# Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Authors: Haidong Liu; Cuiqin Ma
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish several oscillation criteria for a class of second-order neutral delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals. Our results extend and unify a number of other existing results and handle the cases which are not covered by known criteria. The new results we obtain are of significance because the equations we study allow an infinite number of nonlinear terms and even a continuum of nonlinearities.


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## Body

## 1. Introduction

In this paper, we consider the oscillatory behavior of solutions of the following second-order neutral delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals:(1)(r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))+∫aσ(b)‍k(t,s)|x(g(t,s))|θ(s)-1x(g(t,s))Δξ(s)=0,
where t∈[t0,∞)𝕋≔[t0,∞)∩𝕋, t0∈𝕋, 𝕋 is a time scale which is unbounded above, Z(t)=x(t)+p(t)x(τ(t)), α>0 is a constant, and the following conditions are satisfied:(H1)
a,b∈𝕋1, 𝕋1 is another time scale, Crd(𝔻,𝕊) denotes the collection of all functions f:𝔻→𝕊 which are right-dense continuous on 𝔻;(H2)
r(t)∈Crd([t0,∞)𝕋,(0,∞)), R(t,t0)≔∫t0t‍r-1/α(s)Δs, limt→∞R(t,t0)=∞, p(t)∈Crd([t0,∞)𝕋,[0,1)), θ∈Crd([a,b]𝕋1,ℝ) is a strictly increasing and satisfying 0<θ(a)<α<θ(b), k∈Crd([t0,∞)𝕋×[a,b]𝕋1,[0,∞));(H3)
τ(t)∈Crd([t0,∞)𝕋,[t0,∞)𝕋), τ(t)⩽t, for t∈[t0,∞)𝕋, limt→∞τ(t)=∞, δ(t)∈Crd([t0,∞)𝕋,[t0,∞)𝕋), δ(t)⩽t, for t∈[t0,∞)𝕋, limt→∞δ(t)=∞, g(t,s)∈Crd([t0,∞)𝕋×[a,b]𝕋1,[t0,∞)𝕋), limt→∞g(t,s)=∞ for any s∈[a,b]𝕋1;(H4)
δΔ(t)>0 is right-dense continuous on [t0,∞)𝕋, and δ(σ(t))=σ(δ(t)) for all t∈[t0,∞)𝕋, where σ(t) is the forward jump operator on [t0,∞)𝕋;(H5)
f(t,u)∈C([t0,∞)𝕋×ℝ,ℝ) is a continuous function such that uf(t,u)>0, for all u≠0 and there exists a positive right-dense continuous function q(t) defined on [t0,∞)𝕋 such that |f(t,u)|⩾q(t)|uα| for all t∈[t0,∞)𝕋 and for all u∈ℝ;(H6)
ξ:[a,b]𝕋1→ℝ is strictly increasing; ∫aσ(b)‍f(s)Δξ(s) denotes the Riemann-Stieltjes integral of the function f on [a,σ(b)]𝕋1 with respect to ξ.By a solution of (1), we mean a function x(t) such that x(t)+p(t)x(τ(t))∈Crd1(tx,∞)𝕋 and r(t)|[x(t)+p(t)x(τ(t))]Δ|α-1[x(t)+p(t)x(τ(t))]Δ∈Crd1(tx,∞)𝕋,  tx⩾t0 and satisfying (1) for all t⩾tx, where Crd1(tx,∞)𝕋 denotes the set of right-dense continuously Δ-differentiable functions on (tx,∞)𝕋. In the sequel, we will restrict our attention to those solutions of (1) which exist on the half-line [tx,∞)𝕋 and satisfy sup{|x(t)|:t∈[T~,∞)𝕋}>0 for any T~⩾tx. A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of neutral functional equations on time scales, and we refer the reader to the papers [1–10] and the references cited therein. For an introduction to time scale calculus and dynamic equations, we refer the reader to the landmark paper of Hilger [11] and the seminal book by Bohner and Peterson [12] for a comprehensive treatment of the subject.Recently, Saker and O’Regan [13] studied the the quasi-linear equation of the form
(2)(p(t)([y(t)+r(t)y(τ(t))]Δ)γ)Δ+f(t,y(δ(t)))=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt∈𝕋,t⩾t0,
where |f(t,u)|⩾q(t)|uγ|, γ>0 is an odd positive integer.Wu et al. [14] obtained several oscillation criteria for the equation
(3)(r(t)([y(t)+p(t)y(τ(t))]Δ)γ)Δ+f(t,y(δ(t)))=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkit∈𝕋,t⩾t0,
with |f(t,u)|⩾q(t)|u|γ,  γ⩾1 is a quotient of odd positive integers.Chen [15] investigated the following second-order Emden-Fowler neutral delay dynamic equation
(4)(r(t)|xΔ(t)|γ-1xΔ(t))Δ+f(t,y(δ(t)))=0,kkkkkkkkkkkkkkkkkkkkkkkkt∈𝕋,t⩾t0,
with x(t)=y(t)+p(t)y(τ(t)), |f(t,u)|⩾q(t)|u|γ, γ>0 is a constant.It is obvious that (2)–(4) are special cases of (1). In the present paper, we will establish several oscillation criteria for the more general (1), which is of significance because (1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function ξ. Our results extend and unify a number of other existing results and handle the cases which are not covered by known criteria. Finally, two examples are demonstrated to illustrate the efficiency of our work.


## 2. Preliminaries

In the sequel, we denote byLξ[a,b] the set of Riemann-Stieltjes integrable functions on [a,σ(b))𝕋1 with respect to ξ, and we use the convention that ln0=-∞, e-∞=0.Lemma 1 (see [15]).
Suppose that (H4) holds. Let x:𝕋→ℝ. If xΔ exists for all sufficiently large t∈𝕋, then (x(δ(t)))Δ=xΔ(δ(t))δΔ(t) for all sufficiently large t∈𝕋.Lemma 2 (see [12]).
Assume thatx(t) is Δ-differentiable and eventually positive or eventually negative, then
(5)(xα(t))Δ=α{∫01‍[(1-h)x(t)+hx(σ(t))]α-1dh}xΔ(t).Lemma 3 (see [16]).
Suppose thatX and Y are nonnegative, then
(6)γXYγ-1-Xγ⩽(γ-1)Yγ,γ>1,
where equality holds if and only if X=Y.Lemma 4 (see [17]).
Letu(t)∈Crd([a,b]𝕋1,ℝ) and η(t)∈Lξ[a,b] satisfy u(t)⩾0(≢0), η(t)>0 on [a,b]𝕋1, and
(7)∫aσ(b)‍η(s)Δξ(s)=1.
Then,
(8)∫aσ(b)‍η(s)u(s)Δξ(s)⩾exp(∫aσ(b)‍η(s)ln[u(s)]Δξ(s)).Lemma 5 (see [17]).
Letτ(t)∈Crd([t0,∞)𝕋,[t0,∞)𝕋) satisfying 0⩽τ(t)⩽t, and τt0=inf{τ(t):t∈[t0,∞)𝕋}. Assume x(t)∈Crd([τt0,∞)𝕋,(0,∞)) such that r(t)|xΔ(t)|α-1xΔ(t) is nonincreasing on [t0,∞)𝕋, where r(t)∈Crd([t0,∞)𝕋,(0,∞)), α>0 is a constant. Then,
(9)x(τ(t))x(σ(t))⩾R(τ(t),τt0)R(σ(t),τt0).


## 3.  Main Results

Theorem 6.
Assume that(H1)–(H6) hold. If there exist a function ϕ(t)∈Crd1([t0,∞)𝕋,(0,∞)) and a function η(s)∈Lξ[a,b] such that η(s)>0 on [a,b]𝕋1,
(10)∫aσ(b)‍η(s)Δξ(s)=1,(11)∫aσ(b)‍η(s)θ(s)Δξ(s)=α,(12)lim¯s→∞∫t0s‍(M(t)-1(α+1)α+1r(δ(t))(ϕΔ(t))+α+1ϕα(t)(δΔ(t))α)Δt=∞,
where(13)M(t)=ϕ(t)p¯(t)+ϕ(t)×exp([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]∫aσ(b)‍η(s)+ϕ(t)exp×ln([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)η-1(s)k(t,s)+ϕ(t)exp×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s))+ϕ(t)exp×Δξ(s)[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)),p¯(t)=q(t)[1-p(δ(t))]α,(ϕΔ(t))+=max{ϕΔ(t),0},then (1) is oscillatory.Proof.
Suppose that (1) has a nonoscillatory solution x(t), then there exists T0(∈𝕋)⩾t0 such that x(t)≠0 for all t∈[T0,∞)𝕋. Without loss of generality, we assume that x(t)>0, x(τ(t))>0, x(δ(t))>0, and x(g(t,s))>0 for t∈[T0,∞)𝕋, s∈[a,b]𝕋1, because a similar analysis holds for x(t)<0, x(τ(t))<0, x(δ(t))<0, and x(g(t,s))<0. Then, from (1), (H2), and (H5), we get
(14)Z(t)⩾x(t)>0,(r(t)|ZΔ(t)|α-1ZΔ(t))Δ⩽0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt∈[T0,∞)𝕋.
Therefore, r(t)|ZΔ(t)|α-1ZΔ(t) is a nonincreasing function, and ZΔ(t) is eventually of one sign.
We claim that(15)ZΔ(t)>0orZΔ(t)=0,t∈[T0,∞)𝕋.
Otherwise, if there exists a t1(∈𝕋)⩾T0 such that ZΔ(t)<0 for t∈[t1,∞)𝕋, then, from (14), there exists some positive constant K such that
(16)-r(t)(-ZΔ(t))α⩽-K,t∈[t1,∞)𝕋,
that is,
(17)-ZΔ(t)⩾(Kr(t))1/α,t∈[t1,∞)𝕋,
and integrating the above inequality from t1 to t, we have
(18)Z(t)⩽Z(t1)-K1/α(R(t,t0)-R(t1,t0)).
Letting t→∞, from (H2), we get limt→∞Z(t)=-∞, which contradicts (14). Thus, we have proved (15).
We choose someT1(∈𝕋)⩾T0 such that δ(t)⩾T0 for t∈[T1,∞)𝕋. Therefore, from (14), (15), and the fact δ(t)⩽σ(t), we have that
(19)r(σ(t))(ZΔ(σ(t)))α⩽r(δ(t))(ZΔ(δ(t)))α,kkkkkkkkkkkkkkkkkkkkkkkkkit∈[T1,∞)𝕋,
which follows that
(20)ZΔ(δ(t))⩾ZΔ(σ(t))(r(σ(t))r(δ(t)))1/α,t∈[T1,∞)𝕋.
On the other hand, from (1), (H5), and (15), we obtain
(21)(r(t)(ZΔ(t))α)Δ+q(t)(Z(δ(t))-p(δ(t))x(τ(δ(t))))α+∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)⩽0,t∈[T1,∞)𝕋.
Notice (15) and the fact Z(t)⩾x(t), we get
(22)(r(t)(ZΔ(t))α)Δ+p¯(t)Zα(δ(t))+∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)⩽0,t∈[T1,∞)𝕋,
where p¯(t)=q(t)[1-p(δ(t))]α.
Define(23)w(t)=ϕ(t)r(t)(ZΔ(t))αZα(δ(t)),fort∈[T1,∞)𝕋.
Obviously, w(t)>0. From (22), (23), it follows that
(24)wΔ(t)=ϕ(t)Zα(δ(t))(r(t)(ZΔ(t))α)Δ+ϕΔ(t)Zα(δ(t))-ϕ(t)(Zα(δ(t)))ΔZα(δ(t))Zα(δ(σ(t)))×r(σ(t))(ZΔ(σ(t)))α⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)(Zα(δ(t)))Δr(σ(t))(ZΔ(σ(t)))αZα(δ(t))Zα(δ(σ(t))).
Now, we consider the following two cases.
In the first case,α⩾1. By (15), (H4), and Lemmas 1 and 2, we have
(25)(Zα(δ(t)))Δ=α{∫01‍[(1-h)Z(δ(t))+hZ(δ(σ(t)))]α-1dh}×(Z(δ(t)))Δ⩾α(Z(δ(t)))α-1ZΔ(δ(t))δΔ(t).
From (H4), (20), (23)–(25), and the fact that Z(t) is nondecreasing, we obtain
(26)wΔ(t)⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)α(Z(δ(t)))α-1ZΔ(δ(t))δΔ(t)r(σ(t))(ZΔ(σ(t)))αZα(δ(t))Zα(δ(σ(t)))⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)αZΔ(δ(t))δΔ(t)r(σ(t))(ZΔ(σ(t)))αZα+1(δ(σ(t)))⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)r(σ(t))(ZΔ(σ(t)))α+1Zα+1(δ(σ(t)))(r(σ(t))r(δ(t)))1/α=-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)).
In the second case,0<α<1. By (15), (H4), and Lemmas 1 and 2, we get
(27)(Zα(δ(t)))Δ=α{∫01‍[(1-h)Z(δ(t))+hZ(δ(σ(t)))]α-1dh}×(Z(δ(t)))Δ⩾α(Z(δ(σ(t))))α-1ZΔ(δ(t))δΔ(t).
From (H4), (20), (23)-(24), (27), and the fact that Z(t) is nondecreasing, we have(28)wΔ(t)⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)α(Z(δ(σ(t))))α-1ZΔ(δ(t))δΔ(t)r(σ(t))(ZΔ(σ(t)))αZα(δ(t))Zα(δ(σ(t)))⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-ϕ(t)αZΔ(δ(t))δΔ(t)r(σ(t))(ZΔ(σ(t)))αZα+1(δ(σ(t)))⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)).Therefore, for α>0, from (26) and (28), we get
(29)wΔ(t)⩽-ϕ(t)p¯(t)-ϕ(t)Zα(δ(t))∫aσ(b)‍k(t,s)(x(g(t,s)))θ(s)Δξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)).
On the other hand, it is obvious that the conditions in Lemma 5 are satisfied with x(t), τ(t), p(t) replaced by Z(t), g(t,s), and r(t), respectively. So, we have
(30)Z(g(t,s))Z(σ(t))⩾R(g(t,s),gT1)R(σ(t),gT1),
in the view of x(t)⩾(1-p(t))Z(t), we get
(31)Z(δ(t))⩽Z(σ(t))⩽R(σ(t),gT1)R(g(t,s),gT1)Z(g(t,s))⩽R(σ(t),gT1)R(g(t,s),gT1)x(g(t,s))1-p(g(t,s)).
From (29) and (31), we obtain
(32)wΔ(t)⩽-ϕ(t)p¯(t)-ϕ(t)∫aσ(b)‍k(t,s)-ϕ(t)∫aσ(b)‍×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)-ϕ(t)∫aσ(b)‍×[Z(σ(t))]θ(s)-αΔξ(s)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)).
By (10) and (11), we have
(33)∫aσ(b)‍η(s)[θ(s)-α]Δξ(s)=0.
Therefore, by Lemma 4 and (33), we have that for t∈[T1,∞)𝕋(34)∫aσ(b)‍k(t,s)[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)∫aσ(b)‍×[Z(σ(t))]θ(s)-αΔξ(s)=∫aσ(b)‍η(s)η-1(s)k(t,s)×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)×[Z(σ(t))]θ(s)-αΔξ(s)⩾exp([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)∫aσ(b)‍η(s)×ln([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)η-1(s)k(t,s)×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)×[Z(σ(t))]θ(s)-α[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s))Δξ(s))=exp([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)∫aσ(b)‍η(s)×ln([R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)η-1(s)k(t,s)×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s))×Δξ(s)[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s))×exp(ln(Z(σ(t)))∫aσ(b)‍η(s)[θ(s)-α]Δξ(s))=exp(×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)∫aσ(b)‍η(s)×ln(×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s)η-1(s)k(t,s)×[R(g(t,s),gT1)(1-p(g(t,s)))R(σ(t),gT1)]θ(s))×Δξ(s)∫aσ(b)‍).Substituting (34) into (32), we obtain
(35)wΔ(t)⩽-M(t)+ϕΔ(t)ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t))⩽-M(t)+(ϕΔ(t))+ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)),
where M(t) and (ϕΔ(t))+ are defined by (13).
Taking(36)X=[αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/α]α/(α+1)w(σ(t)),γ=α+1α,Y=αα/(α+1)(α+1)α[r(δ(t))(ϕΔ(t))+α+1ϕα(t)(δΔ(t))α]α/(α+1),
by Lemma 3 and (35), we obtain
(37)wΔ(t)⩽-M(t)+1(α+1)α+1r(δ(t))(ϕΔ(t))+α+1ϕα(t)(δΔ(t))α.
Integrating above inequality (37) from T1 to t, we have
(38)w(t)⩽w(T1)-∫T1t‍(M(s)-1(α+1)α+1r(δ(s))(ϕΔ(s))+α+1ϕα(s)(δΔ(s))α)Δs.⩽w(T1)+∫t0T1‍M(s)Δs-∫t0t‍(M(s)-1(α+1)α+1r(δ(s))(ϕΔ(s))+α+1ϕα(s)(δΔ(s))α)Δs.
Since w(t)>0 for t>T1, we have
(39)∫t0t‍(M(s)-1(α+1)α+1r(δ(s))(ϕΔ(s))+α+1ϕα(s)(δΔ(s))α)Δs⩽w(T1)+∫t0T1‍M(s)Δs-w(t)⩽w(T1)+∫t0T1‍M(s)Δs,
which contradicts (12). This completes the proof of Theorem 6.Remark 7.
If we taker(x,s)=0 and use the convention that ln0=-∞, e-∞=0, then Theorems 6 reduces to [15, Theorems 3.1]. If furthermore α⩾1 is a quotient of odd positive integer, then Theorem 6 reduces to [14, Theorem 3.1].Remark 8.
The functionη(t) satisfying (10) and (11) in Theorem 6 can be constructed explicitly for any nondecreasing function ξ. In fact, if we assume that θ(t),1/θ(t)∈Lξ[a,b]𝕋1, and let h=sup{s∈(a,b)𝕋1:θ(s)⩽α},
(40)η1(s)={αθ(s)(∫σ(h)σ(b)‍Δξ(s))-1,s∈[σ(h),b]𝕋,0,s∈[a,σ(h))𝕋,η2(s)={0,s∈[σ(h),b]𝕋,αθ(s)(∫aσ(h)‍Δξ(s))-1,s∈[a,σ(h))𝕋.
It is easy to see that ηi(s)∈Lξ[a.b]𝕋1 and
(41)∫aσ(b)‍θ(s)ηi(s)Δξ(s)=α,i=1,2.
Moreover,(42)∫aσ(b)‍η1(s)Δξ(s)=m1<1,∫aσ(b)‍η2(s)Δξ(s)=m2>1.
Let
(43)η(s,l)=(1-l)η1(s)+lη2(s),fors∈[a,b]𝕋,l∈[0,1].
Then, we obtain that
(44)∫aσ(b)‍θ(s)η(s,l)Δξ(s)=α,∫aσ(b)‍η(s,l)Δξ(s)=(1-l)∫aσ(b)‍η1(s)Δξ(s)+l∫aσ(b)‍η2(s)Δξ(s)=(1-l)m1+lm2.
By the continuous dependence of η(s,l) on l, there exists l*∈(0,1) such that η(s)≔η(s,l*) satisfies
(45)∫aσ(b)‍η(s)Δξ(s)=∫aσ(b)‍η(s,l*)Δξ(s)=1.Remark 9.
Set𝕋1=ℕ, a=1, b=n for n∈ℕ, andξ(s)=s;θ(s)=βs, (s=1,2,…,n) satisfying β1>β2>⋯>βm>α>βm+1>⋯>βn;k(t,s)=qs(t), s=1,2,…,n;g(t,s)=τs(t), s=1,2,…,n;Then, (1) reduces to
(46)(r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))+∑j=1n‍qj(t)|x(τj(t))|βj-1x(τj(t))=0.
So, if we take 𝕋 for some peculiar cases in Theorem 6, we can obtain various results. For example, if we take 𝕋=ℝ, p(t)=0, |f(t,u)|=q(t)|uα|, and n=2 in (46), then Theorem 6 generalizes the results by [18, Theorem 2].
Next, we use the general weighted functions from the classϜ which will be extensively used in the sequel.
Let𝔻≡{(t,s)∈[t0,∞)𝕋×[t0,∞)𝕋:t⩾s⩾t0}, we say that a continuous function H(t,s)∈Crd(𝔻,ℝ) belongs to the class Ϝ if(i)
H(t,t)=0 for t∈[t0,∞)𝕋 and H(t,s)>0 for t>s⩾t0 where t,s∈[t0,∞)𝕋;(ii)
H(t,s) has a nonpositive right-dense continuous Δ-partial derivative HΔs(t,s) with respect to the second variable.Theorem 10.
Assume that(H1)–(H6) hold. If there exist functions H(t,s)∈Ϝ, φ(t)∈Crd1([t0,∞)𝕋,(0,∞)), η(t)∈Lξ[a,b] such that η(s)>0 on [a,b]𝕋1, (10) and (11) hold, and
(47)lim¯t→∞1H(t,t0)∫t0t‍[H(t,s)M(s)-U(t,s)]Δs=∞,
where
(48)U(t,s)=1(α+1)α+1Φ+α+1(t,s)r(δ(s))(ϕ(σ(s)))α+1(H(t,s))αϕα(s)(δΔ(s))α,(49)Φ+(t,s)=max{HΔs(t,s)+H(t,s)(ϕΔ(s))+ϕ(σ(s)),0},M(t) and (ϕΔ(t))+ are defined as in Theorem 6, then(1) is oscillatory.Proof.
We proceed as in the proof of Theorem6 to have (35). From (35), we obtain
(50)M(t)⩽-wΔ(t)+(ϕΔ(t))+ϕ(σ(t))w(σ(t))-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αw(α+1)/α(σ(t)),-αϕ(t)δΔ(t)(ϕ(σ(t)))1+1/α(r(δ(t)))1/αt∈[T1,∞)𝕋.
Multiplying (50) (with t replaced by s) by H(t,s), integrating it with respect to s from T1 to t for t∈(T1,∞)𝕋, and using integration by parts and (i)-(ii), we get
(51)∫T1t‍H(t,s)M(s)Δs⩽-∫T1t‍H(t,s)wΔ(s)Δs+∫T1t‍H(t,s)(ϕΔ(s))+ϕ(σ(s))w(σ(s))Δs-∫T1t‍H(t,s)αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α×w(α+1)/α(σ(s))Δs=H(t,T1)w(T1)+∫T1t‍HΔs(t,s)w(σ(s))Δs+∫T1t‍H(t,s)(ϕΔ(s))+ϕ(σ(s))w(σ(s))Δs-∫T1t‍H(t,s)αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α×w(α+1)/α(σ(s))Δs=H(t,T1)w(T1)+∫T1t‍[(HΔs(t,s)+H(t,s)(ϕΔ(s))+ϕ(σ(s)))w(σ(s))-H(t,s)αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α×w(α+1)/α(σ(s))αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α]Δs⩽H(t,T1)w(T1)+∫T1t‍[αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/αΦ+(t,s)w(σ(s))-H(t,s)αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α×w(α+1)/α(σ(s))αϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α]Δs,
where Φ+(t,s) is defined as in (49).
Taking(52)X=[αH(t,s)ϕ(s)δΔ(s)(ϕ(σ(s)))1+1/α(r(δ(s)))1/α]α/(α+1)w(σ(s)),γ=α+1α,Y=αα/(α+1)(α+1)α[Φ+α+1(t,s)r(δ(s))(ϕ(σ(s)))α+1(H(t,s))αϕα(s)(δΔ(s))α]α/(α+1),
by Lemma 3 and (51), we obtain
(53)∫T1t‍H(t,s)M(s)Δs⩽H(t,T1)w(T1)+∫T1t‍[1(α+1)α+1Φ+α+1(t,s)r(δ(s))(ϕ(σ(s)))α+1(H(t,s))αϕα(s)(δΔ(s))α]Δs⩽H(t,t0)w(T1)+∫T1t‍U(t,s)Δs,
where U(t,s) is defined as in (48).
Then, it follows that(54)∫T1t‍[H(t,s)M(s)-U(t,s)]Δs⩽H(t,t0)w(T1).
Thus, from (54), we get
(55)∫t0t‍[H(t,s)M(s)-U(t,s)]Δs=(∫t0T1‍+∫T1t‍)[H(t,s)M(s)-U(t,s)]Δs⩽∫t0T1‍[H(t,s)M(s)]Δs+H(t,t0)w(T1)⩽H(t,t0)∫t0T1‍M(s)Δs+H(t,t0)w(T1).
Therefore,(56)lim¯t→∞1H(t,t0)∫t0t‍[H(t,s)M(s)-U(t,s)]Δs⩽∫t0T1‍M(s)Δs+w(T1)<∞,
which contradicts (47). This completes the proof of Theorem 10.Remark 11.
In the literature, there are so many results for second-order nonlinear neutral functional dynamic equation; however, to the best of our knowledge, there is no work done attempting to study the neutral functional dynamic equation with an infinite number of nonlinear terms. Hence, our paper seems to be the first one dealing with this untouched problem. Our results not only unify the existing results in the literature, but also extend the existing results to a wider class of dynamic equations.


## 4. Examples

Example 1.
Consider the following dynamic equation:(57)[|(x(t)+11+t2x(τ(t)))Δ|1/2(x(t)+11+t2x(τ(t)))Δ]Δ+1t2(1+q02t2q02t2)s∫12‍|x(g(t,s))|s-1x(g(t,s))Δs=0,t∈𝕋.
In (57), r(t)=1, α=3/2, p(t)=1/(1+t2), f(t,u)=0, a=1, b=2, θ(s)=s, ξ(s)=s, and q0>1 is a constant.
If𝕋=q0ℤ¯={q0n:n∈ℤ}∪{0} and τ(t)=t/q0k0, δ(t)=t/q0, g(t,s)=q0t, and k(t,s)=(1/t2)((1+q02t2)/q02t2)s, where k0 is an arbitrary positive integer, then δΔ(t)=1/q0. We can choose η(t)=1, ϕ(t)=t. Then, it is easy to get that M(t)=1/t, and therefore,
(58)lim¯s→∞∫t0s‍(M(t)-1(α+1)α+1r(δ(t))(ϕΔ(t))+α+1ϕα(t)(δΔ(t))α)Δt=lim¯s→∞∫t0s‍(1t-q03/2(5/2)5/2t3/2)Δt=∞.
Hence, by Theorem 6, (57) is oscillatory.Example 2.
Consider on𝕋=ℝ the following differential equation:
(59)[t1/2|(x(t)+(1-11+t2)x(t-1))′|1/2×(x(t)+(1-11+t2)x(t-1))′t1/2|(x(t)+(1-11+t2)x(t-1))′|1/2]′+βt2(4+t24)3/2|x(t2)|1/2x(t2)+λ(1+2t2)3st2∫01‍|x(t)|3s-1x(t)ds=0.
In (59), r(t)=t1/2, α=3/2, p(t)=1-1/(1+t2),  τ(t)=t-1,  δ(t)=t/2,  β,λ>0 are constants, f(t,x(δ(t))=(β/t2)((4+t2)/4)3/2|x(t/2)|1/2x(t/2), q(t)=(β/t2)((4+t2)/4)3/2,  k(t,s)=λ(1+2t2)3s/t2,    g(t,s)=t,  a=0,  b=1,  θ(s)=3s, and  ξ(s)=s.
We can chooseη(t)=1, ϕ(t)=t. Then, it is easy to get that M(t)=(β+λ)/t, δ′(t)=1/2, and therefore, from Theorem 6,
(60)lim¯s→∞∫t0s‍(M(t)-1(α+1)α+1r(δ(t))(ϕ′(t))+α+1ϕα(t)(δ′(t))α)dt=lim¯s→∞∫t0s‍(β+λt-27/255/2t)dt=(β+λ-27/255/2)lim¯s→∞∫t0s‍1tdt=∞,ifβ+λ>27/255/2.
Hence, by Theorem 6, (59) is oscillatory if β+λ>27/2/55/2.



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*Source: 530457-2013-12-02.xml*