is an interesting problem to investigate equation (1.1) with non-monotone arguments and derive sufficient oscillation conditions, involving lim sup (as condition ($C 1$) for equation (1.2) with one argument), which is the main objective in the following.

Theorem 3.3 ([32])

Assume that there exist non-decreasing functions $σ i ∈C([ t 0 ,+∞))$ such that

$τ i (t)≤ σ i (t)≤t(i=1,…,m),$
(3.8)

and

$lim sup t → + ∞ ∏ j = 1 m [ ∏ i = 1 m ∫ σ j ( t ) t p i ( s ) exp ( ∫ τ i ( s ) σ i ( t ) ∑ i = 1 m p i ( ξ ) × exp ( ∫ τ i ( ξ ) ξ ∑ i = 1 m p i ( u ) d u ) d ξ ) d s ] 1 m > 1 m m .$
(3.9)

Then all solutions of equation (1.1) oscillate.

When $m=1$, that is, in the case of equation (1.2) with one argument, from Theorem 3.3 we have the following corollary.

Corollary 3.1 ([32])

Let

$C:= lim sup t → + ∞ ∫ σ ( t ) t p(s)exp ( ∫ τ ( s ) σ ( t ) p ( ξ ) exp ( ∫ τ ( ξ ) ξ p ( u ) d u ) d ξ ) ds>1.$
(3.10)

Then all solutions of equation (1.2) oscillate.

In the case of monotone arguments we have the following.

Theorem 3.4 ([32])

Let $τ i$ be non-decreasing functions and

$lim sup t → + ∞ ∏ j = 1 m [ ∏ i = 1 m ∫ τ j ( t ) t p i ( s ) exp ( ∫ τ i ( s ) τ i ( t ) ∑ i = 1 m p i ( ξ ) × exp ( ∫ τ i ( ξ ) ξ ∑ i = 1 m p i ( u ) d u ) d ξ ) d s ] 1 m > 1 m m .$

Then all solutions of equation (1.1) oscillate.

Corollary 3.2 ([32])

Let $τ i$ be non-decreasing functions and

$lim sup t → + ∞ ∏ j = 1 m ( ∏ i = 1 m ∫ τ j ( t ) t p i ( s ) d s ) 1 m > 1 m m ,$
(3.11)

then all solutions of equation (1.1) oscillate.

Corollary 3.3 ([32])

Let $τ i$ be non-decreasing functions, $p i (t)≥p(t)$ ($i=1,…,m$) and

$lim sup t → + ∞ ∏ j = 1 m ∫ τ j ( t ) t p(s)ds> 1 m m ,$
(3.12)

then all solutions of equation (1.1) oscillate.

Corollary 3.4 ([32])

Let $τ i$ be non-decreasing functions, $p i (t)≥p=const$ and

$p m lim sup t → + ∞ ∏ i = 1 m ( t − τ i ( t ) ) > 1 m m ,$
(3.13)

then all solutions of equation (1.1) oscillate.

Remark 3.1 It should be pointed out that condition (3.9) of Theorem 3.3 presents for the first time sufficient conditions (in terms of lim sup) for the oscillation of all solutions to equation (1.1) with several non-monotone arguments. They are also independent and essentially improve all the related oscillation conditions in the literature. Even in the case where $m=1$, the improvement is substantial. Observe that (3.10) essentially improves (2.5).

Remark 3.2 Observe that when $m=1$, the above condition (3.11) [(3.12)] reduces to the (classical) condition ($C 1$).

The following examples illustrate the significance of our results.

Example 3.1 (cf. [7, 32])

We consider a generalization of an example presented in [7], where the equation

$x ′ (t)+ 1 e x ( τ ( t ) ) =0,t≥0,$

with the retarded argument

$τ(t):= { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , − 3 t + ( 12 n + 3 ) , t ∈ [ 3 n + 1 , 3 n + 2 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2 , 3 n + 3 ] ,$

was studied. Here we discuss the more general equation

$x ′ (t)+px ( τ ( t ) ) =0,t≥0,p>0,$
(3.14)

and illustrate how our methodology can be utilized to prove the existence of oscillatory solutions for some range of the parameter p. In this case, as in [7], one may choose the function

$σ(t)= { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , 3 n , t ∈ [ 3 n + 1 , 3 n + 2.6 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2.6 , 3 n + 3 ] .$

Now note that, since $τ(t)≤t−1$,

$∫ τ ( t ) t pdu≥ ∫ t − 1 t pdu=p.$

The choice as in [6] of $t n =3n+3$ gives

$C = lim sup t → + ∞ ∫ σ ( t ) t p exp ( ∫ τ ( s ) σ ( t ) p exp ( ∫ τ ( ξ ) ξ p d u ) d ξ ) d s ≥ lim n → + ∞ ∫ 3 n + 2 3 n + 3 p exp ( ∫ 5 s − ( 12 n + 13 ) 3 n + 2 p exp ( p ) d ξ ) d s = 1 5 ( e 5 p e p − 1 ) e − p .$

The inequality

$1 5 ( e 5 p e p − 1 ) e − p >1$

is satisfied for (the numbers that follow are rounded to the third decimal place unless exact) $p∈[0.303,0.358]$. Thus, for $p∈[0.303,0.358]$, condition (3.10) of Corollary 3.1 is satisfied, and therefore all solutions to the above equation (3.14) oscillate. Observe, however, that when $p∈[0.303,0.358]$ in (3.14), we find

$A = lim sup t → ∞ ∫ σ ( t ) t p d s = p ⋅ ( 2.6 ) < 1 , a : = lim inf t → ∞ ∫ τ ( t ) t p d s = p < 1 e$

and

$∫ σ ( 3 n + 3 ) 3 n + 3 p exp { ∫ τ ( s ) σ ( 3 n + 3 ) p d ξ } d s = ∫ 3 n + 2 3 n + 3 p exp { ∫ 5 s − ( 12 n + 13 ) 3 n + 2 p d ξ } d s = 1 5 ( e 5 p − 1 ) < 1 .$

That is, none of the known oscillation conditions ($C 1$), ($C 2$) (and also conditions (2.5), (3.1)-(3.6)) is satisfied.

Remark 3.3 ([32])

It is obvious that if for some $i 0 ∈{1,…,m}$ all solutions of the equation

$x ′ (t)+ p i 0 (t)x ( τ i 0 ( t ) ) =0$

oscillate, then all solutions of equation (1.1) also oscillate.

Example 3.2 ([32])

Let $p, Δ 1 , Δ 2 ∈(0,+∞)$ and consider the sequences ${ t k } k = 1 ∞$ such that $t k ↑+∞$ for $k↑+∞$, $t k +2Δ< t k + 1$ ($k=1,2,…$), where $Δ=max{ Δ i ,i=1,2}$. Choose p, $Δ 1$ and $Δ 2$ such that

$p 2 Δ 1 Δ 2 > 1 4$
(3.15)

and

$p Δ i <1(i=1,2).$
(3.16)

Let $p(t)=p$ for $t∈[ t k , t k +Δ]$ ($k=1,2,…$) and $p(t)=0$ for $t∈ R + ∖ ⋃ k = 1 ∞ [ t k , t k +Δ]$.

According to (3.15), it is obvious that condition (3.13) is fulfilled, where $m=2$ and $τ i (t)=t− Δ i$ ($i=1,2$) a.e., and therefore all solutions to equation (1.1) are oscillatory. However, for the equations

$x ′ (t)+p(t)x(t− Δ i )=0(i=1,2)$

by (3.16), we have

$lim sup t → + ∞ ∫ t − Δ i t p(s)ds<1(i=1,2)$

and

$lim inf t → + ∞ ∫ t − Δ i t p(s)ds=0(i=1,2).$

Remark 3.4 ([32])

In the above mentioned Example 3.2, by a solution we mean an absolutely continuous function which satisfies the corresponding equation almost everywhere.

Example 3.3 ([32])

Consider the equation

$x ′ (t)+ p 1 x ( τ 1 ( t ) ) + p 2 x ( τ 2 ( t ) ) =0,t≥0, p 1 , p 2 >0,$
(3.17)

where

$τ 1 ( t ) = { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , − 3 t + ( 12 n + 3 ) , t ∈ [ 3 n + 1 , 3 n + 2 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2 , 3 n + 3 ] , τ 2 ( t ) = { t − 2 , t ∈ [ 3 n , 3 n + 1 ] , − t + 6 n , t ∈ [ 3 n + 1 , 3 n + 2 ] , 3 t − ( 6 n + 8 ) , t ∈ [ 3 n + 2 , 3 n + 3 ] .$

We can take

$σ 1 ( t ) = { t − 1 , t ∈ [ 3 n , 3 n + 1 ] , 3 n , t ∈ [ 3 n + 1 , 3 n + 2.6 ] , 5 t − ( 12 n + 13 ) , t ∈ [ 3 n + 2.6 , 3 n + 3 ] , σ 2 ( t ) = { t − 2 , t ∈ [ 3 n , 3 n + 1 ] , 3 n − 1 , t ∈ [ 3 n + 1 , 3 n + 2 . 3 ¯ ] , 3 t − ( 6 n + 8 ) , t ∈ [ 3 n + 2 . 3 ¯ , 3 n + 3 ] .$

Note that, since $τ 1 (t)≤t−1$ and $τ 2 (t)≤t−2$, we have

$∫ τ 1 ( t ) t du≥ ∫ t − 1 t du=1, ∫ τ 2 ( t ) t du≥ ∫ t − 2 t du=2.$

Set $P= p 1 exp( p 1 + p 2 )+ p 2 exp(2 p 1 +2 p 2 )$. The choice of $t n =3n+3$ gives

$lim sup t → + ∞ ∏ j = 1 2 ( ∏ i = 1 2 ∫ σ j ( t ) t p i exp ( ∫ τ i ( s ) σ i ( t ) ∑ i = 1 2 p i exp ( ∫ τ i ( ξ ) ξ ( p 1 + p 2 ) d u ) d ξ ) d s ) 1 2 ≥ lim n → + ∞ ∏ j = 1 2 ( ∏ i = 1 2 ∫ σ j ( 3 n + 3 ) 3 n + 3 p i exp ( ∫ τ i ( s ) σ i ( 3 n + 3 ) ∑ i = 1 2 p i exp ( ∫ τ i ( ξ ) ξ ( p 1 + p 2 ) d u ) d ξ ) d s ) 1 2 ≥ lim n → + ∞ ∏ j = 1 2 ( ∫ σ j ( 3 n + 3 ) 3 n + 3 p 1 exp ( ∫ τ 1 ( s ) 3 n + 2 P d ξ ) d s ) 1 2 × ( ∫ σ j ( 3 n + 3 ) 3 n + 3 p 2 exp ( ∫ τ 2 ( s ) 3 n + 1 P d ξ ) d s ) 1 2 = lim n → + ∞ ( ∫ 3 n + 2 3 n + 3 p 1 exp ( ∫ τ 1 ( s ) 3 n + 2 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 2 3 n + 3 p 2 exp ( ∫ τ 2 ( s ) 3 n + 1 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 1 3 n + 3 p 1 exp ( ∫ τ 1 ( s ) 3 n + 2 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 1 3 n + 3 p 2 exp ( ∫ τ 2 ( s ) 3 n + 1 P d ξ ) d s ) 1 2 = lim n → + ∞ ( ∫ 3 n + 2 3 n + 3 p 1 exp ( ∫ 5 s − ( 12 n + 13 ) 3 n + 2 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 2 3 n + 3 p 2 exp ( ∫ 3 s − ( 6 n + 8 ) 3 n + 1 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 1 3 n + 2 p 1 exp ( ∫ − 3 s + ( 12 n + 3 ) 3 n + 2 P d ξ ) d s + ∫ 3 n + 2 3 n + 3 p 1 exp ( ∫ 5 s − ( 12 n + 13 ) 3 n + 2 P d ξ ) d s ) 1 2 × ( ∫ 3 n + 1 3 n + 2 p 2 exp ( ∫ − s + 6 n 3 n + 1 P d ξ ) d s + ∫ 3 n + 2 3 n + 3 p 2 exp ( ∫ 3 s − ( 6 n + 8 ) 3 n + 1 P d ξ ) d s ) 1 2 = : D ( p 1 , p 2 ) .$

Let $p 1 =0.1$. Then, by direct computation, we get

$D> 1 4$

if $p 2 ≥0.158$. That is, when $p 1 =0.1$ and $p 2 ≥0.158$ in equation (3.17), condition (3.9) of Theorem 3.3 is satisfied, and therefore all solutions to this equation oscillate.

Note that since the delays are not monotone, Theorem 3.2 cannot be applied to this example. We now compare our result with Theorem 3.1. Note that

The choice $p 1 =0.1$, $p 2 =0.158$ gives

$lim inf t → ∞ ∫ σ 1 ( t ) t ( p 1 + p 2 )ds= p 1 + p 2 =0.258< 1 e ,$

that is, condition (3.6) is not satisfied.

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Stavroulakis, I.P. A survey on the oscillation of differential equations with several deviating arguments. J Inequal Appl 2014, 399 (2014). https://doi.org/10.1186/1029-242X-2014-399