We study the higher-order neutral dynamic equation {a(t)[(x(t)−p(t)x(τ(t)))Δm]α}Δ+f(t,x(δ(t)))=0 for t∈[t0,∞)𝕋 and obtain some necessary and sufficient conditions for the existence of nonoscillatory
bounded solutions for this equation.

1. Introduction

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers. Thus, ℝ,ℤ,andℕ, that is, the real numbers, the integers, and the natural numbers, are examples of time scales. We assume throughout that the time scale 𝕋 has the topology that it inherits from the real numbers with the standard topology.

The theory of time scale, which has recently received a lot of attention, was introduced by Hilger's landmark paper [1], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is a nonempty closed subset of the real numbers, and the cases when this time scale is equal to the real numbers or to the integers represent the classical theories of differential and difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only the new theory of the so-called “dynamic equations” unifies the theories of differential equations and difference equations but also extends these classical cases to cases “in between”, for example, to the so-called q-difference equations when 𝕋={1,q,q2,…}, which has important applications in quantum theory (see [3]).

On a time scale 𝕋, the forward jump operator, the backward jump operator and the graininess function are defined as σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},μ(t)=σ(t)-t,
respectively. We refer the reader to [2, 4] for further results on time scale calculus.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the papers of [5–20].

In [21] Zhu and Wang studied the existence of nonoscillatory solutions to neutral dynamic equation [x(t)+p(t)x(g(t))]Δ+f(t,x(h(t)))=0.

Karpuz [22] studied the asymptotic behavior of delay dynamic equations having the following form:[x(t)+A(t)x(α(t))]Δ+B(t)F(x(β(t)))-C(t)G(x(γ(t)))=φ(t).
Furthermore, Karpuz in [23] obtained necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation [x(t)+A(t)x(α(t))]Δn+f(t,x(β(t)),x(γ(t)))=φ(t)
and also studied in [24] oscillation of unbounded solutions of a similar type of equations.

Li et al. [25] considered the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form [x(t)+p(t)x(τ0(t))]ΔΔ+q1(t)x(τ1(t))-q2(t)x(τ2(t))=e(t).

In [26, 27], Zhang et al. obtained some sufficient conditions for the existence of nonoscillatory solutions for the following higher-order equations: [x(t)+p(t)x(τ(t))]Δn+f(t,x(t-τ1(t)),…,x(t-τk(t)))=0,[x(t)+p(t)x(τ(t))]Δn+f(t,x(τ1(t)),…,x(τk(t)))=0,

respectively.

Motivated by the above studies, in this paper, we investigate the existence of nonoscillatory solutions of the following higher order neutral dynamic equation:{a(t)[(x(t)-p(t)x(τ(t)))Δm]α}Δ+f(t,x(δ(t)))=0fort∈[t0,∞)𝕋,
where m∈ℕ, α is the quotient of odd positive integers, t0∈𝕋, the time scale interval [t0,∞)𝕋={t∈𝕋:t≥t0}, a∈Crd([t0,∞)𝕋,(0,∞)), p∈C([t0,∞)𝕋,ℝ), τ,δ∈C(𝕋,𝕋) with limt→∞τ(t)=limt→∞δ(t)=∞ and f∈C([t0,∞)𝕋×ℝ,ℝ) satisfying the following conditions:

uf(t,u)>0 for any t∈[t0,∞)𝕋 and u≠0.

f(t,u) is nondecreasing in u for any t∈[t0,∞)𝕋.

Since we are interested in the oscillatory behavior of solutions near infinity, we assume that sup𝕋=∞. By a solution of (1.7) we mean a nontrivial real-valued function x∈Crd([Tx,∞)𝕋,ℝ),Tx≥t0, such that a(t)[(x(t)-p(t)x(τ(t)))Δm]α∈Crd1([Tx,∞)𝕋,ℝ) and satisfies (1.7) on [Tx,∞). The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.7) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

2. Auxiliary Results

We state the following conditions, which are needed in the sequel:

(H1)∫t0∞(a(t))-1/αΔt=∞;

(H2) there exist constants a1,b1∈[0,1) with a1+b1<1 such that -a1≤p(t)≤b1 for all t∈[t0,∞)𝕋;

(H3) there exist constants a2,b2∈(1,∞) such that -a2≤p(t)≤-b2 for all t∈[t0,∞)𝕋;

(H4) there exist constants a3,b3∈(1,∞) such that a3≤p(t)≤b3 for all t∈[t0,∞)𝕋.

Let k be a nonnegative integer and s,t∈𝕋; we define two sequences of functions hk(t,s) and gk(t,s) as follows: hk(t,s)={1ifk=0,∫sthk-1(τ,s)Δτifk≥1,gk(t,s)={1ifk=0,∫stgk-1(σ(τ),s)Δτifk≥1.

By Theorems 1.112 and 1.60 of [2], we have hk(t,s)=(-1)kgk(s,t)forallt,s∈𝕋,hkΔt(t,s)={0ifk=0,hk-1(t,s)ifk≥1,gkΔt(t,s)={0ifk=0,gk-1(σ(t),s)ifk≥1,

where gkΔt(t,s) and hkΔt(t,s) denote for each fixed s the derivatisve of gk(t,s) and hk(t,s) with respect to t, respectively.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B23">23</xref>, <xref ref-type="bibr" rid="B24">24</xref>]).

Assume that s,t∈𝕋 and g∈Crd(𝕋×𝕋,ℝ), then
∫st[∫ηtg(η,ζ)Δζ]Δη=∫st[∫sσ(ζ)g(η,ζ)Δη]Δζ.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B23">23</xref>, <xref ref-type="bibr" rid="B24">24</xref>]).

Let n be a nonnegative integer, h∈Crd(𝕋,[0,∞)), and s∈𝕋. Then each of the following is true:

∫s∞gn(σ(θ),s)h(θ)Δθ<∞ implies that ∫t∞gn(σ(θ),t)h(θ)Δθ<∞ for all t∈𝕋;

∫s∞gn(σ(θ),s)h(θ)Δθ=∞ implies that ∫t∞gn(σ(θ),t)h(θ)Δθ=∞ for all t∈𝕋.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let n be a nonnegative integer, h∈Crd(𝕋,[0,∞)) and s∈𝕋. Then
∫s∞gn(σ(θ),s)h(θ)Δθ<∞

implies that each of the following is true:

∫t∞gj(σ(θ),t)h(θ)Δθ is decreasing for all t∈𝕋 and all 0≤j≤n.

limt→∞∫t∞gj(σ(θ),t)h(θ)Δθ=0 for all 0≤j≤n.

∫t∞gj(σ(θ),t)h(θ)Δθ<∞ for all t∈𝕋 and all 0≤j≤n-1.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let m∈N and fΔm∈Crd([t0,∞)𝕋,ℝ). Then

lim inft→∞fΔm(t)>0 implies limt→∞fΔi(t)=∞ for all 0≤i≤m-1.

lim supt→∞fΔm(t)<0 implies limt→∞fΔi(t)=-∞ for all 0≤i≤m-1.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B29">29</xref>]).

Let z(t) be bounded for t∈[t0,∞)𝕋 with zΔn(t)>0, where n∈N. Then (-1)n-izΔi(t)>0 for 1≤i≤n and
limt→∞zΔi(t)=0for1≤i≤n-1.

In the sequel, write
y(t)=x(t)-p(t)x(τ(t)).

Lemma 2.6.

Assume that p(t) is bounded and (H1) holds. If x(t) is a bounded nonoscillatory solution of (1.7), then x(t)yΔm(t)>0 eventually.

Proof.

Without loss of generality, assume that there is some t1≥t0 such that x(t)>0 and x(δ(t))>0 for t≥t1. From (1.7) we have
{a(t)[yΔm(t)]α}Δ=-f(t,x(δ(t)))<0fort≥t1.
Thus, R(t)=a(t)[yΔm(t)]α is strictly decreasing on [t1,∞)𝕋. If there exists t2≥t1 such that R(t2)<0, then
yΔm(t)≤[R(t2)a(t)]1/αfort≥t2.
Therefore, we have
yΔm-1(t)-yΔm-1(t2)≤∫t2t[R(t2)a(s)]1/αΔs.
By condition (H1), we obtain limt→∞yΔm-1(t)=-∞. Then it follows from Lemma 2.4 that limt→∞y(t)=-∞, which is a contradiction since x(t) and p(t) are bounded. Hence, yΔm(t)=[R(t)/a(t)]1/α>0 for all t≥t1. The proof is completed.

Let BCrd([t0,∞)𝕋,ℝ) be the Banach space of all bounded rd-continuous functions on [t0,∞)𝕋 with sup norm ∥x∥=supt≥t0|x(t)|. Let X⊂BCrd([t0,∞)𝕋,ℝ), we say that X is uniformly Cauchy if for any given ɛ>0, there exists t1>t0 such that for any x∈X, |x(u)-x(v)|<ɛ for all u,v∈[t1,∞)𝕋. X is said to be equicontinuous on [a,b]𝕋 if, for any given ɛ>0, there exists δ>0 such that, for any x∈X and u,v∈[a,b]𝕋 with |u-v|<δ, |x(u)-x(v)|<ɛ. S:X→BCrd([t0,∞)𝕋,ℝ) is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Suppose that X⊂BCrd([t0,∞)𝕋,ℝ) is bounded and uniformly Cauchy. Further, suppose that X is equi-continuous on [t0,t1]𝕋 for any t1∈[t0,∞)𝕋. Then X is relatively compact.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Suppose that X is a Banach space and Ω is a bounded, convex, and closed subset of X. Suppose further that there exist two operators U and S:Ω→X such that

Ux+Sy∈Ω for all x,y∈Ω,

U is a contraction mapping,

S is completely continuous.

Then U+S has a fixed point in Ω.3. Main Results and Examples

Now, we state and prove our main results.

Theorem 3.1.

Assume that (H1) and (H2) hold. Then (1.7) has a nonoscillatory bounded solution x(t) with liminft→∞|x(t)|>0 if and only if there exists some constant K≠0 such that
∫t0∞gm-1(σ(s),t0)[1a(s)∫s∞f(θ,|K|)Δθ]1/αΔs<∞.

Proof.

Sufficiency. Assume that (1.7) has a nonoscillatory bounded solution x(t) on [t0,∞)𝕋 with liminft→∞|x(t)|>0. Without loss of generality, we assume that there is a constant K>0 and some t1≥t0 such that x(t)>K and x(δ(t))>K for t≥t1. It follows from Lemma 2.6 that yΔm(t)>0 for t≥t1. By assumption that x(t) is bounded and condition (H2), we see that y(t) is bounded. Thus, by Lemma 2.5 we get that there exists t2≥t1 such that
(-1)m-kyΔk(t)>0fort≥t≥t2,1≤k≤m.
Integrating (1.7) from t(≥t2) to ∞, we have
a(t)(yΔm(t))α≥∫t∞f(s,x(δ(s)))Δs≥∫t∞f(s,K)Δsfort≥t2.
Therefore, it follows from (3.2) and (3.3) that, for t≥t2,
∫t∞gm-1(σ(θ),t)[1a(θ)∫θ∞f(s,K)Δs]1/αΔθ≤∫t∞yΔm(θ)gm-1(σ(θ),t)Δθ=yΔm-1(θ)gm-1(θ,t)|t∞-∫t∞yΔm-1(θ)gm-2(σ(θ),t)Δθ=-limθ→∞(-1)m-(m-1)yΔm-1(θ)gm-1(θ,t)-∫t∞yΔm-1(θ)gm-2(σ(θ),t)Δθ≤-∫t∞yΔm-1(θ)gm-2(σ(θ),t)Δθ=-yΔm-2(θ)gm-2(θ,t)|t∞+(-1)2∫t∞yΔm-2(θ)gm-3(σ(θ),t)Δθ≤(-1)2∫t∞yΔm-2(θ)gm-3(σ(θ),t)Δθ⋯⋯⋯⋯≤(-1)m-1∫t∞yΔ(θ)Δθ=(-1)m-1y(θ)|t∞<∞.
By Lemma 2.2, we see that (3.1) holds.

Necessity. Suppose that there exists some constant K>0 such that
∫t0∞gm-1(σ(s),t0)A(s)Δs<∞,
where A(s)=[∫s∞f(θ,K)Δθ]1/α/a(s). Then by Lemma 2.3 there exists t1≥t0 such that
∫t∞gm-1(σ(s),t)A(s)Δs<1-b1nK
and min{δ(t),τ(t)}≥t0 for t≥t1, where n>2(1-b1)/(1-b1-a1) is a constant. Let
Ω={x∈BCrd([t0,∞)𝕋,ℝ):n(1-b1-a1)-2(1-b1)nK≤x(t)≤Kfort≥t0}.
It is easy to verify that Ω is a bounded, convex, and closed subset of BCrd([t0,∞)𝕋,ℝ).

Now we define two operators S and T:Ω→BCrd([t0,∞)𝕋,ℝ) as follows:
(Sx)(t)=p(t*)x(τ(t*)),(Tx)(t)=(n-1)(1-b1)nK+(-1)m∫t*∞gm-1(σ(s),t*)A(s,x)Δs,
where u*=max{u,t1} for any u∈[t0,∞)𝕋 and A(s,z)=[∫s∞f(θ,z(δ(θ)))Δθ]1/α/a(s) for any z∈Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.

(1) We will show that Sx+Ty∈Ω for any x,y∈Ω. In fact, for any x,y∈Ω and t≥t0, x(t),y(t)∈[[n(1-b1-a1)-2(1-b1)]K/n,K],
(Sx)(t)+(Ty)(t)=(n-1)(1-b1)nK+p(t*)x(τ(t*))+(-1)m∫t*∞gm-1(σ(s),t*)A(s,x)Δs≤(n-1)(1-b1)nK+b1K+1-b1nK=K,(Sx)(t)+(Ty)(t)=(n-1)(1-b1)nK+p(t*)x(τ(t*))+(-1)m∫t*∞gm-1(σ(s),t*)A(s,x)Δs≥(n-1)(1-b1)nK-a1K-1-b1nK=n(1-b1-a1)-2(1-b1)nK,
which implies that Sx+Ty∈Ω for any x,y∈Ω.

(2) We will show that S is a contraction mapping. Indeed, for any x,y∈Ω and t≥t0, we have
|(Sx)(t)-(Sy)(t)|=|p(t*)x(τ(t*))-p(t*)y(τ(t*))|≤max{a1,b1}∥x-y∥.
Therefore, we have
∥Sx-Sy∥≤max{a1,b1}∥x-y∥,
which implies that S is a contraction mapping.

(3) We will show that T is a completely continuous mapping.

By the proof of (1), we see that [n(1-b1-a1)-2(1-b1)]K/n≤(Tx)(t)≤K for t∈[t0,∞)𝕋. That is, TΩ⊂Ω.

We consider the continuity of T. Let xn∈Ω and ∥xn-x∥→0 as n→∞, then x∈Ω and |xn(t)-x(t)|→0 as n→∞ for any t∈[t0,∞)𝕋. Consequently, for any s∈[t1,∞)𝕋, we have
limn→∞|gm-1(σ(s),t1)[A(s,xn)-A(s,x)]|=0.
Since
|gm-1(σ(s),t1)[A(s,xn)-A(s,x)]|≤2gm-1(σ(s),t1)A(s)
and, for any t∈[t0,∞)𝕋,
|(Txn)(t)-(Tx)(t)|≤∫t1∞gm-1(σ(s),t1)|A(s,xn)-A(s,x)|Δs,
we have
∥Txn-Tx∥≤∫t1∞gm-1(σ(s),t1)|A(s,xn)-A(s,x)|Δs.
By Chapter 5 in [4], we see that the Lebesgue dominated convergence theorem holds for the integral on time scales. Then
limn→∞∥Txn-Tx∥=0,
which implies that T is continuous on Ω.

We show that TΩ is uniformly Cauchy. In fact, for any ɛ>0, take t2>t1 so that
∫t2∞gm-1(σ(s),t2)A(s)Δs<ɛ.
Then for any x∈Ω and u,v∈[t2,∞)𝕋, we have
|(Tx)(u)-(Tx)(v)|<2ɛ,
which implies that TΩ is uniformly Cauchy.

We show that TΩ is equicontinuous on [t0,t2]𝕋 for any t2∈[t0,∞)𝕋. Without loss of generality, we assume t2≥t1. For any ɛ>0, choose δ=ɛ/(1+∫t0∞gm-2(σ(s),t0)A(s)Δs), then when u,v∈[t0,t2] with |u-v|<δ, we have by Lemma 2.1 that for any x∈Ω,
|(Tx)(u)-(Tx)(v)|=|∫u*∞gm-1(σ(s),u*)A(s,x)Δs-∫v*∞gm-1(σ(s),v*)A(s,x)Δs|=|∫u*∞hm-1(u*,σ(s))A(s,x)Δs-∫v*∞hm-1(v*,σ(s))A(s,x)Δs|=|∫u*∞[∫σ(s)u*hm-2(θ,σ(s))A(s,x)Δθ]Δs-∫v*∞[∫σ(s)v*hm-2(θ,σ(s))A(s,x)Δθ]Δs|=|∫u*∞[∫θ∞hm-2(θ,σ(s))A(s,x)Δs]Δθ-∫v*∞[∫θ∞hm-2(θ,σ(s))A(s,x)Δs]Δθ|=|∫u*∞[∫θ∞gm-2(σ(s),θ)A(s,x)Δs]Δθ-∫v*∞[∫θ∞gm-2(σ(s),θ)A(s,x)Δs]Δθ|=|∫u*v*[∫θ∞gm-2(σ(s),θ)A(s,x)Δs]Δθ|≤|∫u*v*[∫t0∞gm-2(σ(s),t0)A(s,x)Δs]Δθ|=|u*-v*|∫t0∞gm-2(σ(s),t0)A(s,x)Δs≤δ∫t0∞gm-2(σ(s),t0)A(s)Δs<ε,

which implies that TΩ is equi-continuous on [t0,t2]𝕋 for any t2∈[t0,∞)𝕋.

By Lemma 2.7, we see that T is a completely continuous mapping. It follows from Lemma 2.8 that there exists x∈Ω such that (U+S)x=x, which is the desired bounded solution of (1.7) with liminft→∞|x(t)|>0. The proof is completed.

Theorem 3.2.

Assume that (H1) and (H3) hold, and that τ has the inverse τ-1∈C(𝕋,𝕋). Then (1.7) has a nonoscillatory bounded solution x(t) with liminft→∞|x(t)|>0 if and only if there exists some constant K≠0 such that (3.1) holds.

Proof.

The proof of sufficiency is similar to that of Theorem 3.1.

Necessity. Suppose that there exists some constant K>0 such that
∫t0∞gm-1(σ(s),t0)A(s)Δs<∞,
where A(s)=[∫s∞f(θ,K)Δθ]1/α/a(s). Then by Lemma 2.3 there exists t1≥t0 such that
∫τ-1(t)∞gm-1(σ(s),τ-1(t))A(s)Δs<b2nK,
and min{δ(τ-1(t)),τ-1(t)}≥t0 for t≥t1, where n>2b2/(b2-1) is a constant. Let
Ω={x∈BCrd([t0,∞)𝕋,ℝ):(n-2)b2-na2nK≤x(t)≤Kfort≥t0}.
It is easy to verify that Ω is a bounded, convex and closed subset of BCrd([t0,∞)𝕋,ℝ).

Now we define two operators S and T:Ω→BCrd([t0,∞)𝕋,ℝ) as follows:
(Sx)(t)=x(τ-1(t*))p(τ-1(t*))+(n-1)b2K-np(τ-1(t*))(Tx)(t)=1p(τ-1(t*))(-1)m-1∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs,
where u*=max{u,t1} for any u∈[t0,∞)𝕋 and A(s,z)=[∫s∞f(θ,z(δ(θ)))Δθ]1/α/a(s) for any z∈Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.

We will show that Sx+Ty∈Ω for any x,y∈Ω. In fact, for any x,y∈Ω and t≥t0, x(t),y(t)∈[[(n-2)b2-n]K/a2n,K],
(Sx)(t)+(Ty)(t)=x(τ-1(t*))p(τ-1(t*))+1-p(τ-1(t*))[x(τ-1(t*))p(τ-1(t*))(n-1)b2nK+(-1)m∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs]≤1b2[(n-1)b2nK+b2nK]=K,(Sx)(t)+(Ty)(t)=x(τ-1(t*))p(τ-1(t*))+1-p(τ-1(t*))[x(τ-1(t*))p(τ-1(t*))(n-1)b2nK+(-1)m∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs]≥1a2[(n-1)b2nK-K-b2nK]=(n-2)b2-na2nK,
and |(Tx)(t)|≤b2K/n, which implies that Sx+Ty∈Ω for any x,y∈Ω and TΩ is uniformly bounded.

Now we show that TΩ is equicontinuous on [t0,t2]𝕋 for any t2∈[t0,∞)𝕋. Without loss of generality, we assume that t2≥t1. Since 1/p(τ-1(t)),τ-1(t) are continuous on [t0,t2]𝕋, so they are uniformly continuous on [t0,t2]𝕋. For any ɛ>0, choose δ>0 such that when u,v∈[t0,t2]𝕋 with |u-v|<δ, we have
|1p(τ-1(u))-1p(τ-1(v))|<ɛ1+∫t0∞gm-1(σ(s),t0)A(s)Δs|τ-1(u)-τ-1(v)|<ɛ1+∫t0∞gm-2(σ(s),t0)A(s)Δs.
Then, it follows from Lemma 2.1 that, for any x∈Ω,
|(Tx)(u)-(Tx)(v)|=|1p(τ-1(u*))∫τ-1(u*)∞gm-1(σ(s),τ-1(u*))A(s,x)Δs-1p(τ-1(v*))∫τ-1(v*)∞gm-1(σ(s),τ-1(v*))A(s,x)Δs|≤|[1p(τ-1(u*))-1p(τ-1(v*))]∫τ-1(u*)∞gm-1(σ(s),τ-1(u*))A(s,x)Δs|+|1p(τ-1(v*))[∫τ-1(u*)∞gm-1(σ(s),τ-1(u*))A(s,x)Δs-∫τ-1(v*)∞gm-1(σ(s),τ-1(v*))A(s,x)Δs]|=|[1p(τ-1(u*))-1p(τ-1(v*))]∫τ-1(u*)∞gm-1(σ(s),τ-1(u*))A(s,x)Δs|+|1p(τ-1(v*))∫τ-1(u*)τ-1(v*)[∫θ∞gm-2(σ(s),θ)A(s,x)Δs]Δθ|≤|[1p(τ-1(u*))-1p(τ-1(v*))]∫t0∞gm-1(σ(s),t0)A(s)Δs|+|1p(τ-1(v*))∫τ-1(u*)τ-1(v*))[∫t0∞gm-2(σ(s),t0)A(s)Δs]Δθ|≤ɛ+|τ-1(u*)-τ-1(v*)|∫t0∞gm-2(σ(s),t0)A(s)Δs<2ε,
which implies that TΩ is equi-continuous on [t0,t2]𝕋 for any t2∈[t0,∞)𝕋.

The rest of the proof is similar to that of Theorem 3.1. The proof is completed.

Theorem 3.3.

Assume that (H1) and (H4) hold and that τ has the inverse τ-1∈C(𝕋,𝕋). Then (1.7) has a nonoscillatory bounded solution x(t) with liminft→∞|x(t)|>0 if and only if there exists some constant K≠0 such that (3.1) holds.

Proof.

The proof of sufficiency is similar to that of Theorem 3.1.

Necessity. Suppose that there exists some constant K>0 such that
∫t0∞gm-1(σ(s),t0)A(s)Δs<∞,
where A(s)=[∫s∞f(θ,K)Δθ]1/α/a(s). Then by Lemma 2.3 there exists t1≥t0 such that
∫τ-1(t)∞gm-1(σ(s),τ-1(t))A(s)Δs<a3-1nK
and min{δ(τ-1(t)),τ-1(t)}≥t0 for t≥t1, where n>2 is a constant. Let
Ω={x∈BCrd([t0,∞)𝕋,ℝ):(n-2)(a3-1)n(b3-1)K≤x(t)≤Kfort≥t0}.
It is easy to verify that Ω is a bounded, convex, and closed subset of BCrd([t0,∞)𝕋,ℝ).

Now we define two operators S and T:Ω→BCrd([t0,∞)𝕋,ℝ) as follows:
(Sx)(t)=x(τ-1(t*))p(τ-1(t*))+(n-1)(a3-1)Knp(τ-1(t*)),(Tx)(t)=1p(τ-1(t*))(-1)m-1∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs,
where u*=max{u,t1} for any u∈[t0,∞)𝕋 and A(s,z)=[∫s∞f(θ,z(δ(θ)))Δθ]1/α/a(s) for any z∈Ω. Now we show that S and T satisfy the conditions in Lemma 2.8.

We will show that Sx+Ty∈Ω for any x,y∈Ω. In fact, for any x,y∈Ω and t≥t0, x(t),y(t)∈[(n-2)(a3-1)K/n(b3-1),K],
(Sx)(t)+(Ty)(t)=x(τ-1(t*))p(τ-1(t*))+1p(τ-1(t*))×[(n-1)(a3-1)nK+(-1)m-1∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs]≤1a3[(n-1)(a3-1)nK+K+a3-1nK]=K,(Sx)(t)+(Ty)(t)=x(τ-1(t*))p(τ-1(t*))+1p(τ-1(t*))×[(n-1)(a3-1)nK+(-1)m-1∫τ-1(t*)∞gm-1(σ(s),τ-1(t*))A(s,x)Δs]≥1b3[(n-1)(a3-1)nK+(n-2)(a3-1)Kn(b3-1)-a3-1nK]=(n-2)(a3-1)Kn(b3-1)
and |(Tx)(t)|≤(a3-1)K/n, which implies that Sx+Ty∈Ω for any x,y∈Ω and TΩ is uniformly bounded. The rest of the proof is similar to that of Theorem 3.2. The proof is completed.

According to the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in [23], we have

Lemma 3.4.

Assume that (H1) holds. Suppose that one of the following holds:

p(t) satisfies condition (H2),

p(t) satisfies condition (H3) and τ has the inverse τ-1∈C(𝕋,𝕋),

p(t) satisfies condition (H4) and τ has the inverse τ-1∈C(𝕋,𝕋).

If x(t) is a bounded nonoscillatory solution of (1.7), then liminft→∞|x(t)|=0 implies that limt→∞x(t)=0.

By Lemma 3.4, Theorem 3.1,Theorem 3.2 and Theorem 3.3, we have.

Corollary 3.5.

Assume that (H1) holds. Suppose that one of the following holds:

p(t) satisfies condition (H2),

p(t) satisfies condition (H3) and τ has the inverse τ-1∈C(𝕋,𝕋),

p(t) satisfies condition (H4) and τ has the inverse τ-1∈C(𝕋,𝕋).

Then every bounded solution of (1.7) is oscillation or converges to zero at infinity if and only if there exists some constant K≠0 such that
∫t0∞gm-1(σ(s),t0)[1a(s)∫s∞f(θ,|K|)Δθ]1/αΔs=∞.Example 3.6.

Let 𝕋={qn:n∈ℤ}∪{0} with q>1. Consider the following higher order dynamic equation:
{tα[(x(t)-pk(t)x(qt))Δm]α}Δ+qgm-1α(q2t,0)-gm-1α(qt,0)(q-1)qt2gm-1α(q2t,0)gm-1α(qt,0)x2r+1(qr+3t)=0fort∈[1,∞)𝕋,
where m,r∈N, α is the quotient of odd positive integers,pk(t)=-2[(-1)kk2+(-1)logqt]/5,, where k=1,2,3, a(t)=tα, τ(t)=qt, δ(t)=q3+rt and f(t,u)=(qgm-1α(q2t,0)-gm-1α(qt,0)/(q-1)qt2gm-1α(q2t,0)gm-1α(qt,0))u2r+1.

It is easy to verify that pk(t) satisfies condition (Hk+1) and τ-1∈C(𝕋,𝕋). On the other hand, we have
∫1∞[1a(t)]1/αΔt=∫1∞[1tα]1/αΔt=∞
andfor any constant K>0,
∫1∞gm-1(σ(s),1)[1a(s)∫s∞f(θ,K)Δθ]1/αΔs=K(2r+1)/α∫1∞gm-1(σ(s),1){1sα∫s∞[-1θgm-1α(σ(θ),0)]Δ}1/αΔs=K(2r+1)/α∫1∞gm-1(σ(s),1)[1sα1sgm-1α(σ(s),0)]1/αΔs=K(2r+1)/α∫1∞1s1+1/αΔs=K2r+1αq1/α(q-1)q1/α-1<∞.
That is, conditions (H1) and (3.1) hold. By Theorem 3.1, Theorem 3.2, and Theorem 3.3, we see that (3.33) has a nonoscillatory bounded solution x(t) with liminft→∞|x(t)|>0.

Remark 3.7.

Results of this paper can be extended to the case with several delays easily.

Acknowledgment

Project Supported by NSF of China (10861002) and NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA014781) and SF of Education Department of Guangxi (200911MS212) and Innovation Project of Guangxi Graduate Education 2010105930701M43.

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