# Nonoscillation of Second-Order Dynamic Equations with Several Delays

Authors: Elena Braverman; Başak Karpuz
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Existence of nonoscillatory solutions for the second-order dynamic equation(A0xΔ)Δ(t)+∑i∈[1,n]ℕAi(t)x(αi(t))=0 for t∈[t0,∞)𝕋  is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case
A0(t)≡1 for t∈[t0,∞)ℝ and for second-order nondelay difference equations (αi(t)=t+1 for t∈[t0,∞)ℕ). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary A0 and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.


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## Body

## 1. Introduction

This paper deals with second-order linear delay dynamic equations on time scales. Differential equations of the second order have important applications and were extensively studied; see, for example, the monographs of Agarwal et al. [1], Erbe et al. [2], Győri and Ladas [3], Ladde et al. [4], Myškis [5], Norkin [6], Swanson [7], and references therein. Difference equations of the second order describe finite difference approximations of second-order differential equations, and they also have numerous applications.We study nonoscillation properties of these two types of equations and some of their generalizations. The main result of the paper is that under some natural assumptions for a delay dynamic equation the following four assertions are equivalent: nonoscillation of solutions of the equation on time scales and of the corresponding dynamic inequality, positivity of the fundamental function, and the existence of a nonnegative solution for a generalized Riccati inequality. The equivalence of oscillation properties of the differential equation and the corresponding differential inequality can be applied to obtain new explicit nonoscillation and oscillation conditions and also to prove some well-known results in a different way. A generalized Riccati inequality is used to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the well-known Sturm-Picone comparison theorem for a second-order ordinary differential equation; see  [7, Section 1.1]. Applying positivity of the fundamental function, positive solutions of two equations can be compared. There are many results of this kind for delay differential equations of the first-order and only a few for second-order equations. Myškis [5] obtained one of the first comparison theorems for second-order differential equations. The results presented here are generalizations of known nonoscillation tests even for delay differential equations (when the time scale is the real line).The paper also contains conditions on the initial function and initial values which imply that the corresponding solution is positive. Such conditions are well known for first-order delay differential equations; however, to the best of our knowledge, the only paper concerning second-order equations is [8].From now on, we will without furthermore mentioning suppose that the time scale𝕋 is unbounded from above. The purpose of the present paper is to study nonoscillation of the delay dynamic equation(1.1)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=f(t)fort∈[t0,∞)T,
where n∈ℕ, t0∈𝕋, f∈Crd([t0,∞)𝕋,ℝ) is the forcing term, A0∈Crd([t0,∞)𝕋,ℝ+), and for all i∈[1,n]ℕ, Ai∈Crd([t0,∞)𝕋,ℝ) is the coefficient corresponding to the function αi, where αi≤σ on [t0,∞)𝕋.In this paper, we follow the method employed in [8] for second-order delay differential equations. The method can also be regarded as an application of that used in [9] for first-order dynamic equations.As a special case, the results of the present paper allow to deduce nonoscillation criteria for the delay differential equation(1.2)(A0x′)′(t)+∑i∈[1,n]NAi(t)x(αi(t))=0fort∈[t0,∞)R,
in the case A0(t)≡1 for t∈[t0,∞)ℝ, they coincide with theorems in [8]. The case of a “quickly growing” function A0 when the integral of its reciprocal can converge is treated separately.Let us recall some known nonoscillation and oscillation results for the ordinary differential equations(1.3)(A0x′)′(t)+A1(t)x(t)=0fort∈[t0,∞)R,(1.4)x′′(t)+A1(t)x(t)=0fort∈[t0,∞)R,
where A1 is nonnegative, which are particular cases of (1.2) with n=1, α1(t)=t, and A0(t)≡1 for all t∈[t0,∞)ℝ.In [10], Leighton proved the following well-known oscillation test for (1.4); see [10, 11].Theorem A (see [10]).
Assume that(1.5)∫t0∞1A0(η)dη=∞,∫t0∞A1(η)dη=∞,
then (1.3) is oscillatory.This result for (1.4) was obtained by Wintner in [12] without imposing any sign condition on the coefficient A1.In [13], Kneser proved the following result.Theorem B (see [13]).
Equation (1.4) is nonoscillatory if t2A1(t)≤1/4 for all t∈[t0,∞)ℝ, while oscillatory if t2A1(t)>λ0/4 for all t∈[t0,∞)ℝ and some λ0∈(1,∞)𝕋.In [14], Hille proved the following result, which improves the one due to Kneser; see also [14–16].Theorem C (see [14]).
Equation (1.4) is nonoscillatory if
(1.6)t∫t∞A1(η)dη≤14∀t∈[t0,∞)R,
while it is oscillatory if
(1.7)t∫t∞A1(η)dη>λ04∀t∈[t0,∞)Randsomeλ0∈(1,∞)R.Another particular case of (1.1) is the second-order delay difference equation(1.8)Δ(A0Δx)(k)+∑i∈[1,n]NAi(k)x(αi(k))=0for∈[k0,∞)N,
to the best of our knowledge, there are very few nonoscillation results for this equation; see, for example, [17]. However, nonoscillation properties of the nondelay equations(1.9)Δ(A0Δx)(k)+A1(k)x(k+1)=0fork∈[k0,∞)N,(1.10)Δ2x(k)+A1(k)x(k+1)=0fork∈[k0,∞)N
have been extensively studied in [1, 18–22]; see also [23]. In particular, the following result is valid.Theorem D.
Assume that(1.11)∑j=k0∞A1(j)=∞,
then (1.10) is oscillatory.The following theorem can be regarded as the discrete analogue of the nonoscillation result due to Kneser.Theorem E.
Assume thatk(k+1)A1(k)≤1/4 for all k∈[k0,∞)ℕ, then (1.10) is nonoscillatory.Hille's result in [14] also has a counterpart in the discrete case. In [22], Zhou and Zhang proved the nonoscillation part, and in [24], Zhang and Cheng justified the oscillation part which generalizes Theorem E.Theorem F (see [22, 24]).
Equation (1.10) is nonoscillatory if
(1.12)k∑j=k∞A1(j)≤14∀k∈[k0,∞)N,
while is oscillatory if
(1.13)k∑j=k∞A1(j)>λ04∀k∈[k0,∞)Nandsomeλ0∈(1,∞)R.In [23], Tang et al. studied nonoscillation and oscillation of the equation(1.14)Δ2x(k)+A1(k)x(k)=0fork∈[k0,∞)N,
where {A1(k)} is a sequence of nonnegative reals and obtained the following theorem.Theorem G (see [23]).
Equation (1.14) is nonoscillatory if (1.12) holds, while is it oscillatory if (1.13) holds.These results together with some remarks on theq-difference equations will be discussed in Section 7. The readers can find some nonoscillation results for second-order nondelay dynamic equations in the papers [20, 25–29], some of which generalize some of those mentioned above.The paper is organized as follows. In Section2, some auxiliary results are presented. In Section 3, the equivalence of the four above-mentioned properties is established. Section 4 is dedicated to comparison results. Section 5 includes some explicit nonoscillation and oscillation conditions. A sufficient condition for existence of a positive solution is given in Section 6. Section 7 involves some discussion and states open problems. Section 7 as an appendix contains a short account on the fundamentals of the time scales theory.


## 2. Preliminary Results

Consider the following delay dynamic equation:(2.1)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=f(t)fort∈[t0,∞)T,x(t0)=x1,xΔ(t0)=x2,x(t)=φ(t)fort∈[t-1,t0)T,
where n∈ℕ, 𝕋 is a time scale unbounded above, t0∈𝕋, x1,x2∈ℝ are the initial values, φ∈Crd([t-1,t0)𝕋,ℝ) is the initial function, such that φ has a finite left-sided limit at the initial point t0 provided that it is left dense, f∈Crd([t0,∞)𝕋,ℝ) is the forcing term, A0∈Crd([t0,∞)𝕋,ℝ+), and for all i∈[1,n]ℕ, Ai∈Crd([t0,∞)𝕋,ℝ) is the coefficient corresponding to the function αi∈Crd([t0,∞)𝕋,𝕋), which satisfies αi(t)≤σ(t) for all t∈[t0,∞)𝕋 and limt→∞αi(t)=∞. Here, we denoted(2.2)αmin⁡(t):=min⁡i∈[1,n]N{αi(t)}fort∈[t0,∞)T,t-1:=inf⁡t∈[t0,∞)T{αmin⁡(t)},
then t-1 is finite, since αmin asymptotically tends to infinity.Definition 2.1.
A functionx:[t-1,∞)𝕋→ℝ with x∈Crd1([t0,∞)𝕋,ℝ) and a derivative satisfying A0xΔ∈Crd1([t0,∞)𝕋,ℝ) is called a solution of (2.1) if it satisfies the equation in the first line of (2.1) identically on [t0,∞)𝕋 and also the initial conditions in the second line of (2.1).For a given functionφ∈Crd([t-1,t0)𝕋,ℝ) with a finite left-sided limit at the initial point t0 provided that it is left-dense and x1,x2∈ℝ, problem (2.1) admits a unique solution satisfying x=φ on [t-1,t0)𝕋 with x(t0)=x1 and xΔ(t0)=x2 (see [30] and [31, Theorem 3.1]).Definition 2.2.
A solution of (2.1) is called eventually positive if there exists s∈[t0,∞)𝕋 such that x>0 on [s,∞)𝕋, and if (-x) is eventually positive, then x is called eventually negative. If (2.1) has a solution which is either eventually positive or eventually negative, then it is called nonoscillatory. A solution, which is neither eventually positive nor eventually negative, is called oscillatory, and (2.1) is said to be oscillatory provided that every solution of (2.1) is oscillatory.For convenience in the notation and simplicity in the proofs, we suppose that functions vanish out of their specified domains, that is, letf:D→ℝ be defined for some D⊂ℝ, then it is always understood that f(t)=χD(t)f(t) for t∈ℝ, where χD is the characteristic function of the set D⊂ℝ defined by χD(t)≡1 for t∈D and χD(t)≡0 for t∉D.Definition 2.3.
Lets∈𝕋 and s-1:=inft∈[s,∞)𝕋{αmin(t)}. The solutions 𝒳1=𝒳1(·,s) and 𝒳2=𝒳2(·,s) of the problems
(2.3)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=0fort∈[s,∞)T,xΔ(s)=1A0(s),x(t)≡0fort∈[s-1,s]T,(2.4)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=0fort∈[s,∞)T,xΔ(s)=0,x(t)=χ{s}(t)fort∈[s-1,s]T,
which satisfy 𝒳1(·,s),𝒳2(·,s)∈Crd1([s,∞)𝕋,ℝ), are called the first fundamental solution and the second fundamental solution of (2.1), respectively.The following lemma plays the major role in this paper; it presents a representation formula to solutions of (2.1) by the means of the fundamental solutions 𝒳1 and 𝒳2.Lemma 2.4.
Letx be a solution of (2.1), then x can be written in the following form:
(2.5)x(t)=x2X1(t,t0)+x1X2(t,t0)+∫t0tX1(t,σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη
for t∈[t0,∞)𝕋.Proof.
Fort∈[t-1,∞)𝕋, let
(2.6)y(t):={∫t0tX1(t,σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δηfort∈[t0,∞)T,φ(t)fort∈[t1,t0)T.
We recall that 𝒳1(·,t0) and 𝒳2(·,t0) solve (2.3) and (2.4), respectively. To complete the proof, let us demonstrate that y solves
(2.7)(A0yΔ)Δ(t)+∑i∈[1,n]NAi(t)y(αi(t))=f(t)fort∈[t0,∞)T,y(t0)=0,yΔ(t0)=0,y(t)=φ(t)fort∈[t-1,t0)T.
This will imply that the function z defined by z:=x2𝒳1(·,t0)+x1𝒳2(·,t0)+y on [t0,∞)𝕋 is a solution of (2.1). Combining this with the uniqueness result in [31, Theorem 3.1] will complete the proof. For all t∈[t0,∞)𝕋, we can compute that
(2.8)yΔ(t)=∫t0tX1Δ(t,σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη+X1(σ(t),σ(t))[f(t)-∑i∈[1,n]NAi(t)φ(αi(t))]=∫t0tX1Δ(t,σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη.
Therefore, y(t0)=0, yΔ(t0)=0, and y=φ on [t-1,t0)𝕋, that is, y satisfies the initial conditions in (2.7). Differentiating yΔ after multiplying by A0 and using the properties of the first fundamental solution 𝒳1, we get
(2.9)(A0yΔ)Δ(t)=∫t0t(A0X1Δ(⋅,σ(η)))Δ(t)[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη+A0σ(t)X1Δ(σ(t),σ(t))[f(t)-∑i∈[1,n]NAi(t)φ(αi(t))]=-∑j∈[1,n]NAj(t)∫t0αj(t)X1(αj(t),σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη-∑i∈[1,n]NAi(t)φ(αi(t))+f(t)
for all t∈[t0,∞)𝕋. For t∈[t0,∞)𝕋, set I(t)={i∈[1,n]ℕ:χ[t0,∞)𝕋(αi(t))=1} and J(t):={i∈[1,n]ℕ:χ[t-1,t0)𝕋(αi(t))=1}. Making some arrangements, for all t∈[t0,∞)𝕋, we find
(2.10)(A0yΔ)Δ(t)=-∑j∈I(t)Aj(t)∫t0αj(t)X1(αj(t),σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη-∑j∈J(t)Aj(t)∫t0αj(t)X1(αj(t),σ(η))[f(η)-∑i∈[1,n]NAi(η)φ(αi(η))]Δη-∑i∈[1,n]NAi(t)φ(αi(t))+f(t),
and thus
(2.11)(A0yΔ)Δ(t)=-∑j∈I(t)Aj(t)∫t0αj(t)X1(αj(t),σ(η))f(η)Δη-∑j∈J(t)Ai(t)φ(αi(t))+f(t)=-∑j∈I(t)Aj(t)y(αj(t))-∑j∈J(t)Aj(t)y(αj(t))+f(t),
which proves that y satisfies (2.7) on [t0,∞)𝕋 since I(t)∩J(t)=∅ and I(t)∪J(t)=[1,n]ℕ for each t∈[t0,∞)𝕋. The proof is therefore completed.Next, we present a result from [9] which will be used in the proof of the main result.Lemma 2.5 (see [9, Lemma 2.5]).
Lett0∈𝕋 and assume that K is a nonnegative Δ-integrable function defined on {(t,s)∈𝕋×𝕋:t∈[t0,∞)𝕋,s∈[t0,t]𝕋}. If f,g∈Crd([t0,∞)𝕋,ℝ) satisfy
(2.12)f(t)=∫t0tK(t,η)f(η)Δη+g(t)∀t∈[t0,∞)T,
then g(t)≥0 for all t∈[t0,∞)𝕋 implies f(t)≥0 for all t∈[t0,∞)𝕋.


## 3. Nonoscillation Criteria

Consider the delay dynamic equation(3.1)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=0fort∈[t0,∞)T
and its corresponding inequalities(3.2)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))≤0fort∈[t0,∞)T,(3.3)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))≥0fort∈[t0,∞)T.We now prove the following result, which plays a major role throughout the paper.Theorem 3.1.
Suppose that the following conditions hold:(A1)
A0∈Crd([t0,∞)𝕋,ℝ+),(A2)
fori∈[1,n]ℕ, Ai∈Crd([t0,∞)𝕋,ℝ0+),(A3)
fori∈[1,n]ℕ, αi∈Crd([t0,∞)𝕋,𝕋) satisfies αi(t)≤σ(t) for all t∈[t0,∞)𝕋 and limt→∞αi(t)=∞,then the following conditions are equivalent: (i)
the second-order dynamic equation (3.1) has a nonoscillatory solution,(ii)
the second-order dynamic inequality (3.2) has an eventually positive solution and/or (3.3) has an eventually negative solution,(iii)
there exist a sufficiently larget1∈[t0,∞)𝕋 and a function Λ∈Crd1([t1,∞)𝕋,ℝ) with Λ/A0∈ℛ+([t1,∞)𝕋,ℝ) satisfying the first-order dynamic Riccati inequality
(3.4)ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(t,αi(t))≤0∀t∈[t1,∞)T,(iv)
the first fundamental solution𝒳1 of (3.1) is eventually positive, that is, there exists a sufficiently large t1∈[t0,∞)𝕋 such that 𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋.Proof.
The proof follows the scheme: (i)⇒(ii)⇒(iii)⇒(iv)⇒(i).
(i)⇒(ii) This part is trivial, since any eventually positive solution of (3.1) satisfies (3.2) too, which indicates that its negative satisfies (3.3).
(ii)⇒(iii) Let x be an eventually positive solution of (3.2), then there exists t1∈[t0,∞)𝕋 such that x(t)>0 for all t∈[t1,∞)𝕋. We may assume without loss of generality that x(t1)=1 (otherwise, we may proceed with the function x/x(t1), which is also a solution since (3.2) is linear). Let
(3.5)Λ(t):=A0(t)xΔ(t)x(t)fort∈[t1,∞)T,
then evidently Λ∈Crd1([t1,∞)𝕋,ℝ) and
(3.6)1+μ(t)Λ(t)A0(t)=1+μ(t)xΔ(t)x(t)=xσ(t)x(t)>0∀t∈[t1,∞)T,
which proves that Λ/A0∈ℛ+([t1,∞)𝕋,ℝ). This implies that the exponential function eΛ/A0(·,t1) is well defined and is positive on the entire time scale [t1,∞)𝕋; see [32, Theorem 2.48]. From (3.5), we see that Λ satisfies the ordinary dynamic equation
(3.7)xΔ(t)=Λ(t)A0(t)x(t)fort∈[t1,∞)T,x(t1)=1,
whose unique solution is
(3.8)x(t)=eΛ/A0(t,t1)∀t∈[t1,∞)T,
see [32, Theorem 2.77]. Hence, using (3.8), for all t∈[t1,∞)𝕋, we get
(3.9)xΔ(t)=Λ(t)A0(t)eΛ/A0(t,t1),(A0xΔ)Δ(t)=(ΛeΛ/A0(⋅,t1))Δ(t)=ΛΔ(t)eΛ/A0(t,t1)+Λσ(t)eΛ/A0Δ(t,t1)=ΛΔ(t)eΛ/A0(t,t1)+1A0(t)Λσ(t)Λ(t)eΛ/A0(t,t1),
which gives by substituting into (3.2) and using [32, Theorem 2.36] that
(3.10)0≥ΛΔ(t)eΛ/A0(t,t1)+1A0(t)Λσ(t)Λ(t)eΛ/A0(t,t1)+∑i∈[1,n]NAi(t)eΛ/A0(αi(t),t1)=eΛ/A0(t,t1)[ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi(t)eΛ/A0(αi(t),t1)eΛ/A0(t,t1)]=eΛ/A0(t,t1)[ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(t,αi(t))]
for all t∈[t1,∞)𝕋. Since the expression in the brackets is the same as the left-hand side of (3.4) and eΛ/A0(·,t1)>0 on [t1,∞)𝕋, the function Λ is a solution of (3.4).
(iii)⇒(iv) Consider the initial value problem
(3.11)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=f(t)fort∈[t1,∞)T,xΔ(t1)=0,x(t)≡0fort∈[t-1,t1]T.
Denote
(3.12)g(t):=A0(t)xΔ(t)-Λ(t)x(t)fort∈[t1,∞)T,
where x is any solution of (3.11) and Λ is a solution of (3.4). From (3.12), we have
(3.13)xΔ(t)=Λ(t)A0(t)x(t)+g(t)A0(t)fort∈[t1,∞)T,x(t1)=0,
whose unique solution is
(3.14)x(t)=∫t1teΛ/A0(t,σ(η))g(η)A0(η)Δη∀t∈[t1,∞)T,
see [32, Theorem 2.77]. Now, for all t∈[t1,∞)𝕋, we compute that
(3.15)x(t)=e⊖(Λ/A0)(σ(t),t)[∫t1σ(t)eΛ/A0(σ(t),σ(η))g(η)A(η)Δη-μ(t)eΛ/A0(σ(t),σ(t))g(t)A0(t)]=A0(t)A0(t)+μ(t)Λ(t)[xσ(t)-μ(t)g(t)A0(t)]=1A0(t)+μ(t)Λ(t)[A0(t)xσ(t)-μ(t)g(t)],
and similarly
(3.16)x(αi(t))=e⊖(Λ/A0)(σ(t),αi(t))×[∫t1σ(t)eΛ/A0(σ(t),σ(η))g(η)A(η)Δη-∫αi(t)σ(t)eΛ/A0(σ(t),σ(η))g(η)A0(η)Δη]=e⊖(Λ/A0)(σ(t),αi(t))[xσ(t)-∫αi(t)σ(t)eΛ/A0(σ(t),σ(η))g(η)A0(η)Δη]=e⊖(Λ/A0)(σ(t),αi(t))xσ(t)-∫αi(t)σ(t)eΛ/A0(αi(t),σ(η))g(η)A0(η)Δη
for i∈[1,n]ℕ. From (3.12) and (3.15), we have
(3.17)(A0xΔ)Δ(t)=(Λx+g)Δ(t)=ΛΔ(t)xσ(t)+Λ(t)xΔ(t)+gΔ(t)=ΛΔ(t)xσ(t)+Λ2(t)A0(t)x(t)+Λ(t)A0(t)g(t)+gΔ(t)
for all t∈[t1,∞)𝕋. We substitute (3.14), (3.15), (3.16), and (3.17) into (3.11) and find that
(3.18)f(t)=[ΛΔ(t)xσ(t)+Λ2(t)A0(t)x(t)+∑i∈[1,n]NAi(t)x(αi(t))]+Λ(t)A0(t)g(t)+gΔ(t)=[ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(σ(t),αi(t))]xσ(t)-[μ(t)Λ2(t)A0(t)(A0(t)+μ(t)Λ(t))g(t)+∑i∈[1,n]NAi(t)∫αi(t)σ(t)eΛ/A0(αi(t),σ(η))g(η)A0(η)Δη]+Λ(t)A0(t)g(t)+gΔ(t)=[ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(σ(t),αi(t))]×[1+μ(t)Λ(t)A0(t)]∫t1σ(t)eΛ/A0(t,σ(η))g(η)A0(η)Δη-∑i∈[1,n]NAi(t)∫αi(t)σ(t)eΛ/A0(αi(t),σ(η))g(η)A0(η)Δη+Λ(t)A0(t)+μ(t)Λ(t)g(t)+gΔ(t)
for all t∈[t1,∞)𝕋. Then, (3.18) can be rewritten as
(3.19)gΔ(t)=-Λ(t)A0(t)+μ(t)Λ(t)g(t)+Υ(t)∫t1σ(t)eΛ/A0(t,σ(η))g(η)A0(η)Δη+∑i∈[1,n]NAi(t)∫αi(t)σ(t)eΛ/A0(αi(t),σ(η))g(η)A0(η)Δη+f(t)
for all t∈[t1,∞)𝕋, where
(3.20)Υ(t):=-[1+μ(t)Λ(t)A0(t)][ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(σ(t),αi(t))]
for t∈[t1,∞)𝕋. We now show that Υ≥0 on [t1,∞)𝕋. Indeed, by using (3.4) and the simple useful formula (A.2), we get
(3.21)Υ(t)=-[(1+μ(t)Λ(t)A0(t))ΛΔ(t)+1A0(t)Λ2(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(t,αi(t))]=-[ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(t,αi(t))]≥0
for all t∈[t1,∞)𝕋. On the other hand, from (3.11) and (3.12), we see that g(t1)=0. Then, by [32, Theorem 2.77], we can write (3.19) in the equivalent form
(3.22)g=Hg+hon⁡[t1,∞)T,
where, for t∈[t1,∞)𝕋, we have defined
(3.23)(Hg)(t):=∫t1te-Λ/(A0+μΛ)(t,σ(η))[∑i∈[1,n]NAi(η)∫αi(η)σ(η)eΛ/A0(αi(η),σ(ζ))g(ζ)A0(ζ)ΔζΥ(η)∫t1σ(η)eΛ/A0(σ(η),σ(ζ))g(ζ)A0(ζ)Δζ+∑i∈[1,n]NAi(η)∫αi(η)σ(η)eΛ/A0(αi(η),σ(ζ))g(ζ)A0(ζ)Δζ]Δη,(3.24)h(t):=∫t1teΛ/A0(t,σ(ζ))f(η)Δη.
Note that Λ/A0∈ℛ+([t1,∞)𝕋,ℝ) implies -Λ/(A0+μΛ)∈ℛ+([t1,∞)𝕋,ℝ) (indeed, we have 1-μΛ/(A0+μΛ)=A0/(A0+μΛ)>0 on [t1,∞)𝕋), and thus the exponential function e⊖(Λ/A0)(·,t1) is also well defined and positive on the entire time scale [t1,∞)𝕋, see [32, Exercise 2.28]. Thus, f≥0 on [t1,∞)𝕋 implies h≥0 on [t1,∞)𝕋. For simplicity of notation, for s,t∈[t1,∞)𝕋, we let
(3.25)K1(t,s):=1A0(s)∫ste-Λ/(A0+μΛ)(t,σ(η))Υ(η)eΛ/A0(σ(η),σ(s))Δη,K2(t,s):=1A0(s)∫ste-Λ/(A0+μΛ)(t,σ(η))∑i∈[1,n]NAi(η)χ[αi(η),∞)T(s)eΛ/A0(σ(η),σ(s))Δη.
Using the change of integration order formula in [33, Lemma 1], for all t∈[t1,∞)𝕋, we obtain
(3.26)∫t1t∫t1σ(η)e-Λ/(A0+μΛ)(t,σ(η))Υ(η)eΛ/A0(σ(η),σ(ζ))g(ζ)A0(ζ)ΔζΔη=∫t1t∫ζte-Λ/(A0+μΛ)(t,σ(η))Υ(η)eΛ/A0(σ(η),σ(ζ))g(ζ)A0(ζ)ΔηΔζ=∫t1tK1(t,ζ)g(ζ)Δζ,
and similarly
(3.27)∫t1t∫t1σ(η)e-Λ/(A0+μΛ)(t,σ(η))∑i∈[1,n]NAi(η)χ[αi(η),∞)T(ζ)eΛ/A0(σ(η),σ(ζ))g(ζ)A0(ζ)ΔζΔη=∫t1tK2(t,ζ)g(ζ)Δζ.
Therefore, we can rewrite (3.23) in the equivalent form of the integral operator
(3.28)(Hg)(t)=∫t1t[K1(t,η)+K2(t,η)]g(η)Δηfort∈[t1,∞)T,
whose kernel is nonnegative. Consequently, using (3.22), (3.24), and (3.28), we obtain that f≥0 on [t1,∞)𝕋 implies h≥0 on [t1,∞)𝕋; this and Lemma 2.5 yield that g≥0 on [t1,∞)𝕋. Therefore, from (3.14), we infer that if f≥0 on [t1,∞)𝕋, then x≥0 on [t1,∞)𝕋 too. On the other hand, by Lemma 2.4, x has the following representation:
(3.29)x(t)=∫t1tX1(t,σ(η))f(η)Δηfort∈[t1,∞)T.
Since x is eventually nonnegative for any eventually nonnegative function f, we infer that the kernel 𝒳1 of the integral on the right-hand side of (3.29) is eventually nonnegative. Indeed, assume to the contrary that x≥0 on [t1,∞)𝕋 but 𝒳1 is not nonnegative, then we may pick t2∈[t1,∞)𝕋 and find s∈[t1,t2)𝕋 such that 𝒳1(t2,σ(s))<0. Then, letting f(t):=-min{𝒳1(t2,σ(t)),0}≥0 for t∈[t1,∞)𝕋, we are led to the contradiction x(t2)<0, where x is defined by (3.29). To prove that 𝒳1 is eventually positive, set x(t):=𝒳1(t,s) for t∈[t0,∞)𝕋, where s∈[t1,∞)𝕋, to see that x≥0 and (A0xΔ)Δ≤0 on [s,∞)𝕋, which implies A0xΔ is nonincreasing on [s,∞)𝕋. So that, we may let t1∈[t0,∞)𝕋 so large that xΔ (i.e., A0xΔ) is of fixed sign on [s,∞)𝕋⊂[t1,∞)𝕋. The initial condition and (A1) together with xΔ(s)=1/A0(s)>0 imply that xΔ>0 on [s,∞)𝕋. Consequently, we have x(t)=𝒳1(t,s)>𝒳1(s,s)=0 for all t∈(s,∞)𝕋⊂[t1,∞)𝕋.
(iv)⇒(i) Clearly, 𝒳1(·,t0) is an eventually positive solution of (3.1).
The proof is completed.Let us introduce the following condition:(A4)
A0∈Crd([t0,∞)𝕋,ℝ+) with(3.30)∫t0∞1A0(η)Δη=∞.Remark 3.2.
It is well known that (A4) ensures existence oft1∈[t0,∞)𝕋 such that x(t)xΔ(t)≥0 for all t∈[t1,∞)𝕋, for any nonoscillatory solution x of (3.1). This fact follows from the formula
(3.31)x(t)=x(s)+A0(s)xΔ(s)∫st1A0(η)Δη-∫st1A0(η)[∫sη∑i∈[1,n]NAi(ζ)x(αi(ζ))Δζ]Δη
for all t∈[t0,∞)𝕋, obtained by integrating (3.1) twice, where s∈[t0,∞)𝕋. In the case when (A4) holds, (iii) of Theorem 3.1 can be assumed to hold with Λ∈Crd1([t1,∞)𝕋,ℝ0+), which means that any positive (negative) solution is nondecreasing (nonincreasing).Remark 3.3.
Let (A4) hold and existt1∈[t0,∞)𝕋 and the function Λ∈Crd1([t1,∞)𝕋,ℝ0+) satisfying inequality (3.4), then the assertions (i), (iii), and (iv) of Theorem 3.1 are also valid on [t1,∞)𝕋.Remark 3.4.
It should be noted that (3.4) is also equivalent to the inequality
(3.32)ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(σ(t),αi(t))≤0∀t∈[t1,∞)T,
see (3.20) and compare with [26, 28, 29, 34].Example 3.5.
For𝕋=ℝ, (3.4) has the form
(3.33)Λ′(t)+1A0(t)Λ2(t)+∑i∈[1,n]NAi(t)exp⁡{-∫αi(t)tΛ(η)A0(η)dη}≤0∀t∈[t1,∞)R,
see [8] for the case A0(t)≡1, t∈[t0,∞)ℝ, and [35] for n=1, α1(t)=t, t∈[t0,∞)ℝ.Example 3.6.
For𝕋=ℕ, (3.4) becomes
(3.34)ΔΛ(k)+Λ2(k)A0(k)+Λ(k)+∑i∈[1,n]NAi(k)∏j=αi(k)kA0(j)A0(j)+Λ(j)≤0∀k∈[k1,∞)N,
where the product over the empty set is assumed to be equal to one; see [1, 18] (or (1.10)) for n=1, α1(k)=k+1, k∈[k0,∞)ℕ, and [20] for n=1, A0(k)≡1, α1(k)=k+1, k∈[k0,∞)ℕ. It should be mentioned that in the literature all the results relating difference equations with discrete Riccati equations consider only the nondelay case. This result in the discrete case is therefore new.Example 3.7.
For𝕋=qℤ¯ with q∈(1,∞)ℝ, under the same assumption on the product as in the previous example, condition (3.4) reduces to the inequality
(3.35)DqΛ(t)+Λ2(t)A0(t)+(q-1)tΛ(t)+∑i∈[1,n]NAi(t)∏η=log⁡q(αi(t))log⁡q(t)A0(qη)A0(qη)+(q-1)qηΛ(qη)≤0
for all t∈[t1,∞)qℤ¯.


## 4. Comparison Theorems

Theorem3.1 can be employed to obtain comparison nonoscillation results. To this end, together with (3.1), we consider the second-order dynamic equation(4.1)(A0xΔ)Δ(t)+∑i∈[1,n]NBi(t)x(αi(t))=0fort∈[t0,∞)T,
where Bi∈Crd([t0,∞)𝕋,ℝ) for i∈[1,n]ℕ.The following theorem establishes the relation between the first fundamental solution of the model equation with positive coefficients and comparison (4.1) with coefficients of arbitrary signs.Theorem 4.1.
Suppose that (A2), (A3), (A4), and the following condition hold:(A5)
fori∈[1,n]ℕ, Bi∈Crd([t0,∞)𝕋,ℝ) with Ai(t)≥Bi(t) for all t∈[t0,∞)𝕋.Assume further that (3.4) admits a solution Λ∈Crd1([t1,∞)𝕋,ℝ0+) for some t1∈[t0,∞)𝕋, then the first fundamental solution 𝒴1 of (4.1) satisfies 𝒴1(t,s)≥𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋, where 𝒳1 denotes the first fundamental solution of (3.1).Proof.
We consider the initial value problem(4.2)(A0xΔ)Δ(t)+∑i∈[1,n]NBi(t)x(αi(t))=f(t)fort∈[t0,∞)T,xΔ(t0)=0,x(t)≡0fort∈[t-1,t0]T,
where f∈Crd([t0,∞)𝕋,ℝ). Let g∈Crd([t1,∞)𝕋,ℝ), and define the function x as
(4.3)x(t)=∫t1tX1(t,σ(η))g(η)Δη∀t∈[t1,∞)T.
By the Leibnitz rule (see [32, Theorem 1.117]), for all t∈[t1,∞)𝕋, we have
(4.4)xΔ(t)=∫t1tX1Δ(t,σ(η))g(η)Δη,(4.5)(A0xΔ)Δ(t)=∫t1t(A0X1Δ(⋅,σ(η)))Δ(t)g(η)Δη+g(t).
Substituting (4.3) and (4.5) into (4.2), we get
(4.6)f(t)=∫t1t(A0X1Δ(⋅,σ(η)))Δ(t)g(η)Δη+∑i∈[1,n]NBi(t)∫t1αi(t)X1(αi(t),σ(η))g(η)Δη+g(t)=∑i∈[1,n]N[Bi(t)-Ai(t)]∫t1αi(t)X1(αi(t),σ(η))g(η)Δη+g(t)=∑i∈[1,n]N[Bi(t)-Ai(t)]∫t1tX1(αi(t),σ(η))g(η)Δη+g(t),
where in the last step, we have used the fact that 𝒳1(t,σ(s))≡0 for all t∈[t1,∞)𝕋 and all s∈[t,∞)𝕋. Therefore, we obtain the operator equation
(4.7)g=Hg+fon[t1,∞)T,
where
(4.8)(Hg)(t):=∫t1t∑i∈[1,n]NX1(αi(t),σ(η))[Ai(t)-Bi(t)]g(η)Δηfort∈[t1,∞)T,
whose kernel is nonnegative. An application of Lemma 2.5 shows that nonnegativity of f implies the same for g, and thus x is nonnegative by (4.3). On the other hand, by Lemma 2.4, x has the representation
(4.9)x(t)=∫t0tY1(t,σ(η))f(η)Δη∀t∈[t0,∞)T.
Proceeding as in the proof of the part (iii)⇒(iv) of Theorem 3.1, we conclude that the first fundamental solution 𝒴1 of (4.1) satisfies 𝒴1(t,s)≥0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋. To complete the proof, we have to show that 𝒴1(t,s)≥𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋. Clearly, for any fixed s∈[t1,∞)𝕋 and all t∈[s,∞)𝕋, we have
(4.10)(A0Y1Δ(⋅,s))Δ(t)+∑i∈[1,n]NAi(t)Y1(αi(t),s)=∑i∈[1,n]N[Ai(t)-Bi(t)]Y1(αi(t),s),
which by the solution representation formula yields that
(4.11)Y1(t,s)=X1(t,s)+∫stX1(t,σ(η))∑i∈[1,n]N[Ai(η)-Bi(η)]Y1(αi(η),s)Δη≥X1(t,s)
for all t∈[s,∞)𝕋. This completes the proof since the first fundamental solution 𝒳1 satisfies 𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋 by Remark 3.3.Corollary 4.2.
Suppose that (A1), (A2), (A3), and (A5) hold, and (3.1) has a nonoscillatory solution on [t1,∞)𝕋⊂[t0,∞)𝕋, then (4.1) admits a nonoscillatory solution on [t2,∞)𝕋⊂[t1,∞)𝕋.Corollary 4.3.
Assume that (A2) and (A3) hold.(i)
If (A1) holds and the dynamic inequality(4.12)(A0xΔ)Δ(t)+∑i∈[1,n]NAi+(t)x(αi(t))≤0fort∈[t0,∞)T,
where Ai+(t):=max{Ai(t),0} for t∈[t0,∞)𝕋 and i∈[1,n]ℕ, has a positive solution on [t0,∞)𝕋, then (3.1) also admits a positive solution on [t1,∞)𝕋⊂[t0,∞)𝕋.(ii)
If (A4) holds and there exist a sufficiently larget1∈[t0,∞)𝕋 and a function Λ∈Crd1([t1,∞)𝕋,ℝ0+) satisfying the inequality
(4.13)ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi+(t)e⊖(Λ/A0)(t,αi(t))≤0∀t∈[t1,∞)T,
then the first fundamental solution 𝒳1 of (3.1) satisfies 𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋.Proof.
Consider the dynamic equation(4.14)(A0xΔ)Δ(t)+∑i∈[1,n]NAi+(t)x(αi(t))=0fort∈[t0,∞)T.
Theorem 3.1 implies that for this equation the assertions (i) and (ii) hold. Since for all i∈[1,n]ℕ, we have Ai(t)≤Ai+(t) for all t∈[t0,∞)𝕋, the application of Corollary 4.2 and Theorem 4.1 completes the proof.Now, let us compare the solutions of problem (2.1) and the following initial value problem:(4.15)(A0xΔ)Δ(t)+∑i∈[1,n]NBi(t)x(αi(t))=g(t)fort∈[t0,∞)T,x(t0)=y1,xΔ(t0)=y2,x(t)=ψ(t)fort∈[t-1,t0)T,
where y1,y2∈ℝ are the initial values, ψ∈Crd([t-1,t0)𝕋,ℝ) is the initial function such that ψ has a finite left-sided limit at the initial point t0 provided that it is left dense, g∈Crd([t0,∞)𝕋,ℝ) is the forcing term.Theorem 4.4.
Suppose that (A2), (A3), (A4), (A5), and the following condition hold:(A6)
f,g∈Crd([t0,∞)𝕋,ℝ) and φ,ψ∈Crd([t-1,t0)𝕋,ℝ) satisfy(4.16)f(t)-∑i∈[1,n]NBi(t)φ(αi(t))≤g(t)-∑i∈[1,n]NBi(t)ψ(αi(t))∀t∈[t0,∞)T.
Moreover, let (2.1) have a positive solution x on [t0,∞)𝕋, y1=x1, and y2≥x2, then the solution y of (4.15) satisfies y(t)≥x(t) for all t∈[t0,∞)𝕋.Proof.
By Theorem3.1 and Remark 3.3, we can assume that Λ∈Crd([t0,∞)𝕋,ℝ0+) is a solution of the dynamic Riccati inequality (3.4), then by (A5), the function Λ is also a solution of the dynamic Riccati inequality
(4.17)ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NBi(t)e⊖(Λ/A0)(t,αi(t))≤0∀t∈[t0,∞)T,
which is associated with (4.15). Hence, by Theorem 3.1 and Remark 3.3, the first fundamental solution 𝒴1 of (4.15) satisfies 𝒴1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t0,∞)𝕋. Rewriting (2.1) in the form
(4.18)(A0xΔ)Δ(t)+∑i∈[1,n]NBi(t)x(αi(t))=f(t)-∑i∈[1,n]N[Ai(t)-Bi(t)]x(αi(t)),t∈[t0,∞)Tx(t0)=x1,xΔ(t0)=x2,x(t)=φ(t),t∈[t-1,t0)T,
applying Lemma 2.4, and using (A6), we have
(4.19)x(t)=x2Y1(t,t0)+x1Y2(t,t0)+∫t0tY1(t,σ(η))×[f(η)-∑i∈[1,n]N[Ai(η)-Bi(η)]χ[t0,∞)T(αi(η))x(αi(η))-∑i∈[1,n]NBi(η)φ(αi(η))]Δη≤y2Y1(t,t0)+y1Y2(t,t0)+∫t0tY1(t,σ(η))[g(η)-∑i∈[1,n]NBi(η)ψ(αi(η))]Δη=y(t)
for all t∈[t0,∞)𝕋. This completes the proof.Remark 4.5.
IfBi∈Crd([t0,∞)𝕋,ℝ0+) for i∈[1,n]ℕ, f(t)≤g(t) for all t∈[t0,∞)𝕋 and φ(t)≥ψ(t) for all t∈[t-1,t0)𝕋, then (A6) holds.The following example illustrates Theorem4.4 for the quantum time scale 𝕋=2ℤ¯.Example 4.6.
Let2ℤ¯:={2k:k∈ℤ}∪{0}, and consider the following initial value problems:
(4.20)D2(Id2Z¯D2x)(t)+2t4x(t4)=-1t4fort∈[1,∞)2Z¯,D2x(1)=1,x(t)≡1fort∈[14,1]2Z¯,
where Id2ℤ¯ is the identity function on 2ℤ¯, that is, Id2ℤ¯(t)=t for t∈2ℤ¯, and
(4.21)D2x(t)=1t(x(2t)-x(t))fort∈2Z¯,(4.22)D2(Id2Z¯D2x)(t)+1t4x(t4)=1t4fort∈[1,∞)2Z¯,D2x(1)=1,x(t)≡1fort∈[14,1]2Z¯.
Denoting by x and y the solutions of (4.20) and (4.22), respectively, we obtain y(t)≥x(t) for all t∈[1,∞)2ℤ¯ by Theorem 4.4. For the graph of the first 10 iterates, see Figure 1.Figure 1
The graph of 10 iterates for the solutions of (4.20) and (4.22) illustrates the result of Theorem 4.4, here y(t)>x(t) for all t∈(1,∞)2ℤ¯.As an immediate consequence of Theorem4.4, we obtain the following corollary.Corollary 4.7.
Suppose that (A1), (A2), and (A3) hold and that (3.1) is nonoscillatory, then, for f∈Crd([t0,∞)𝕋,ℝ0+), the dynamic equation
(4.23)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=f(t)fort∈[t0,∞)T
is also nonoscillatory.We now consider the following dynamic equation:(4.24)(A0xΔ)Δ(t)+∑i∈[1,n]NAi(t)x(αi(t))=g(t)fort∈[t0,∞)T,x(t0)=y1,xΔ(t0)=y2,x(t)=ψ(t)fort∈[t-1,t0)T,
where the parameters are the same as in (4.15).We obtain the most complete result if we compare solutions of (2.1) and (4.24) by omitting the condition (A2) and assuming that the solution of (2.1) is positive.Corollary 4.8.
Suppose that (A3), (A4), and the following condition hold:(A7)
f,g∈Crd([t0,∞)𝕋,ℝ) and φ,ψ∈Crd([t-1,t0)𝕋,ℝ) satisfy(4.25)f(t)-∑i∈[1,n]NAi(t)φ(αi(t))≤g(t)-∑i∈[1,n]NAi(t)ψ(αi(t))∀t∈[t0,∞)T.
If x is a positive solution of (2.1) on [t0,∞)𝕋 with x1=y1 and y2≥x2, then for the solution y of (4.24), one has y(t)≥x(t) for all t∈[t0,∞)𝕋.Proof.
Corollary4.3 and Remark 3.3 imply that the first fundamental solution 𝒳1 associated with (2.1) (and (4.24)) satisfies 𝒳1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t0,∞)𝕋. Hence, the claim follows from the solution representation formula.Remark 4.9.
If at least one of the inequalities in the statements of Theorem4.4 and Corollary 4.8 is strict, then the conclusions hold with the strict inequality too.Let us compare equations with different coefficients and delays. Now, we consider(4.26)(A0xΔ)Δ(t)+∑i∈[1,n]NBi(t)x(βi(t))=0fort∈[t0,∞)T.Theorem 4.10.
Suppose that (A2), (A4), (A5), and the following condition hold:(A8)
fori∈[1,n]ℕ, βi∈Crd([t0,∞)𝕋,𝕋) satisfies βi(t)≤αi(t) for all t∈[t0,∞)𝕋 and limt→∞βi(t)=∞.
Assume further that the first-order dynamic Riccati inequality (3.4) has a solution Λ∈Crd1([t1,∞)𝕋,ℝ0+) for some t1∈[t0,∞)𝕋, then the first fundamental solution 𝒴1 of (4.26) satisfies 𝒴1(t,s)>0 for all t∈(s,∞)𝕋 and all s∈[t1,∞)𝕋.Proof.
Note that(A5) implies Ai(t)≥Bi+(t) for all t∈[t0,∞)𝕋 and i∈[1,n]ℕ, then we have
(4.27)0≥ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(σ(t),αi(t))≥ΛΔ(t)+Λ2(t)A0(t)+μ(t)Λ(t)+∑i∈[1,n]NBi+(t)e⊖(Λ/A0)(σ(t),βi(t))
for all t∈[t1,∞)𝕋. The reference to Corollary 4.3 (ii) concludes the proof.Remark 4.11.
If the condition(A4) in Theorem 4.1, Theorem 4.4, Corollary 4.8, and Theorem 4.10 is replaced with (A1), then the claims of the theorems are valid eventually.Let us introduce the function(4.28)αmax⁡(t):=max⁡i∈[1,n]N{αi(t)}fort∈[t0,∞)T.Corollary 4.12.
Suppose that (A1), (A2), (A3), and (A5) hold. If(4.29)(A0xΔ)Δ(t)+(∑i∈[1,n]NAi(t))x(αmax⁡(t))=0fort∈[t0,∞)T
is nonoscillatory, then (4.1) is also nonoscillatory.Remark 4.13.
The claim of Corollary4.12 is also true when αmax is replaced by σ.


## 5. Explicit Nonoscillation and Oscillation Results

Theorem 5.1.
Suppose that (A1), (A2), and (A3) hold and that(5.1)σ(t)2tA0(t)+μ(t)+2tσ(t)∑i∈[1,n]NAi(t)e⊖(1/(2IdTA0))(σ(t),αi(t))≤1∀t∈[t1,∞)T,
where t1∈[t0,∞)𝕋 and Id𝕋 is the identity function on 𝕋, then (3.1) is nonoscillatory.Proof.
The statement of the theorem yields thatΛ(t)=1/(2t) for t∈[t0,∞)𝕋+ is a positive solution of the Riccati inequality (3.32).Next, let us apply Theorem5.1 to delay differential equations.Corollary 5.2.
LetA0∈C([t0,∞)ℝ,ℝ+), for i∈[1,n]ℕ, Ai∈C([t0,∞)ℝ,ℝ0+), and αi∈C([t0,∞)ℝ,ℝ) such that αi(t)≤t for all t∈[t0,∞)ℝ and limt→∞αi(t)=∞. If
(5.2)12A0(t)+2t2∑i∈[1,n]NAi(t)exp⁡{-∫αi(t)t12ηA0(η)dη}≤1∀t∈[t1,∞)R
for some t1∈[t0,∞)ℝ, then (1.2) is nonoscillatory.Now, let us proceed with the discrete case.Corollary 5.3.
Let{A0(k)} be a positive sequence, for i∈[1,n]ℕ, let {Ai(k)} be a nonnegative sequence, and let {αi(k)} be a divergent sequence such that αi(k)≤k+1 for all k∈[k0,∞)ℕ. If
(5.3)k+12kA0(k)+1+2k(k+1)∑i∈[1,n]NAi(k)∏j=αi(k)k2jA0(j)2jA0(j)+1≤1∀k∈[k1,∞)N
for some k1∈[k0,∞)ℕ, then (1.8) is nonoscillatory.Let us introduce the function(5.4)A(t,s):=∫st1A0(η)Δηfors,t∈[t0,∞)T.Theorem 5.4.
Suppose that (A1), (A2), and (A3) hold, and for everyt1∈[t0,∞)𝕋, the dynamic equation
(5.5)(A0xΔ)Δ(t)+1A(αmax⁡(t),t1)(∑i∈[1,n]NAi(t)A(αi(t),t1))x(αmax⁡(t))=0,t∈[t2,∞)T
is oscillatory, where t2∈[t1,∞)𝕋 satisfies αmin(t)>t1 for all t∈[t2,∞)𝕋, then (3.1) is also oscillatory.Proof.
Assume to the contrary that (3.1) is nonoscillatory, then there exists a solution x of (3.1) such that x>0, (A0xΔ)Δ≤0 on [t1,∞)𝕋⊂[t0,∞)𝕋. This implies that A0xΔ is nonincreasing on [t1,∞)𝕋, then it follows that
(5.6)x(t)≥x(t)-x(t1)=∫t1t1A0(η)A0(η)xΔ(η)Δη≥A(t,t1)A0(t)xΔ(t)∀t∈[t1,∞)T,
or simply by using (5.4),
(5.7)x(t)-A(t,t1)A0(t)xΔ(t)≥0∀t∈[t1,∞)T.
Now, let
(5.8)ψ(t):=x(t)A(t,t1)fort∈(t1,∞)T.
By the quotient rule, (5.4) and (5.7), we have
(5.9)ψΔ(t)=A(t,t1)A0(t)xΔ(t)-x(t)A(σ(t),t1)A(t,t1)A0(t)≤0∀t∈(t1,∞)T,
proving that ψ is nonincreasing on (t1,∞)𝕋. Therefore, for all i∈[1,n]ℕ, we obtain
(5.10)x(αmax⁡(t))A(αmax⁡(t),t1)=ψ(αmax⁡(t))≤ψ(αi(t))=x(αi(t))A(αi(t),t1)∀t∈[t2,∞)T,
where t2∈[t1,∞)𝕋 satisfies αmin(t)>t1 for all t∈[t2,∞)𝕋. Using (5.10) in (3.1), we see that x solves
(5.11)(A0xΔ)Δ(t)+1A(αmax⁡(t),t1)(∑i∈[1,n]NAi(t)A(αi(t),t1))x(αmax⁡(t))≤0∀t∈[t2,∞)T,
which shows that (5.5) is also nonoscillatory by Theorem 3.1. This is a contradiction, and the proof is completed.The following theorem can be regarded as the dynamic generalization of Leighton's result (Theorem A).Theorem 5.5.
Suppose that (A2), (A3), and (A4) hold and that(5.12)∫t2∞∑i∈[1,n]NAi(η)e⊖(1/(A0A(⋅,t1)))(σ(η),αi(η))Δη=∞,
where t2∈(t1,∞)⊂[t0,∞)𝕋, then every solution of (3.1) is oscillatory.Proof.
Assume to the contrary that (3.1) is nonoscillatory. It follows from Theorem 3.1 and Remark 3.2 that (3.4) has a solution Λ∈Crd([t0,∞)𝕋,ℝ0+). Using (3.5) and (5.7), we see that
(5.13)Λ(t)≤1A(t,t1)∀t∈[t2,∞)T,
which together with (3.4) implies that
(5.14)ΛΔ(t)+∑i∈[1,n]NAi(t)e⊖(1/(A0A(⋅,t1)))(σ(t),αi(t))≤0∀t∈[t2,∞)T.
Integrating the last inequality, we get
(5.15)Λ(t)-Λ(t2)+∫t2t∑i∈[1,n]NAi(η)e⊖(1/(A0A(⋅,t1)))(σ(η),αi(η))Δη≤0∀t∈[t2,∞)T,
which is in a contradiction with (5.12). This completes the proof.We conclude this section with applications of Theorem5.5 to delay differential equations and difference equations.Corollary 5.6.
LetA0∈C([t0,∞)ℝ,ℝ+), for i∈[1,n]ℕ, Ai∈C([t0,∞)ℝ,ℝ0+), and αi∈C([t0,∞)ℝ,ℝ) such that αi(t)≤t for all t∈[t0,∞)ℝ and limt→∞αi(t)=∞. If
(5.16)lim⁡t→∞A(t,t0)=∞,∫t0∞∑i∈[1,n]NAi(η)A(αi(η),t0)A(η,t0)dη=∞,
where
(5.17)A(t,s):=∫st1A0(η)dηfors,t∈[t0,∞)R,
then (1.2) is oscillatory.Corollary 5.7.
Let{A0(k)} be a positive sequence, for i∈[1,n]ℕ, let {Ai(k)} be a nonnegative sequence and let {αi(k)} be a divergent sequence such that αi(k)≤k+1 for all k∈[k0,∞)ℕ. If
(5.18)lim⁡k→∞A(k,k0)=∞,∑j=k0∞∑i∈[1,n]NAi(j)∏l=αi(j)jA0(l)A(l,k0)A0(l)A(l,k0)+1=∞,
where
(5.19)A(k,l):=∑j=lk-11A0(j)forl,k∈[k0,∞)N,
then (1.8) is oscillatory.


## 6. Existence of a Positive Solution

Theorem 6.1.
Suppose that (A2), (A3), and (A4) hold,f∈Crd([t0,∞)𝕋,ℝ0+), and the first-order dynamic Riccati inequality (3.4) has a solution Λ∈Crd1([t0,∞)𝕋,ℝ0+). Moreover, suppose that there exist x1,x2∈ℝ+ such that φ(t)≤x1 for all t∈[t-1,t0)𝕋 and x2≥Λ(t0)x1/A0(t0), then (2.1) admits a positive solution x such that x(t)≥x1 for all t∈[t0,∞)𝕋.Proof.
First assume thaty is the solution of the following initial value problem:
(6.1)(A0yΔ)Δ(t)+∑i∈[1,n]NAi(t)y(αi(t))=0fort∈[t0,∞)T,yΔ(t0)=Λ(t0)A0(t0)x1,y(t)≡x1fort∈[t-1,t0]T.
Denote
(6.2)z(t):={x1eΛ/A0(t,t0)fort∈[t0,∞)T,x1fort∈[t-1,t0)T,
then, by following similar arguments to those in the proof of the part (ii)⇒(iii) of Theorem 3.1, we obtain
(6.3)g(t):=(A0zΔ)Δ(t)+∑i∈[1,n]NAi(t)z(αi(t))=x1eΛ/A0(t,t0)[ΛΔ(t)+1A0(t)Λσ(t)Λ(t)+∑i∈[1,n]NAi(t)e⊖(Λ/A0)(t,αi(t))]≤0
for all t∈[t0,∞)𝕋. So z is a solution to
(6.4)(A0zΔ)Δ(t)+∑i∈[1,n]NAi(t)z(αi(t))=g(t)fort∈[t0,∞)T,zΔ(t0)=Λ(t0)A0(t0)x1,z(t)≡x1fort∈[t-1,t0]T.
Theorem 4.4 implies that y(t)≥z(t)≥x1>0 for all t∈[t0,∞)𝕋. By the hypothesis of the theorem, Theorem 4.4, and Corollary 4.8, we have x(t)≥y(t)≥x1>0 for all t∈[t0,∞)𝕋. This completes the proof for the case f≡0 and g≡0 on [t0,∞)𝕋.
The general case wheref≢0 on [t0,∞)𝕋 is also a consequence of Theorem 4.4.Let us illustrate the result of Theorem6.1 with the following example.Example 6.2.
Letℕ0:={k:k∈ℕ0}, and consider the following delay dynamic equation:
(6.5)(IdN0xΔ)Δ(t)+18tt2+1(x(t)+12x(t2-1))=1tt2+1,t∈[1,∞)N0,xΔ(1)=2,x(t)≡2fort∈[0,1]N0,
then (5.1) takes the form Φ(t)≤1 for all t∈[1,∞)ℕ0, where the function Φ is defined by
(6.6)Φ(t):=12t2+(t2+1-t)(t2+1+t22(1+t2-12(t2-1)+(t-t2-1)))fort∈[1,∞)R
and is decreasing on [1,∞)ℝ and thus is not greater than Φ(1)≈0.79, that is, Theorem 5.1 holds. Theorem 6.1 therefore ensures that the solution is positive on [1,∞)ℕ0. For the graph of 15 iterates, see Figure 2.Figure 2
The graph of 15 iterates for the solution of (6.5) illustrates the result of Theorem 6.1.


## 7. Discussion and Open Problems

We start this section with discussion of explicit nonoscillation conditions for delay differential and difference equations. Let us first consider the continuous case. Corollary5.6 with n=1 and α1(t)=t for t∈[t0,∞)ℝ reduces to Theorem A. Nonoscillation part of Kneser's result for (1.4) follows from Corollary 5.2 by letting n=1, A0(t)≡1, and α1(t)=t for t∈[t0,∞)ℝ. Theorem E is obtained by applying Corollary 5.3 to (1.10).Known nonoscillation tests for difference equations can also be deduced from the results of the present paper. In [18, Lemma 1.2], Chen and Erbe proved that (1.9) is nonoscillatory if and only if there exists a sequence {Λ(k)} with A0(k)+Λ(k)>0 for all k∈[k1,∞)ℕ and some k1∈[k0,∞)ℕ satisfying(7.1)ΔΛ(k)+Λ2(k)A0(k)+Λ(k)+A1(k)≤0∀k∈[k1,∞)N.
Since this result is a necessary and sufficient condition, the conclusion of Theorem F could be deduced from(7.2)ΔΛ(k)+Λ2(k)1+Λ(k)+A1(k)≤0∀k∈[k1,∞)N,
which is a particular case of (7.1) with A0(k)≡1 for k∈[k0,∞)ℕ. We present below a short proof for the nonoscillation part only. Assuming (1.12) and letting(7.3)Λ(k):=14(k-1)+∑j=k∞A1(j)fork∈[k1,∞)N⊂[2,∞)N,
we get(7.4)14(k-1)+14k≥Λ(k)≥14(k-1)∀k∈[k1,∞)N,
and this yields(7.5)ΔΛ(k)+Λ2(k)1+Λ(k)+A1(k)≤-14k2(4k-3)<0∀k∈[k1,∞)N.
That is, the discrete Riccati inequality (7.2) has a positive solution implying that (1.10) is nonoscillatory. It is not hard to prove that (1.13) implies nonexistence of a sequence {Λ(k)} satisfying the discrete Riccati inequality (7.2) (see the proof of [23, Lemma 3]). Thus, oscillation/nonoscillation results for (1.10) in [21] can be deduced from nonexistence/existence of a solution for the discrete Riccati inequality (7.2); see also [20].An application of Theorem3.1 with Λ(t):=λ/t for t∈[t0,∞)qℤ¯+ and λ∈ℝ+ implies the following result for quantum scales.Example 7.1.
Let𝕋=qℤ¯:={qk:k∈ℤ}∪{0} with q∈(1,∞)ℝ. If there exist λ∈ℝ0+ and t1∈[t0,∞)qℤ¯+ such that
(7.6)λ2A0(t)+(q-1)λ+t2∑i∈[1,n]NAi(t)∏η=log⁡q(αi(t))log⁡q(t)A0(qη)A0(qη)+(q-1)λ≤λq,t∈[t1,∞)qZ¯,
then the delay q-difference equation
(7.7)Dq(A0Dqx)(t)+∑i∈[1,n]NAi(t)x(αi(t))=0fort∈[t0,∞)qZ¯
is nonoscillatory.In [36], Bohner and Ünal studied nonoscillation and oscillation of the q-difference equation(7.8)Dq2x(t)+aqt2x(qt)=0fort∈[t0,∞)qZ¯,
where a∈ℝ0+, and proved that (7.7) is nonoscillatory if and only if(7.9)a≤1(q+1)2.
For the above q-difference equation, (7.6) reduces to the algebraic inequality(7.10)λ21+(q-1)λ+aq≤λqorλ2-(1-(q-1)a)λ+a≤0,
whose discriminant is (1-(q-1)a)2-4a=(q-1)2a2-(q+1)a+1. The discriminant is nonnegative if and only if(7.11)a≥q+2q+1q2-2q+1=1(q-1)2ora≤q-2q+1q2-2q+1=1(q+1)2.
If the latter one holds, then the inequality (7.6) holds with an equality for the value(7.12)λ:=12(1-(q-1)a+(1-(q-1)a)2-4a).
It is easy to check that this value is not less than 2/(q+1)2, that is, the solution is nonnegative. This gives us the nonoscillation part of [36, Theorem 3].Let us also outline connections to some known results in the theory of second-order ordinary differential equations. For example, the Sturm-Picone comparison theorem is an immediate corollary of Theorem4.10 if we remark that a solution Λ∈Crd1([t1,∞)𝕋,ℝ) of the inequality (3.32) satisfying Λ/A0∈ℛ+([t1,∞)𝕋,ℝ) is also a solution of (3.32) with Bi instead of Ai for i=0,1.Proposition 7.2 (see [28, 32, 36]).
Suppose thatB0(t)≥A0(t)>0, A1(t)≥0, and A1(t)≥B1(t) for all t∈[t0,∞)𝕋, then nonoscillation of
(7.13)(A0xΔ)Δ(t)+A1(t)xσ(t)=0fort∈[t0,∞)T
implies nonoscillation of
(7.14)(B0xΔ)Δ(t)+B1(t)xσ(t)=0fort∈[t0,∞)T.The following result can also be regarded as another generalization of the Sturm-Picone comparison theorem. It is easily deduced that there is a solutionΛ∈Crd1([t1,∞)𝕋,ℝ0+) of the inequality (3.4).Proposition 7.3.
Suppose that (A4) and the conditions of Proposition7.2 are fulfilled, then nonoscillation of
(7.15)(A0xΔ)Δ(t)+A1(t)x(t)=0fort∈[t0,∞)T
implies the same for
(7.16)(B0xΔ)Δ(t)+B1(t)x(t)=0fort∈[t0,∞)T.Finally, let us present some open problems. To this end, we will need the following definition.Definition 7.4.
A solutionx of (3.1) is said to be slowly oscillating if for every t1∈[t0,∞)𝕋 there exist t2∈(t1,∞)𝕋 with αmin(t)≥t1 for all t∈[t2,∞)𝕋 and t3∈[t2,∞)𝕋 such that x(t1)xσ(t1)≤0, x(t2)xσ(t2)≤0, x(t)>0 for all t∈(t1,t2)𝕋.Following the method of [8, Theorem 10], we can demonstrate that if (A1), (A2) with positive coefficients and (A3) hold, then the existence of a slowly oscillating solution of (3.1) which has infinitely many zeros implies oscillation of all solutions.(P1)
Generally, will existence of a slowly oscillating solution imply oscillation of all solutions? To the best of our knowledge, slowly oscillating solutions have not been studied for difference equations yet, the only known result is [9, Proposition 5.2].All the results of the present paper are obtained under the assumptions that all coefficients of (3.1) are nonnegative, and if some of them are negative, it is supposed that the equation with the negative terms omitted has a positive solution.(P2)
Obtain sufficient nonoscillation conditions for (3.1) with coefficients of an arbitrary sign, not assuming that all solutions of the equation with negative terms omitted are nonoscillatory. In particular, consider the equation with one oscillatory coefficient.(P3)
Describe the asymptotic and the global properties of nonoscillatory solutions.(P4)
Deduce nonoscillation conditions for linear second-order impulsive equations on time scales, where both the solution and its derivative are subject to the change at impulse points (and these changes can be matched or not). The results of this type for second-order delay differential equations were obtained in [37].(P5)
Consider the same equation on different time scales. In particular, under which conditions will nonoscillation of (1.8) imply nonoscillation of (1.2)?(P6)
Obtain nonoscillation conditions for neutral delay second-order equations. In particular, for difference equations some results of this type (a necessary oscillation conditions) can be found in [17].(P7)
In the present paper, all parameters of the equation are rd-continuous which corresponds to continuous delays and coefficients for differential equations. However, in [8], nonoscillation of second-order equations is studied under a more general assumption that delays and coefficients are Lebesgue measurable functions. Can the restrictions of rd-continuity of the parameters be relaxed to involve, for example, discontinuous coefficients which arise in the theory of impulsive equations?



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