R E S E A R C H
Open Access
Oscillatory behavior of second-order
nonlinear neutral differential equations with
distributed deviating arguments
Tongxing Li
1*, Blanka Baculíková
2and Jozef Džurina
2Dedicated to Professor Ivan Kiguradze
*Correspondence:
1Qingdao Technological University,
Feixian, Shandong 273400, P.R. China
Full list of author information is available at the end of the article
Abstract
We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results.
MSC: 34C10; 34K11
Keywords: oscillation; neutral differential equation; second-order equation; distributed deviating argument
1 Introduction
This paper is concerned with oscillation of the second-order nonlinear functional differ-ential equation
r(t)z(t)α–z(t)+
b
a
q(t,ξ)xg(t,ξ)α–xg(t,ξ)dσ(ξ) = , (.)
wheret≥t> ,α≥ is a constant, andz:=x+p·x◦τ. Throughout, we assume that the following hypotheses hold:
(H) I:= [t,∞),r,p∈C(I,R),r(t) > , andp(t)≥;
(H) q∈C(I×[a,b], [,∞))andq(t,ξ)is not eventually zero on any[tμ,∞)×[a,b],tμ∈I;
(H) g∈C(I×[a,b], [,∞)),lim inft→∞g(t,ξ) =∞, andg(t,a)≤g(t,ξ)forξ∈[a,b];
(H) τ∈C(I,R),τ(t) > ,limt→∞τ(t) =∞, andg(τ(t),ξ) =τ[g(t,ξ)];
(H) σ ∈C([a,b],R)is nondecreasing and the integral of (.) is taken in the sense of Riemann-Stieltijes.
By a solution of (.), we mean a functionx∈C([tx,∞),R) for sometx≥t, which has the properties thatz∈C([t
x,∞),R),r|z|α–z∈C([tx,∞),R), and satisfies (.) on [tx,∞).
We restrict our attention to those solutionsxof (.) which exist on [tx,∞) and satisfy sup{|x(t)|:t≥T}> for anyT ≥tx. A solutionxof (.) is termed oscillatory if it is
neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (.) is said to be oscillatory if all its solutions oscillate.
As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed net-works containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see []. Hence, there has been much research activity concerning oscillatory and nonoscillatory behav-ior of solutions to different classes of neutral differential equations, we refer the reader to [–] and the references cited therein.
In the following, we present some background details that motivate our research. Re-cently, Baculíková and Lacková [], Džurina and Hudáková [], Liet al.[, ], and Sun et al. [] established some oscillation criteria for the second-order half-linear neutral differential equation
r(t)z(t)α–z(t)+q(t)xδ(t)α–xδ(t)= , wherez:=x+p·x◦τ,
≤p(t) < or p(t) > .
Baculíková and Džurina [, ] and Li et al. [] investigated oscillatory behavior of a
second-order neutral differential equation
r(t)x(t) +p(t)xτ(t)+q(t)xσ(t)= , where
≤p(t)≤p<∞ and τ(t)≥τ> . (.)
Ye and Xu [] and Yu and Fu [] considered oscillation of the second-order differential equation
x(t) +p(t)x(t–τ)+
b
a
q(t,ξ)xg(t,ξ)dσ(ξ) = .
Assuming ≤p(t) < , Thandapani and Piramanantham [], Wang [], Xu and Weng
[], and Zhao and Meng [] studied oscillation of an equation
r(t)x(t) +p(t)x(t–τ)+
b
a
q(t,ξ)fxg(t,ξ)dσ(ξ) = .
As yet, there are few results regarding the study of oscillatory properties of (.) under
the conditionsp(t)≥ orlimt→∞p(t) =∞. Thereinto, Li and Thandapani [] obtained
several oscillation results for (.) in the case where (.) holds,σ(ξ) =ξ, and
∞ t
dt
r/α(t)=∞. (.)
or
∞
t dt
r/α(t)<∞. (.)
All functional inequalities are assumed to hold eventually, that is, they are satisfied for all tlarge enough.
2 Main results
In what follows, we use the following notation for the convenience of the reader:
Q(t,ξ) :=minq(t,ξ),qτ(t),ξ , d+(t) :=max
,d(t) ,
φ(t) :=αp
[h(t)]h(t)
p[h(t)] –
τ(t)
τ(t), ζ(t) :=
ρ+(t)
ρ(t) +φ(t),
ϕ(t) :=
ρ+(t)
ρ(t)
α+
+p
α[h(t)](ζ
+(t))α+
τ(t) , and δ(t) :=
∞
η(t) ds r/α(s),
whereh,ρ, andηwill be specified later.
Theorem . Assume(H)-(H), (.),and let g(t,a)∈C(I,R),g(t,a) > ,g(t,a)≤t,and g(t,a)≤τ(t)for t∈I.Suppose further that there exists a real-valued function h∈C(I,R) such that p[g(t,ξ)]≤p[h(t)]for t∈Iandξ ∈[a,b].If there exists a real-valued function
ρ∈C(I, (,∞))such that
lim sup t→∞
t
t
ρ(s)
b
a Q(s,ξ) dσ(ξ)
α– –
r[g(s,a)]ϕ(s) (α+ )α+(g(s,a))α
ds=∞, (.)
then(.)is oscillatory.
Proof Letxbe a nonoscillatory solution of (.). Without loss of generality, we assume that there exists at∈Isuch thatx(t) > ,x[τ(t)] > , andx[g(t,ξ)] > for allt≥tand
ξ∈[a,b]. Thenz(t) > . Applying (.), one has, for all sufficiently larget,
r(t)z(t)α–z(t)+
b
a
q(t,ξ)xαg(t,ξ)dσ(ξ)
+
b
a
qτ(t),ξpαh(t)xαgτ(t),ξdσ(ξ)
+p
α[h(t)] τ(t)
rτ(t)zτ(t)α–zτ(t)= .
Using the inequality (see [, Lemma ])
the definition ofz,g(τ(t),ξ) =τ[g(t,ξ)], andp[g(t,ξ)]≤p[h(t)], we conclude that
r(t)z(t)α–z(t)+ α–
b
a
Q(t,ξ)zαg(t,ξ)dσ(ξ)
+p
α[h(t)] τ(t)
rτ(t)zτ(t)α–zτ(t)≤. (.)
By virtue of (.), we get
r(t)z(t)α–z(t)≤, t≥t. (.)
Thus,r|z|α–zis nonincreasing. Now we have two possible cases for the sign ofz: (i)z<
eventually, or (ii)z> eventually.
(i) Assume thatz(t) < fort≥t≥t. Then we have by (.) r(t)z(t)α–z(t)≤r(t)z(t)
α–
z(t) < , t≥t, which yields
z(t)≤z(t) –r/α(t)z(t)
t
t
r–/α(s) ds.
Then we obtainlimt→∞z(t) = –∞due to (.), which is a contradiction.
(ii) Assume thatz(t) > fort≥t≥t. It follows from (.) andg(t,ξ)≥g(t,a) that
r(t)z(t)α+p
α[h(t)] τ(t)
rτ(t)zτ(t)α
+
α–z
α
g(t,a)
b
a
Q(t,ξ) dσ(ξ)≤. (.)
We define a Riccati substitution
ω(t) :=ρ(t)r(t)(z
(t))α
(z[g(t,a)])α, t≥t. (.)
Thenω(t) > . From (.) andg(t,a)≤t, we have
zg(t,a)≥r(t)/rg(t,a)/αz(t). (.)
Differentiating (.), we get
ω(t) =ρ(t)r(t)(z
(t))α
(z[g(t,a)])α +ρ(t)
(r(t)(z(t))α)
(z[g(t,a)])α
–αρ(t)r(t)(z
(t))αzα–[g(t,a)]z[g(t,a)]g(t,a)
(z[g(t,a)])α . (.)
Therefore, by (.), (.), and (.), we see that
ω(t)≤ρ
(t)
ρ(t)ω(t) +ρ(t)
(r(t)(z(t))α)
(z[g(t,a)])α –
αg(t,a)
ρ/α(t)r/α[g(t,a)]ω
Similarly, we introduce another Riccati transformation:
υ(t) :=ρ(t)r[τ(t)](z
[τ(t)])α
(z[g(t,a)])α , t≥t. (.)
Thenυ(t) > . From (.) andg(t,a)≤τ(t), we obtain
zg(t,a)≥rτ(t)/rg(t,a)/αzτ(t). (.)
Differentiating (.), we have
υ(t) =ρ(t)r[τ(t)](z
[τ(t)])α
(z[g(t,a)])α +ρ(t)
(r[τ(t)](z[τ(t)])α)
(z[g(t,a)])α
–αρ(t)r[τ(t)](z
[τ(t)])αzα–[g(t,a)]z[g(t,a)]g(t,a)
(z[g(t,a)])α . (.)
Therefore, by (.), (.), and (.), we find
υ(t)≤ρ
(t)
ρ(t)υ(t) +ρ(t)
(r[τ(t)](z[τ(t)])α)
(z[g(t,a)])α –
αg(t,a)
ρ/α(t)r/α[g(t,a)]υ
(α+)/α
(t). (.)
Combining (.) and (.), we get
ω(t) +p
α[h(t)] τ(t) υ
(t)
≤ρ(t)(r(t)(z
(t))α)+pα[h(t)]
τ(t) (r[τ(t)](z[τ(t)])
α)
(z[g(t,a)])α +
ρ(t)
ρ(t)ω(t)
– αg
(t,a)
ρ/α(t)r/α[g(t,a)]ω
(α+)/α(t) +p α[h(t)]
τ(t)
ρ(t)
ρ(t)υ(t)
–p
α[h(t)] τ(t)
αg(t,a)
ρ/α(t)r/α[g(t,a)]υ
(α+)/α
(t).
It follows from (.) that
ω(t) +p
α[h(t)] τ(t) υ
(t)≤–ρ(t)
α–
b
a
Q(t,ξ) dσ(ξ) +ρ
+(t)
ρ(t)ω(t)
– αg
(t,a)
ρ/α(t)r/α[g(t,a)]ω
(α+)/α
(t) +p
α[h(t)] τ(t)
ρ+(t)
ρ(t)υ(t)
–p
α[h(t)] τ(t)
αg(t,a)
ρ/α(t)r/α[g(t,a)]υ
(α+)/α
(t).
Integrating the latter inequality fromttot, we obtain
ω(t) –ω(t) +
pα[h(t)] τ(t) υ(t) –
pα[h(t
)]
τ(t)
υ(t)
≤–
t
t
ρ(s) α–
b
a
+
t
t
ρ+(s)
ρ(s)ω(s) –
αg(s,a)
ρ/α(s)r/α[g(s,a)]ω
(α+)/α
(s)
ds
+
t
t
pα[h(s)] τ(s)
ρ+(s)
ρ(s) +φ(s)
+
υ(s) – αg
(s,a)
ρ/α(s)r/α[g(s,a)]υ
(α+)/α(s)
ds. (.)
Define
A:=
αg(t,a)
ρ/α(t)r/α[g(t,a)] α/(α+)
ω(t) and
B:=
α α+
ρ+(t)
ρ(t)
αg(t,a)
ρ/α(t)r/α[g(t,a)]
–α/(α+)α .
Using the inequality
α+
α AB
/α–A(α+)/α≤ αB
(α+)/α, forA≥ andB≥, (.)
we get
ρ+(t)
ρ(t)ω(t) –
αg(t,a)
ρ/α(t)r/α[g(t,a)]ω
(α+)/α
(t)≤
(α+ )α+
r[g(t,a)](ρ+(t))α+ (ρ(t)g(t,a))α .
On the other hand, define
A:=
αg(t,a)
ρ/α(t)r/α[g(t,a)] α/(α+)
υ(t) and
B:=
α α+ ζ+(t)
αg(t,a)
ρ/α(t)r/α[g(t,a)]
–α/(α+)α .
Then we have by (.)
ζ+(t)υ(t) –
αg(t,a)
ρ/α(t)r/α[g(t,a)]υ
(α+)/α
(t)≤
(α+ )α+
r[g(t,a)](ζ+(t))α+ρ(t)
(g(t,a))α .
Thus, from (.), we get
ω(t) –ω(t) +
pα[h(t)] τ(t) υ(t) –
pα[h(t
)]
τ(t)
υ(t)
≤–
t
t
ρ(s)
b
a Q(s,ξ) dσ(ξ)
α– –
r[g(s,a)] (α+ )α+(g(s,a))α ×
ρ+(s)
ρ(s)
α+
+p
α[h(s)](ζ
+(s))α+
τ(s)
ds,
which contradicts (.). This completes the proof.
Assuming (.), wherepandτare constants, we obtain the following result.
that
lim sup t→∞
t
t
ρ
(s)abQ(s,ξ) dσ(ξ)
α– –
+pα
τ (α+ )α+
r[g(s,a)](ρ+(s))α+ (ρ(s)g(s,a))α
ds=∞, (.)
then(.)is oscillatory.
Proof As above, letxbe an eventually positive solution of (.). Proceeding as in the proof of Theorem ., we have z(t) > , (.), and (.) for all sufficiently large t. Using (.), (.), and (.), we obtain
r(t)z(t)α+p
α
τ
rτ(t)zτ(t)α
+
α–z
αg(t,a) b
a
Q(s,ξ) dσ(ξ)≤. (.)
The remainder of the proof is similar to that of Theorem ., and hence it is omitted.
Theorem . Suppose we have(H)-(H), (.),and letτ(t)≤t and g(t,a)≥τ(t)for t∈I. Assume also that there exists a real-valued function h∈C(I,R)such that p[g(t,ξ)]≤ p[h(t)]for t∈Iandξ∈[a,b].If there exists a real-valued functionρ∈C(I, (,∞))such that
lim sup t→∞
t
t
ρ(s)
b
a Q(s,ξ) dσ(ξ)
α– –
r[τ(s)]ϕ(s) (α+ )α+(τ(s))α
ds=∞, (.)
then(.)is oscillatory.
Proof Letxbe a nonoscillatory solution of (.). Without loss of generality, we assume that there exists at∈Isuch thatx(t) > ,x[τ(t)] > , andx[g(t,ξ)] > for allt≥tandξ ∈ [a,b]. As in the proof of Theorem ., we obtain (.) and (.). In view of (.),r|z|α–z is nonincreasing. Now we have two possible cases for the sign ofz: (i)z< eventually, or (ii)z> eventually.
(i) Suppose thatz(t) < fort≥t≥t. Then, with a proof similar to the proof of case (i) in Theorem ., we obtain a contradiction.
(ii) Suppose thatz(t) > fort≥t≥t. We define a Riccati substitution
ω(t) :=ρ(t)r(t)(z
(t))α
(z[τ(t)])α , t≥t. (.)
Thenω(t) > . From (.) andτ(t)≤t, we have
zτ(t)≥r(t)/rτ(t)/αz(t). (.)
Differentiating (.), we obtain
ω(t) =ρ(t)r(t)(z
(t))α
(z[τ(t)])α +ρ(t)
(r(t)(z(t))α)
(z[τ(t)])α
–αρ(t)r(t)(z
(t))αzα–[τ(t)]z[τ(t)]τ(t)
Therefore, by (.), (.), and (.), we see that
ω(t)≤ρ
(t)
ρ(t)ω(t) +ρ(t)
(r(t)(z(t))α)
(z[τ(t)])α –
ατ(t)
ρ/α(t)r/α[τ(t)]ω
(α+)/α
(t). (.)
Similarly, we introduce another Riccati substitution:
υ(t) :=ρ(t)r[τ(t)](z
[τ(t)])α
(z[τ(t)])α , t≥t. (.)
Thenυ(t) > . Differentiating (.), we have
υ(t) =ρ(t)r[τ(t)](z
[τ(t)])α
(z[τ(t)])α +ρ(t)
(r[τ(t)](z[τ(t)])α)
(z[τ(t)])α
–αρ(t)r[τ(t)](z
[τ(t)])αzα–[τ(t)]z[τ(t)]τ(t)
(z[τ(t)])α . (.)
Therefore, by (.) and (.), we get
υ(t) =ρ
(t)
ρ(t)υ(t) +ρ(t)
(r[τ(t)](z[τ(t)])α)
(z[τ(t)])α –
ατ(t)
ρ/α(t)r/α[τ(t)]υ
(α+)/α
(t). (.)
Combining (.) and (.), we have
ω(t) +p
α[h(t)] τ(t) υ
(t)≤ρ(t)(r(t)(z(t))α)+
pα[h(t)]
τ(t) (r[τ(t)](z[τ(t)])
α)
(z[τ(t)])α +
ρ(t)
ρ(t)ω(t)
– ατ
(t)
ρ/α(t)r/α[τ(t)]ω
(α+)/α(t) +p α[h(t)]
τ(t)
ρ(t)
ρ(t)υ(t)
–p
α[h(t)] τ(t)
ατ(t)
ρ/α(t)r/α[τ(t)]υ
(α+)/α
(t).
It follows from (.) andg(t,a)≥τ(t) that
ω(t) +p
α[h(t)] τ(t) υ
(t)≤–ρ(t)
α–
b
a
Q(t,ξ) dσ(ξ) +ρ
+(t)
ρ(t)ω(t)
– ατ
(t)
ρ/α(t)r/α[τ(t)]ω
(α+)/α
(t) +p
α[h(t)] τ(t)
ρ+(t)
ρ(t)υ(t)
–p
α[h(t)] τ(t)
ατ(t)
ρ/α(t)r/α[τ(t)]υ
(α+)/α
(t).
Integrating the latter inequality fromttot, we obtain
ω(t) –ω(t) +
pα[h(t)] τ(t) υ(t) –
pα[h(t
)]
τ(t)
υ(t)
≤–
t
t
ρ(s) α–
b
a
Q(s,ξ) dσ(ξ) ds+
t
t
ρ+(s)
ρ(s)ω(s) –
ατ(s)
ρ/α(s)r/α[τ(s)]ω
(α+)/α(s)
ds
+
t
t
pα[h(s)] τ(s)
ρ+(s)
ρ(s) +φ(s)
+
υ(s) – ατ
(s)
ρ/α(s)r/α[τ(s)]υ
(α+)/α
(s)
Define
A:=
ατ(t)
ρ/α(t)r/α[τ(t)] α/(α+)
ω(t) and
B:=
α α+
ρ+(t)
ρ(t)
ατ(t)
ρ/α(t)r/α[τ(t)]
–α/(α+)α .
Using inequality (.), we have
ρ+(t)
ρ(t)ω(t) –
ατ(t)
ρ/α(t)r/α[τ(t)]ω
(α+)/α(t)≤
(α+ )α+
r[τ(t)](ρ+(t))α+ (ρ(t)τ(t))α .
On the other hand, define
A:=
ατ(t)
ρ/α(t)r/α[τ(t)] α/(α+)
υ(t) and
B:=
α α+ ζ+(t)
ατ(t)
ρ/α(t)r/α[τ(t)]
–α/(α+)α .
Then, by (.), we obtain
ζ+(t)υ(t) –
ατ(t)
ρ/α(t)r/α[τ(t)]υ
(α+)/α
(t)≤
(α+ )α+
r[τ(t)](ζ+(t))α+ρ(t)
(τ(t))α .
Thus, from (.), we get
ω(t) –ω(t) +
pα[h(t)] τ(t) υ(t) –
pα[h(t
)]
τ(t)
υ(t)
≤–
t
t
ρ(s)
b
a Q(s,ξ) dσ(ξ)
α– –
r[τ(s)] (α+ )α+(τ(s))α ×
ρ+(s)
ρ(s)
α+
+p
α[h(s)](ζ
+(s))α+
τ(s)
ds,
which contradicts (.). This completes the proof.
Assuming we have (.), wherepandτare constants, we get the following result.
Theorem . Suppose we have(H)-(H), (.), (.),and letτ(t)≤t and g(t,a)≥τ(t)for t∈I.If there exists a real-valued functionρ∈C(I, (,∞))such that
lim sup t→∞
t
t
ρ
(s)abQ(s,ξ) dσ(ξ)
α– –
(α+ )α+
+p
α
τ
r[τ(s)](ρ+(s))α+ (τρ(s))α
ds=∞,
(.)
then(.)is oscillatory.
(.), and (.), we have (.) for all sufficiently larget. The rest of the proof is similar to
that of Theorem ., and so it is omitted.
In the following, we present some oscillation criteria for (.) in the case where (.) holds.
Theorem . Suppose we have(H)-(H), (.), (.),and let g(t,a)∈C(I,R),g(t,a) > , g(t,a)≤τ(t)≤t for t∈I,and g(t,ξ)≤g(t,b)forξ∈[a,b].Assume further that there exists a real-valued functionρ∈C(I, (,∞))such that(.)is satisfied.If there exists a real-valued functionη∈C(I,R)such thatη(t)≥t,η(t)≥g(t,b),η(t) > for t∈I,and
lim sup t→∞
t
t
b
aQ(s,ξ) dσ(ξ)
α– δ
α
(s) –
+p
α
τ
α α+
α+ η(s)
δ(s)r/α[η(s)]
ds=∞,
(.)
then(.)is oscillatory.
Proof Letxbe a nonoscillatory solution of (.). Without loss of generality, we assume that there exists at∈Isuch thatx(t) > ,x[τ(t)] > , andx[g(t,ξ)] > for allt≥tand
ξ∈[a,b]. Thenz(t) > . As in the proof of Theorem ., we get (.). By virtue of (.), we have (.). Thus,r|z|α–zis nonincreasing. Now we have two possible cases for the sign ofz: (i)z< eventually, or (ii)z> eventually.
(i) Suppose thatz(t) > fort≥t≥t. Then, by the proof of Theorem ., we obtain a contradiction to (.).
(ii) Suppose thatz(t) < fort≥t≥t. It follows from (.), (.), andg(t,ξ)≤g(t,b) that
–r(t)–z(t)α+p
α
τ
–rτ(t)–zτ(t)α
+
α–z
α
g(t,b)
b
a
Q(t,ξ) dσ(ξ)≤. (.)
We define the functionuby
u(t) := –r(t)(–z
(t))α
zα[η(t)] , t≥t. (.)
Thenu(t) < . Noting thatr(–z)αis nondecreasing, we get
z(s)≤r /α(t)
r/α(s)z
(t), s≥t≥t
.
Integrating this inequality fromη(t) tol, we obtain
z(l)≤zη(t)+r/α(t)z(t)
l
η(t) ds r/α(s).
Lettingl→ ∞, we have
That is,
–δ(t)r
/α(t)z(t)
z[η(t)] ≤.
Thus, we get by (.)
–δα(t)u(t)≤. (.)
Similarly, we define another functionvby
v(t) := –r[τ(t)](–z
[τ(t)])α
zα[η(t)] , t≥t. (.)
Thenv(t) < . Noting thatr(–z)αis nondecreasing andτ(t)≤t, we get
r(t)–z(t)α≥rτ(t)–zτ(t)α.
Thus, < –v(t)≤–u(t). Hence, by (.), we see that
–δα(t)v(t)≤. (.)
Differentiating (.), we obtain
u(t) =(–r(t)(–z
(t))α)zα[η(t)] +αr(t)(–z(t))αzα–[η(t)]z[η(t)]η(t)
zα[η(t)] .
By (.) andη(t)≥t, we havez[η(t)]≤(r(t)/r[η(t)])/αz(t), and so
u(t)≤(–r(t)(–z
(t))α)
zα[η(t)] –α η(t) r/α[η(t)]
–u(t)(α+)/α. (.)
Similarly, we see that
v(t)≤(–r[τ(t)](–z
[τ(t)])α)
zα[η(t)] –α η(t) r/α[η(t)]
–v(t)(α+)/α. (.)
Combining (.) and (.), we get
u(t) +p
α
τ
v(t)≤ (–r(t)(–z
(t))α)
zα[η(t)] +
pα
τ
(–r[τ(t)](–z[τ(t)])α)
zα[η(t)]
–α η
(t)
r/α[η(t)]
–u(t)(α+)/α–αp
α
τ
η(t) r/α[η(t)]
–v(t)(α+)/α. (.)
Using (.), (.), andg(t,b)≤η(t), we obtain
u(t) +p
α
τ
v(t)≤–
b
a Q(t,ξ) dσ(ξ)
α– –α
η(t) r/α[η(t)]
–u(t)(α+)/α
–αp
α
τ
η(t) r/α[η(t)]
Multiplying (.) byδα(t) and integrating the resulting inequality fromt
tot, we have
u(t)δα(t) –u(t)δα(t) +α
t
t
δα–(s)η(s)u(s)
r/α[η(s)] ds+α t
t
η(s)δα(s)
r/α[η(s)]
–u(s)(α+)/αds
+p
α
τ
v(t)δα(t) – p
α
τ
v(t)δα(t) +
αpα
τ
t
t
δα–(s)η(s)v(s) r/α[η(s)] ds
+αp
α
τ
t
t
η(s)δα(s)
r/α[η(s)]
–v(s)(α+)/αds+
t
t
b
a Q(s,ξ) dσ(ξ)
α– δ
α
(s) ds≤.
Set
A:= –
η(t)δα(t)
r/α[η(t)] (α+)/α
u(t) and
B:=
α α+
δα–(t)η(t) r/α[η(t)]
η(t)δα(t)
r/α[η(t)]
–α/(α+)α .
Using inequality (.), we get
δα–(t)η(t)u(t) r/α[η(t)] +
η(t)δα(t)
r/α[η(t)]
–u(t)(α+)/α≥–
α
α α+
α+ η(t)
δ(t)r/α[η(t)].
Similarly, we set
A:= –
η(t)δα(t)
r/α[η(t)] (α+)/α
v(t) and
B:=
α α+
δα–(t)η(t) r/α[η(t)]
η(t)δα(t)
r/α[η(t)]
–α/(α+)α .
Then we have by (.)
δα–(t)η(t)v(t) r/α[η(t)] +
η(t)δα(t)
r/α[η(t)]
–v(t)(α+)/α≥–
α
α α+
α+ η(t)
δ(t)r/α[η(t)].
Thus, from (.) and (.), we find
t
t
b
a Q(s,ξ) dσ(ξ)
α– δ
α
(s) –
+p
α
τ
α α+
α+ η(s)
δ(s)r/α[η(s)]
ds
≤u(t)δα(t) + pα
τ
v(t)δα(t) + + pα
τ ,
which contradicts (.). This completes the proof.
With a proof similar to the proof of Theorems . and ., we obtain the following result.
Theorem . Suppose we have(H)-(H), (.), (.),and letτ(t)≤t,g(t,a)≥τ(t)for t∈I,and g(t,ξ)≤g(t,b)forξ ∈[a,b].Assume also that there exists a real-valued func-tion ρ∈C(I, (,∞))such that (.)is satisfied.If there exists a real-valued function
η∈C(I,R)such thatη(t)≥t,η(t)≥g(t,b),η(t) > for t∈I,and(.)holds,then(.)
3 Applications and discussion
In this section, we provide three examples to illustrate the main results.
Example . Consider the second-order neutral functional differential equation
x(t) +x(t– π)+
π
–π
x[t+ξ] dξ= , t≥. (.)
Letα= ,a= –π/,b=π/,r(t) = ,p(t) = ,τ(t) =t– π,q(t,ξ) = ,g(t,ξ) =t+ξ,σ(ξ) =
ξ, andρ(t) = . ThenQ(t,ξ) =min{q(t,ξ),q(τ(t),ξ)}= ,g(t,a) = ,g(t,a) =t– π/≤t+ξ
forξ∈[–π/,π/], andg(t,a)≤τ(t)≤t. Moreover, lettingτ= , then
lim sup t→∞
t
t
ρ(s)abQ(s,ξ) dσ(ξ)
α– –
(α+ )α+
+p
α
τ
r[g(s,a)](ρ+(s))α+ (ρ(s)g(s,a))α
ds
= πlim sup t→∞
t
ds=∞.
Hence, by Theorem ., (.) is oscillatory. As a matter of fact, one such solution isx(t) =
sint.
Example . Consider the second-order neutral functional differential equation
x(t) +tx(t–β)+
ξ+
t x[t+ξ] dξ= , t≥, (.)
whereβ≥ is a constant. Letα= ,a= ,b= ,r(t) = ,p(t) =t,τ(t) =t–β,q(t,ξ) = (ξ+ )/t,g(t,ξ) =t+ξ,σ(ξ) =ξ, andρ(t) = . ThenQ(t,ξ) =min{q(t,ξ),q(τ(t),ξ)}= (ξ+ )/t, g(t,a) =g(t, ) =t≤t+ξforξ∈[, ],τ(t) =t–β≤t, andg(t,a)≥τ(t) fort≥. Further, settingh(t) =t+ ,
φ(t) = αp
[h(t)]h(t)
p[h(t)] –
τ(t)
τ(t) = t+ ,
ζ(t) =ρ
+(t)
ρ(t) +φ(t) = t+ ,
and
ϕ(t) =
ρ+(t)
ρ(t)
α+
+p
α[h(t)](ζ
+(t))α+
τ(t) = t+ .
Therefore, we have
lim sup t→∞
t
t
ρ(s)
b
a Q(s,ξ) dσ(ξ)
α– –
r[τ(s)]ϕ(s) (α+ )α+(τ(s))α
ds
=lim sup t→∞
t
ξ+ s dξ–
(s+ )
ds=lim sup t→∞
t
s–
(s+ )
ds=∞.
Example . Consider the second-order neutral functional differential equation
tx(t) +p(t)x(t–β)+
(ξ+ )x[t+ξ] dξ= , t≥, (.)
where ≤p(t)≤p,pandβ are positive constants. Letα = ,a= , b= ,r(t) =t,
τ(t) =t–β,q(t,ξ) =ξ+ ,g(t,ξ) =t+ξ,σ(ξ) =ξ,ρ(t) = , andη(t) =t+ . ThenQ(t,ξ) =
min{q(t,ξ),q(τ(t),ξ)}=ξ+ ,τ= ,g(t,a) =g(t, ) =t≤t+ξforξ∈[, ],τ(t) =t–β≤t, g(t,a)≥τ(t) fort≥, andδ(t) = /t. Further,
lim sup t→∞
t
t
ρ
(s)abQ(s,ξ) dσ(ξ)
α– –
(α+ )α+
+p
α
τ
r[τ(s)](ρ+(s))α+ (τρ(s))α
ds
=
lim supt→∞
t
ds=∞
and
lim sup t→∞
t
t
b
aQ(s,ξ) dσ(ξ)
α– δ
α
(s) –
+p
α
τ
α α+
α+
η(s)
δ(s)r/α[η(s)]
ds
=
–
+p
lim sup
t→∞
t
ds
s+ =∞, ifp< .
Hence, by Theorem ., (.) is oscillatory when ≤p(t)≤p< .
Remark . In this paper, we establish some new oscillation theorems for (.) in the
case wherep is finite or infinite onI. The criteria obtained extend the results in []
and improve those reported in []. Similar results can be presented under the as-sumption that <α≤. In this case, using [, Lemma ], one has to replaceQ(t,ξ) :=
min{q(t,ξ),q(τ(t),ξ)} with Q(t,ξ) := α–min{q(t,ξ),q(τ(t),ξ)} and proceed as above. It would be interesting to find another method to investigate (.) in the case where g(τ(t),ξ) ≡τ[g(t,ξ)].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. They all read and approved the final version of the manuscript.
Author details
1Qingdao Technological University, Feixian, Shandong 273400, P.R. China.2Department of Mathematics, Faculty of
Electrical Engineering and Informatics, Technical University of Košice, Letná 9, Košice, 042 00, Slovakia.
Acknowledgements
The authors express their sincere gratitude to the 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.
Received: 16 January 2014 Accepted: 6 March 2014 Published:24 Mar 2014
References
1. Hale, JK: Theory of Functional Differential Equations. Springer, New York (1977)
2. Agarwal, RP, Bohner, M, Li, W-T: Nonoscillation and Oscillation Theory for Functional Differential Equations. Dekker, New York (2004)
4. Baculíková, B, Džurina, J: Oscillation theorems for second order neutral differential equations. Comput. Math. Appl.
61, 94-99 (2011)
5. Baculíková, B, Džurina, J: Oscillation theorems for second order nonlinear neutral differential equations. Comput. Math. Appl.62, 4472-4478 (2011)
6. Baculíková, B, Lacková, D: Oscillation criteria for second order retarded differential equations. Stud. Univ. Zilina Math. Ser.20, 11-18 (2006)
7. Baculíková, B, Li, T, Džurina, J: Oscillation theorems for second-order superlinear neutral differential equations. Math. Slovaca63, 123-134 (2013)
8. Candan, T: The existence of nonoscillatory solutions of higher order nonlinear neutral equations. Appl. Math. Lett.25, 412-416 (2012)
9. Candan, T, Dahiya, RS: Existence of nonoscillatory solutions of higher order neutral differential equations with distributed deviating arguments. Math. Slovaca63, 183-190 (2013)
10. Candan, T, Karpuz, B, Öcalan, Ö: Oscillation of neutral differential equations with distributed deviating arguments. Nonlinear Oscil.15, 65-76 (2012)
11. Dix, JG, Karpuz, B, Rath, R: Necessary and sufficient conditions for the oscillation of differential equations involving distributed arguments. Electron. J. Qual. Theory Differ. Equ.2011, 1-15 (2011)
12. Džurina, J, Hudáková, D: Oscillation of second order neutral delay differential equations. Math. Bohem.134, 31-38 (2009)
13. Hasanbulli, M, Rogovchenko, YuV: Oscillation criteria for second order nonlinear neutral differential equations. Appl. Math. Comput.215, 4392-4399 (2010)
14. Karpuz, B, Öcalan, Ö, Öztürk, S: Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations. Glasg. Math. J.52, 107-114 (2010)
15. Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral differential equations. J. Indian Math. Soc.79, 97-106 (2012)
16. Li, T, Han, Z, Zhang, C, Li, H: Oscillation criteria for second-order superlinear neutral differential equations. Abstr. Appl. Anal.2011, 367541 (2011)
17. Li, T, Rogovchenko, YuV, Zhang, C: Oscillation of second-order neutral differential equations. Funkc. Ekvacioj56, 111-120 (2013)
18. Li, T, Sun, S, Han, Z, Han, B, Sun, Y: Oscillation results for second-order quasi-linear neutral delay differential equations. Hacet. J. Math. Stat.42, 131-138 (2013)
19. Li, T, Thandapani, E: Oscillation of second-order quasi-linear neutral functional dynamic equations with distributed deviating arguments. J. Nonlinear Sci. Appl.4, 180-192 (2011)
20. Li, WN: Oscillation of higher order delay differential equations of neutral type. Georgian Math. J.7, 347-353 (2000) 21. ¸Senel, MT, Candan, T: Oscillation of second order nonlinear neutral differential equation. J. Comput. Anal. Appl.14,
1112-1117 (2012)
22. Sun, S, Li, T, Han, Z, Li, H: Oscillation theorems for second-order quasilinear neutral functional differential equations. Abstr. Appl. Anal.2012, 819342 (2012)
23. Thandapani, E, Piramanantham, V: Oscillation criteria of second order neutral delay dynamic equations with distributed deviating arguments. Electron. J. Qual. Theory Differ. Equ.2010, 1-15 (2010)
24. Wang, P: Oscillation criteria for second order neutral equations with distributed deviating arguments. Comput. Math. Appl.47, 1935-1946 (2004)
25. Xu, Z, Weng, P: Oscillation of second-order neutral equations with distributed deviating arguments. J. Comput. Appl. Math.202, 460-477 (2007)
26. Ye, L, Xu, Z: Interval oscillation of second order neutral equations with distributed deviating argument. Adv. Dyn. Syst. Appl.1, 219-233 (2006)
27. Yu, Y, Fu, X: Oscillation of second order neutral equation with continuous distributed deviating argument. Rad. Mat.7, 167-176 (1991)
28. Zhang, C, Agarwal, RP, Bohner, M, Li, T: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. (2014, in press)
29. Zhang, C, Baculíková, B, Džurina, J, Li, T: Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments. Math. Slovaca (2014, in press)
30. Zhao, J, Meng, F: Oscillation criteria for second-order neutral equations with distributed deviating argument. Appl. Math. Comput.206, 485-493 (2008)
10.1186/1687-2770-2014-68