New Oscillation Criteria for Second Order Half-Linear Neutral Type Dynamic Equations on Time Scales ()
1. Introduction
In this paper, we are concerned with the oscillatory behavior of solutions of second-order half-linear neutral type dynamic equation with distributed deviating arguments of the form
(r(t)ψ(x(t))|(x(t)+p(t)x(t−τ))Δ|α−1(x(t)+p(t)x(t−τ))Δ)Δ +∫baq(t,ξ)f(x(g(t,ξ)))Δξ=0 (1.1)
where
τ is nonnegative integers. By a solution of (1.1), we mean a nontrivial real-valued function
x∈C1rd[Tx,∞),Tx≥t0 which satisfies Equation (1.1) on
[Tx,∞), where
Crd is the space of rd-continuous functions. A solution
x(t) of Equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative and non-oscillatory otherwise. Equation (1.1) is called oscillatory if all its solutions are oscillatory. Throughout this paper, we will assume the following hypotheses:
(A1)
p(t) is positive,
0≤p(t)≤p≤+∞, where p is a constant;
(A2)
r(t)>0,
ψ:ℝ→(0,∞) ;
(A3)
q∈Crd([0,∞)T×[a,b],[0,∞)) and
g∈Crd([0,∞)T×[a,b],[0,∞)) satisfies
t≥g(t,ξ) for
ξ∈[a,b] and
limt→∞ming(t,ξ)=∞ ;
(A4)
f∈C(ℝ,ℝ) such that
xf(x)>0 for
x≠0 and
f(u)/uα≥K>0.
A time scale
T is an arbitrary nonempty closed subset of the real numbers. For any
t∈T, we define the forward and backward jump operators by
σ(t):=inf{s∈T:s>t}, ρ(t):=sup{s∈T:s<t},
respectively. The graininess function
μ:T→[t0,∞) is defined by
μ:=σ(t)−t.
If
f:T→ℝ is Δ-differentiable at
t∈T, then f is continuous at t. Furthermore, we assume that
g:T→R is Δ-differentiable. The following formulas are useful:
f(σ(t))=f(t)+μ(t)fΔ(t) ;
(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)) ;
(fg)Δ(t)=fΔ(t)g(t)+f(t)gΔ(t)g(t)g(σ(t)) ;
∫bafΔ(t)Δt=F(b)−F(a).
where
a,b∈T. If
T=ℝ, we have
σ(t)=t, μ(t)=0, fΔ(t)=f′(t), ∫bafΔ(t)Δt=∫bafΔ(t)dt,
and (1.1) becomes the second-order half-linear differential equation with distributed deviating arguments:
(r(t)ψ(x(t))|(x(t)+p(t)x(t−τ))′|α−1(x(t)+p(t)x(t−τ))′)′ +b∫aq(s,ξ)f(x(g(s,ξ)))dξ=0. (1.2)
If
T=ℕ, we have
σ(t)=t+1, μ(t)=1, fΔ(t)=Δf(t), ∫bafΔ(t)Δt=b−1∑t=afΔ(t),
and (1.1) becomes the second-order half-linear difference equation with distributed deviating arguments:
Δ(anψ(xn)|Δ(xn+pnxn−τ)|α−1Δ(xn+pnxn−τ)) +b∑ξ=aq(n,ξ)f(x(g(n,ξ)))=0, (1.3)
In recent years, there has been an increasing interest in the study of the oscillatory behavior of solutions of dynamic equations. We refer to the papers [1] - [16] and the references cited therein.
In [1] Bohner et al. proved several theorems provided sufficient conditions for oscillation of all solutions of the second order Emden-Fowler dynamic equations of the form
(p(t)xΔ(t))Δ+q(t)xγ(σ(t))=0.
They studied both the cases
∞∫t0Δsp(s)=∞ and ∞∫t0Δsp(s)<∞.
In [2] Baoguo et al. discussed the oscillatory behavior of second-order linear dynamic equations:
(r(t)xΔ(t))Δ+p(t)xσ(t)=0.
In [3] Grace et al. discussed the oscillation criteria of second order nonlinear dynamic equations:
(a(t)(xΔ(t))α)Δ+q(t)xβ(t)=0.
In [4], by a Riccati transformation technique, Tripathy, obtained some oscillation results for nonlinear neutral second-order dynamic equations of the form
(r(t)((y(t)+p(t)y(t−τ))Δ)γ)Δ+q(t)|y(t−δ)|γsgny(t−δ)=0.
In [5], Chen et al. studied the oscillatory and asymptotic properties of second-order nonlinear neutral dynamic equations of the form
(r(t)|(x(t)+p(t)x(τ(t)))Δ|α−1(x(t)+p(t)x(τ(t)))Δ)Δ+q(t)|x(t)|β−1x(t)=0.
They studied both the cases
∞∫t0(1r(s))1αΔs=∞ and ∞∫t0(1r(s))1αΔs<∞. (1.4)
In [6] by a generalized Riccati transformation technique, Chen studied the oscillatory of second-order dynamic equations
(r(t)|xΔ(t)|α−1xΔ(t))Δ+q(t)|x(t)|β−1x(t)=0,
when
α,β are constants.
In [7] by a generalized Riccati transformation technique, Zhang et al. obtained some new oscillation results for second-order neutral delay dynamic equation of the form
(r(t)(x(t)+p(t)x(τ(t)))Δ)Δ+q(t)x(δ(t))=0.
In [8] under condition (1.4), Li et al. considered nonlinear second order neutral dynamic equations of the form
(r(t)((y(t)+p(t)y(t−τ))Δ)α)Δ+q(t)yα(t−δ)=0.
In [9] Li et al. studied the oscillatory for second-order half-linear delay damped dynamic equations on time scales of the form
(r(t)|xΔ(t)|α−1xΔ(t))Δ+b(t)|xΔ(t)|α−1xΔ(t)+p(t)|x(δ(t))|α−1x(δ(t))=0.
In [10] under condition (1.4) and by generalized Riccati transformation technique and the integral averaging, Zhang et al. obtained some new oscillation criteria of second-order nonlinear delay dynamic equations on time scales of the form
(r(t)(xΔ(t))γ)Δ+q(t)f(x(τ(t)))=0.
In this paper, we will consider both the case when
∞∫t0(r(s)ψ(x(s)))−1αΔs=∞, (1.5)
holds and the case when
∞∫t0(r(s)ψ(x(s)))−1αΔs<∞, (1.6)
holds. For more details, see [13] [14] [15] [16]. When
T=ℕ, we refer the reader to [17] [18] [19] [20] and the references cited therein.
The details of the proofs of results for non-oscillatory solutions will be carried out only for eventually positive solutions, since the arguments are similar for eventually negative solutions.
The paper is organized as follows. In Section 2, we will state and prove the main oscillation theorems and we provide some examples to illustrate the main results.
2. Main Results
In this section, we establish some new oscillation criteria for the Equation (1.1). We begin with some useful lemmas, which will be used later.
Lemma 2.1. Let
x(t) be a non-oscillatory solution of Equation (1.1). Then there exists a
t≥t0 such that
z(t)≥0,
zΔ(t)≥0 and
(r(t)ψ(x(t))|zΔ(t)|α−1zΔ(t))Δ≤0 for
t≥t0. (2.1)
Proof. Let
x(t) is eventually positive solution of equation(1.1), we may assume that
x(t)>0,
x(t−τ)>0, and
x(g(t,ξ))>0 for
t≥t0,
ξ∈[a,b]. Set
z(t)=x(t)+p(t)x(t−τ). By, assumption (A1), we have
z(t)>0, and from Equation (1.1), we get
(r(t)ψ(x(t))|zΔ(t)|α−1zΔ(t))Δ=−∫baq(t,ξ)f(x(g(t,ξ)))Δξ≤0. (2.2)
Therefore,
r(t)ψ(x(t))|zΔ(t)|α−1zΔ(t) is non-increasing function. Now we have two possible cases for
zΔ(t) either
zΔ(t)<0 eventually or
zΔ(t)>0 eventually. Suppose that
zΔ(t)<0 for
t≥t0. Then from (2.2), there is an integer
t1 such that
zΔ(t1)<0 and
r(t)ψ(x(t))(zΔ(t))α≤a(t1)ψ(x(t1))(zΔ(t1))α, for t≥t1. (2.3)
Dividing by
r(t)ψ(x(t)) and integrating the last inequality from
t1 tot, we obtain
z(t)≤z(t1)+(r(t1)ψ(x(t1)))1αzΔ(t1)t∫t11(r(s)ψ(x(s)))1αΔs for t≥t1.
This implies that
z(t)→−∞ as
t→∞, by (1.5), which is a contradiction the fact that
z(t) is positive. Then
zΔ(t)>0. This completes the proof of Lemma 2.1.
□
Lemma 2.2. Assume that
α≥1,
x1,x2∈[0,∞). Then
xα1+xα2≥12α−1(x1+x2)α.
Proof. The proof can be found in [11].
Lemma 2.3. Assume that
0<α≤1,
x1,x2∈[0,∞). Then
xα1+xα2≥(x1+x2)α. (2.4)
Proof. The proof can be found in [12].
Throughout this subsection we assume that there exists a double functions
{H(t,s)|t≥s≥0} and
h(t,s) such that
1)
H(t,t)=0 for
t≥0,
2)
H(t,s)>0 for
t>s>0,
3) H has a nonpositive continuous ∆-partial derivative
HΔs(t,s) with respect to the second variable, and satisfies
h(t,s)=−HΔs(t,s)√H(t,s).
In the following results, we shall use the following notation
R(t):=1(r(G(t))ψ(x(G(t))))1αμα−1(G(t)), Θ(t):=β(t)R(t)(βσ(t))1+1α,
φ(t):=21−αβ(t)Γ(t)(βσ(t))2, r+:=max{0,r}, ϑ(t,s):=((βΔ(t))+βσ(t)−h(t,s)√H(t,s)),
ϕ(t):=(βσ(t))α+1(1+α)1+α(η(t)H(t,s)βσ(t)−h(t,s)√H(t,s))1+α(H(t,s)β(t)R(t))α,
Ψ(t):=Kβ(t)∫baq(t,ξ)(1−p(g(t,ξ)))αΔξ−(β(t)a(t)ψ(x(t))A(t))Δ +β(t)R(t)(rσ(t)ψ(xσ(t))Aσ(t))1+1α,
η(t):=βΔ(t)+αβ(t)R(t)(1+1α)(rσ(t)ψ(xσ(t))Aσ(t))1α.
Next, we state and prove the main theorems.
Theorem 2.1. Let
α≥1 and (1.5) holds. Further, assume that there exists a positive non-decreasing rd-continuous ∆-differentiable function
β(t), such that for any
t1∈N, there exists an integer
t2>t1, with
limt→∞supt∫t0(β(s)K2α−1∫baQ(s,ξ)Δξ−1(α+1)α+1(1+pα)(βΔ(s))α+1(β(s)R(s))α)Δs=∞, (2.5)
where
Q(t,ξ)=min{q(t,ξ),(q(t,ξ)−τ)}. Then every solution of Equation (1.1) is oscillatory.
Proof. Assume that Equation (1.1) has a non-oscillatory solution, say
x(t)>0,
x(t−τ)>0 and
x(g(t,ξ))>0 for all
t≥t0. From Equation (1.1), Lemma 2.2 and condition (A4) there exists
t2≥t1 such that for
t≥t2, we get
[(r(t)ψ(x(t))(zΔ(t))α)Δ+pα[(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δ]]+K2α−1∫baQ(t,ξ)zα(g(t,ξ))Δξ≤0. (2.6)
Further, it is clear from (A3)
g(t,ξ)≥min{g(t,a),g(t,b)}≡G(t), ξ∈[a,b].
Thus
[(r(t)ψ(x(t))(zΔ(t))α)Δ+pα[(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δ]]+Kzα(G(t))2α−1∫baQ(t,ξ)Δξ≤0. (2.7)
Define
ω(t):=β(t)r(t)ψ(x(t))(zΔ(t))αzα(G(t)). (2.8)
Then
ω(t)>0. From (2.8), we have
ωΔ(t)=βΔ(t)rσ(t)ψ(xσ(t))((zσ(t))Δ)αzα(Gσ(t))+β(t)(r(t)ψ(x(t))(zΔ(t))α)Δzα(G(t)) −β(t)rσ(t)ψ(xσ(t))((zσ(t))Δ)α(zα(G(t)))Δzα(Gσ(t))zα(G(t)). (2.9)
Since
zΔ(t)>0, and By using the inequality
xα−yα≥αyα−1(x−y) for x≠y>0 and α≥1,
we have
(zα(G(t)))Δ=zα(G(σ(t)))−zα(G(t))μα(G(t))≥αzα−1(G(t))μα−1(G(t))(zΔ(G(t))),α≥1. (2.10)
Substitute from (2.10) in (2.9), we have
ωΔ(t)≤βΔ(t)βσ(t)ωσ(t)+β(t)(r(t)ψ(x(t))(zΔ(t))α)Δzα(G(t)) −αβ(t)rσ(t)ψ(xσ(t))((zσ(t))Δ)αzα−1(G(t))(zΔ(G(t)))z2α(Gσ(t))μα−1(G(t)). (2.11)
By Lemma (2.1), since
r(t)ψ(x(t))|zΔ(t)|α−1zΔ(t)=r(t)ψ(x(t))(zΔ(t))α is decreasing function then
r(t)ψ(x(t))(zΔ(t))α≤r(G(t))ψ(x(G(t)))(zΔ(G(t)))α. Then it follows that
zΔ(G(t))zΔ(t)≥(r(t)ψ(x(t))r(G(t))ψ(x(G(t))))1α. (2.12)
It follows from (2.11) and (2.12) that
ωΔ(t)≤βΔ(t)βσ(t)ωσ(t)+β(t)(r(t)ψ(x(t))(zΔ(t))α)Δzα(G(t)) −αβ(t)R(t)(βσ(t))α+1α(ωσ(t))α+1α. (2.13)
Similarly, define another function
v(t) by
v(t):=β(t)a(t−τ)ψ(x(t−τ))(zΔ(t−τ))αzα(G(t)). (2.14)
Then
v(t)>0. From (2.14), we have
vΔ(t)=βΔ(t)βσ(t)vσ(t)+β(t)(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δzα(G(t)) −β(t)rσ(t−τ)ψ(xσ(t−τ))((zσ(t−τ))Δ)α(zα(G(t)))Δzα(Gσ(t))zα(G(t)). (2.15)
From (2.10), (2.14), (2.15) and (2.12), we have
vΔ(t)≤β(t)(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δzα(G(t))+βΔ(t)βσ(t)vσ(t) −αβ(t)R(t)(βσ(t))α+1α(vσ(t))α+1α. (2.16)
From (2.13) and (2.16), we obtain
ωΔ(t)+pαvΔ(t)≤β(t)[(r(t)ψ(x(t))(zΔ(t))α)Δ+pα(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δ]zα(G(t)) +βΔ(t)βσ(t)ωσ(t)−αβ(t)R(t)(βσ(t))α+1α(ωσ(t))α+1α +pα[βΔ(t)βσ(t)vσ(t)−αβ(t)R(t)(βσ(t))α+1α(vσ(t))α+1α]. (2.17)
From (2.7) and (2.17), we have
ωΔ(t)+pαvΔ(t)≤−β(t)K2α−1∫baQ(t,ξ)Δξ+βΔ(t)βσ(t)ωσ(t)−αβ(t)R(t)(βσ(t))α+1α(ωσ(t))α+1α +pα[βΔ(t)βσ(t)vσ(t)−αβ(t)R(t)(βσ(t))α+1α(vσ(t))α+1α]. (2.18)
Using (2.18) and the inequality
Bu−Auα+1α≤αα(α+1)α+1Bα+1Aα, A>0, (2.19)
we have
ωΔ(t)+pαvΔ(t)≤−β(t)K2α−1∫baQ(t,ξ)Δξ+1(α+1)α+1(βΔ(t))α+1(β(t)R(t))α +pα(α+1)α+1(βΔ(t))α+1(β(t)R(t))α.
Integrating the last inequality from
t2 to t, we obtain
t∫t2(β(s)K2α−1∫baQ(s,ξ)Δξ−1(α+1)α+1(1+pα)(βΔ(s))α+1(β(s)R(s))α)Δs≤ω(t2)+pαv(t2).
which yields
t∫t2(β(s)K2α−1∫baQ(s,ξ)Δξ−1(α+1)α+1(1+pα)(βΔ(s))α+1(β(s)R(s))α)Δs≤c1,
where
c1>0 is a finite constant. But, this contradicts (2.5). This completes the proof of Theorem 2.1.
□
Corollary 2.1. If
T=ℕ, then (2.5) becomes
limn→∞sup∑m−1s=0(Kρs2α−1∑bξ=a Qs,ξ−1(α+1)α+1(1+pα)(Δρs)α+1(ρsRs)α)=∞. (
¯2.5 )
Then every solution of Equation (1.2) is oscillatory.
Example 2.1. Consider the nonlinear delay dynamic equation
Δ(nn+1Δxn)+∑1ξ=0λξn2xn(ξ+x2n)=0,n≥1,
where
an=nn+1,
ψ(xn)=1,
pn=0,
α=1,
q(n,ξ)=λξn2 If we take
ρn=n,
K=1 then we have
Rl=n+1n,
∑nl=n0(Kρl2α−1∑bξ=a Ql,ξ−(1+pα)((l+1)−l)α+1(α+1)α+1(lRl)α)=∑nl=1(λll2−l4l(l+1))=∑nl=1(λl−14(l+1))≥∑nl=1(4λ−14l)→∞
as
n→∞ if
λ>14 Thus Corollary 2.1 asserts that every solution of (3.1) is oscillatory when
λ>14.
Theorem 2.2. Let
0<α≤1 and (1.5) holds. Further, assume that there exists a positive non-decreasing function
β(t), such that for any
t1, there exists an integer
t2>t1, with
limt→∞supt∫t0(Kβ(s)b∫a Q(s,ξ)Δξ−1(α+1)α+1(1+pα)(βΔ(s))α+1(β(s)R(s))α)Δs=∞.
Then Equation (1.1) is oscillatory.
Proof. The proof is similar to that of Theorem 2.1 and hence the details are omitted.
Theorem 2.3. Assume that
α≥1 and (1.5) holds. Let
β(t) be a positive rd-continuous ∆-differentiable function. Furthermore, we assume that there exists a double function
{H(t,s)|t≥s≥0}. If
limt→∞sup1H(t,0)t∫0(H(t,s)β(s)K2α−1b∫ξ=a Q(s,ξ)Δξ−(1+pα)(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s))Δs=∞. (2.20)
Then every solution of Equation (1.1) is oscillatory.
Proof. Proceeding as in Theorem 2.1 we assume that Equation (1.1) has a non-oscillatory solution, say
x(t)>0,
x(t−τ)>0 and
x(g(t,ξ))>0 for all
t≥t0. From the proof of Theorem 2.1, we find that (2.18) holds for all
t≥t2. From (2.18), we have
β(t)K2α−1b∫a Q(s,ξ)Δξ≤−ωΔ(t)−pαvΔ(t)+βΔ(t)βσ(t)ωσ(t)−αΘ(t)(ωσ(t))α+1α +pα[βΔ(t)βσ(t)vσ(t)−αΘ(t)(vσ(t))α+1α].
Therefore, we have
t∫t2 H(t,s)β(s)K2α−1b∫a Q(s,ξ)ΔξΔs≤−t∫t2 H(t,s)ωΔ(s)Δs−pαt∫t2 H(t,s)vΔ(s)Δs +t∫t2 H(t,s)(βΔ(s))+βσ(s)ωσ(s)Δs−αt∫t2 H(t,s)Θ(s)(ωσ(s))α+1αΔs +pαt∫t2 H(t,s)(βΔ(s))+βσ(s)vσ(s)Δs−αpαt∫t2 H(t,s)Θ(s)(vσ(s))α+1αΔs,
which yields after integrating by parts
t∫t2 H(t,s)β(s)K2α−1b∫a Q(s,ξ)ΔξΔs≤H(t,t2)ω(t2)+t∫t2 H(t,s)ϑ(t,s)ωσ(s)Δs −αt∫t2 H(t,s)Θ(s)(ωσ(s))α+1αΔs+pαH(t,t2)v(t2) +pαt∫t2 H(t,s)ϑ(t,s)vσ(s)Δs−αpαt∫t2 H(t,s)Θ(s)(vσ(s))α+1αΔs.
From (2.19), we have
t∫t2 H(t,s)β(s)K2α−1b∫a Q(s,ξ)ΔξΔs≤H(t,t2)ω(t2)+t∫t21(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s)Δs +pαH(t,t2)v(t2)+pαt∫t21(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s)Δs.
Then,
t∫t2(H(t,s)β(s)K2α−1b∫a Q(s,ξ)Δξ−(1+pα)1(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s))Δs≤H(t,t2)ω(t2)+pαH(t,t2)v(t2),
which implies
t∫t2(H(t,s)β(s)K2α−1b∫a Q(s,ξ)Δξ−(1+pα)1(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s))Δs≤H(t,0)|ω(t2)|+pαH(t,0)|v(t2)|.
Hence,
t∫0(H(t,s)β(s)K2α−1b∫a Q(s,ξ)Δξ−(1+pα)1(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s))Δs≤H(t,0){t∫0|β(s)K2α−1b∫a Q(s,ξ)Δξ|Δs+|ω(t2)|+pα|v(t2)|}.
Hence,
limt→∞sup1H(t,0)t∫0(H(t,s)β(s)K2α−1b∫a Q(s,ξ)Δξ−(1+pα)1(α+1)α+1H(t,s)ϑα+1(t,s)Θα(s))Δs≤t∫0|β(s)K2α−1b∫a Q(s,ξ)Δξ|Δs+|ω(t2)|+pα|v(t2)|<∞,
which is contrary to (2.20). This completes the proof of Theorem 2.3.
□
Corollary 2.2. If
T=ℕ, then (2.20) becomes
limm→∞sup1Hm,0∑m−1n=0(Hm,nρnK2α−1∑bξ=a Qn,ξ−(1+pα)1(α+1)α+1ϑα+1m,nHm,nΘαn)=∞. (
¯2.20 )
Then every solution of Equation (1.2) is oscillatory.
Corollary 2.3. If
T=ℝ, then (2.20) becomes
limt→∞sup1H(t,0)t∫0(H(t,s)β(s)K2α−1b∫a Q(s,ξ)dξ−(1+pα)(α+1)α+1H(t,s)ϑα+1(t,s)Θα1(s))ds=∞, (
¯¯2.20 )
where
Θ1(t):=J(t)G′(t)(β(t))1α, J(t):=(r(G(t))ψ(x(G(t))))−1α.
Then every solution of Equation (1.3) is oscillatory.
Example 2.2. Consider the differential equation
(t4t2+2cos2(lnt)t2+2cos2(lnt)x′(t))′+1∫0(ξt)x[t+ξ]dξ=0 for t≥t0=1,
If we take
β(t)=1 and
H(t,s)=(t−s)2, then we have
limt→∞sup1H(t,t0)t∫t0(H(t,s)β(s)K2α−1b∫a Q(s,ξ)dξ
−(1+pα)1(α+1)α+1H(t,s)ϑα+1(t,s)Θα1(s))ds=limt→∞sup1(t−1)2t∫1{(t−s)22s−2s4s2+2cos2(lns)s2+2cos2(lns)}ds≥limt→∞sup1(t−1)2t∫1{(t−s)22s−2s}ds=limt→∞sup1(t−1)2(t−74t2+12t2lnt+34)=∞.
Hence, this equation is oscillatory by Corollary 2.4.
Theorem 2.4. Let
0<α≤1 and (1.5) holds. Further, assume that there exists a positive rd-continuous ∆-differentiable function
β(t), such that for any
t1, there exists an integer
t2>t1, with
limt→∞sup1H(t,0)t∫0(β(s)H(t,s)Kb∫ξ=a Q(s,ξ)Δξ−(1+pα)1(α+1)α+1ϑα+1(t,s)H(t,s)Θα(s))Δs=∞.
Then Equation (1.1) is oscillatory.
Proof. The proof is similar to that of Theorem 2.3 and hence the details are omitted.
Corollary 2.4. If
T=ℝ, then the condition of Theorem 2.4 becomes
limt→∞sup1H(t,0)t∫0(H(t,s)β(s)Kb∫a Q(s,ξ)dξ−(1+pα)(α+1)α+1H(t,s)ϑα+1(t,s)Θα1(s))ds=∞.
Then every solution of Equation (1.3) is oscillatory.
Example 2.3. Consider the differential Equation
(x(t)+1t+2x(t−1))′′+1∫0γ(t−ξ+2)t2(t−ξ+1)x(t−ξ)dξ=0,
where
α=1,
a(t)=ψ(x)=1,
p(t)=1t+2,
q(t,ξ)=γ(t−ξ+2)t2(t−ξ+1),
g(t,ξ)=t−ξ and
f(x)=x. If we take
p=1,
H(t,s)=(t−s)2 and
β(t)≡1, then
limt→∞sup1(t−t0)2t∫t0(t−s)2(β(s)Kb∫a Q(s,ξ)dξ−(1+pα)1(α+1)α+1ϑα+1(t,s)Θα1(s))ds=limt→∞sup1(t−t0)2t∫t0(t−s)2(γs−14(2t−s)2)ds=∞.
Hence, by Corollary 2.4, this equation is oscillatory.
Theorem 2.5. Let
α≥1 and (1.5) holds. Further, assume that there exists a positive rd-continuous ∆-differentiable function
β(t), such that for any
t1, there exists an integer
t2>t1, with
limt→∞supt∫t0(β(l)K2α−1b∫a Q(l,ξ)Δξ−(1+pα)23−α(βΔ(l))2β(l)Γ(l))Δl=∞, (2.21)
where
Γ(l):=1r(G(l))ψ(x(G(l))).
Then every solution of Equation (1.1) oscillatory.
Proof. Assume that Equation (1.1) has a non-oscillatory solution, say
x(t)>0,
x(t−τ)>0 and
x(g(t,ξ))>0 for all
t≥t0. By Lemma 2.1, we have (2.1) and from Theorem 2.1, we have (2.7). Define
ω(t) and
v(t) by (2.8) and (2.14) respectively. Proceeding as in the proof of Theorem 2.1, we obtain (2.9) and (2.15). By using the inequality
xα−yα≥21−α(x−y)α for
x≥y>0 and
α≥1, we have
(zα(G(t)))Δ=zα(Gn+1)−zα(Gn)μα(Gn)≥21−α(zΔ(G(t)))α,α≥1. (2.22)
Substitute from (2.22) in (2.9), we have
ωΔ(t)≤βΔ(t)βσ(t)ωσ(t)+β(t)(r(t)ψ(x(t))(zΔ(t))α)Δzα(G(t)) −21−αβ(t)rσ(t)ψ(xσ(t))((zσ(t))Δ)α(zΔ(G(t)))αz2α(Gσ(t)). (2.23)
From (2.12), we have
ωΔ(t)≤βΔ(t)βσ(t)ωσ(t)+β(t)(r(t)ψ(x(t))(zΔ(t))α)Δzα(G(t)) −21−αβ(t)Γ(t)(βσ(t))2(ωσ(t))2. (2.24)
On the other hand, from (2.15), we have
vΔ(t)≤β(t)(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δzα(G(t))+βΔ(t)βσ(t)vσ(t) −21−αβ(t)Γ(t)(βσ(t))2(vσ(t))2. (2.25)
From (2.24) and (2.25), we obtain
ωΔ(t)+pαvΔ(t)≤β(t)[(r(t)ψ(x(t))(zΔ(t))α)Δ+pα(r(t−τ)ψ(x(t−τ))(zΔ(t−τ))α)Δ]zα(G( t )
(2.26)
From (2.7) and (2.26), we have
(2.27)
Using the inequality
in (2.27), we have
(2.28)
Integrating (2.28) from
to t, we obtain
which yields
where
is a finite constant. Taking lim sup in the above inequality, we obtain a contradiction with (2.21). This completes the proof of Theorem 2.5.
Corollary 2.5. If
, then (2.21) becomes
(
)
Then every solution of Equation (1.2) oscillatory.
Example 2.4. Consider the nonlinear neutral dynamic equation
where
,
,
,
,
. If we take
,
, then, we have
,
if
. By Corollary 2.8 every solution of this equation is oscillatory when
.
Theorem 2.6. Let
and (1.5) holds. Further, assume that there exists a positive rd-continuous ∆-differentiable function
, such that for any
, there exists an integer
, with
Then Equation (1.1) is oscillatory.
Proof. The proof is similar to that of Theorem 2.5 and hence the details are omitted.
Theorem 2.7. Assume that
and (1.5) holds. Let
be a positive rd-continuous ∆-differentiable function. Furthermore, we assume that there exists a double function
. If
(2.29)
Then every solution of Equation (1.1) is oscillatory.
Proof. Proceeding as in Theorem 2.5 we assume that Equation (1.1) has a non-oscillatory solution, say
,
and
for all
. From the proof of Theorem 2.5, we find that (2.27) holds for all
. From (2.27), we have
(2.30)
Therefore, we have
which yields after summing by parts
Using the inequality
, we have
(2.31)
The rest of the proof is similar that of Theorem 2.3 and hence the details are omitted. This completes the proof of Theorem 2.7.
Corollary 2.6. If
, then (2.29) becomes
(
)
Then every solution of Equation (1.2) is oscillatory.
Theorem 2.8. Let
and (1.5) holds. Further, assume that there exists a positive rd-continuous ∆-differentiable function
, such that for any
, there exists an integer
, with
Then Equation (1.1) is oscillatory.
Proof. The proof is similar to that of Theorem 2.7 and hence the details are omitted.
Theorem 2.9. Let (1.5) holds. Assume that there exists a positive non-decreasing rd-continuous ∆-differentiable function
such that for any
, there exists an integer
, with
(2.32)
Then every solution of Equation (1.1) is oscillatory.
Proof. Assume that Equation (1.1) has a non-oscillatory solution, say
,
and
for all
. From Equation (1.1), From (2.1) and the fact that
, we see that
(2.33)
Further, it is clear form (A3) that
which in view of (2.1) leads to
Using the above inequality together with (2.1), (2.33), (A3) and (A4) in Equation (1.1) for
, we get
(2.34)
Define the function
by the generalized Riccati substitution
(2.35)
It follows that
From (2.34) = and (2.35), we have
(2.36)
First: we consider the case when
. By using the inequality
we have
Substituting in (2.36), we have
(2.37)
From (2.12) and (2.37), we find
(2.38)
Second: we consider the case when
. By using the inequality
We may write
Substituting in (2.36), we have
From (2.12) and by Lemma (2.1), since
is decreasing function, we have
(2.39)
Thus, we again obtain (2.38). However, from (2.35) we see that
(2.40)
Then, by using the inequality
we may write Equation (2.40) as follows
Substituting back in (2.38), we have
(2.41)
Thus,
Therefore, we have
which yields after summing by parts
Hence
From (2.19),
and
, we obtain
which implies
which is contrary to (2.32). This completes the proof of Theorem 2.9.
Theorem 2.10. Let (1.6) and (2.5) hold. Assume that
be as defined as Theorem 2.1. If
(2.42)
then every solution of Equation (1.1) either oscillates or tends to zero
Proof. Assume that Equation (1.1) has a non-oscillatory solution. Without loss of generality, we may assume that
,
and
for all
. Proceeding as in the proof of Lemma 2.1, we have (2.2) holds. Therefore,
is non-increasing function. Now we have two possible cases for
either
eventually or
eventually. If
, The proof is similar to that of Theorem 2.1 and hence is omitted. Suppose that
for
. Since
is a positive decreasing solution of Equation (1.1), then
. Now we claim that
. If
then
for
. Therefore from (A4) and (1.1), we have
Integrating the above inequality from
tot, we obtain
where
. Dividing by
and integrating the last inequality from
tot, we obtain
Condition (2.42) implies that
as
which is contradiction with the fact that
. Then
. i.e.
. Since
then
. The proof is complete.
3. Conclusion
We established some new sufficient conditions for the oscillation of all solutions of this equation. Our results not only unify the oscillation of second order nonlinear differential and difference equations but also can be applied to different types of time scales with
. Our results improved and expanded some known results, see e.g. the following results:
Remark 3.1. If
,
,
,
,
,
, then we extended and improved Theorems in [1].
Remark 3.2. If
,
,
,
,
, then we generalized the results in [3].
Remark 3.3. If
,
,
,
, then we extended and improved Theorems in [4].
Remark 3.4. If
,
,
, then we reduced to Theorems in [5].
Remark 3.5. If
,
,
,
,
, then we reduced to a special case in [6].
Remark 3.6. If
,
,
,
,
, then we reduced to a special case in [7].
Acknowledgements
The authors would like to thank the anonymous referees very much for valuable suggestions, corrections and comments, which results in a great improvement in the original manuscript.