Nonoscillation of Second-Order Dynamic Equations with Several Delays

Abstract

Existence of nonoscillatory solutions for the second-order dynamic equation (𝐴₀𝑥^Δ)^Δ(𝑡)+∑_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥(𝛼_𝑖(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 (𝛼_𝑖(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.

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 scale 𝕋 is unbounded from above. The purpose of the present paper is to study nonoscillation of the delay dynamic equation𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₀,∞_𝕋,(1.1) where 𝑛∈ℕ, 𝑡₀∈𝕋, 𝑓∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) is the forcing term, 𝐴₀∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺), and for all 𝑖∈[1,𝑛]_ℕ, 𝐴_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) is the coefficient corresponding to the function 𝛼_𝑖, where 𝛼_𝑖≤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𝐴₀𝑥^^(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_ℝ,(1.2) 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𝐴𝑥₀𝑥^^(𝑡)+𝐴₁(𝑡)𝑥(𝑡)=0for𝑡∈𝑡₀,∞_ℝ,(1.3)_(𝑡)+𝐴₁(𝑡)𝑥(𝑡)=0for𝑡∈𝑡₀,∞_ℝ,(1.4) 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 ^∞𝑡₀1𝐴₀(𝜂)d𝜂=∞,^∞𝑡₀𝐴₁(𝜂)d𝜂=∞,(1.5) 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 𝑡^∞𝑡𝐴₁(𝜂)d𝜂≤14∀𝑡∈𝑡⁰,∞_ℝ,(1.6) while it is oscillatory if 𝑡^∞𝑡𝐴₁(𝜂)d𝜂>𝜆⁰4∀𝑡∈𝑡⁰,∞_ℝandsome𝜆₀∈(1,∞)_ℝ.(1.7)

Another particular case of (1.1) is the second-order delay difference equationΔ𝐴₀Δ𝑥(𝑘)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑘)𝑥𝛼_𝑖(𝑘)=0for∈𝑘₀,∞_ℕ,(1.8) 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 equationsΔ𝐴Δ₀Δ𝑥(𝑘)+𝐴₁(𝑘)𝑥(𝑘+1)=0for𝑘∈𝑘₀,∞_ℕ,(1.9)₂𝑥(𝑘)+𝐴₁(𝑘)𝑥(𝑘+1)=0for𝑘∈𝑘₀,∞_ℕ(1.10) have been extensively studied in [1, 18–22]; see also [23]. In particular, the following result is valid.

Theorem D. Assume that ^∞_𝑗=𝑘0𝐴₁(𝑗)=∞,(1.11) 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 that 𝑘(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 𝑘^∞_𝑗=𝑘𝐴₁(𝑗)≤14∀𝑘∈𝑘⁰,∞_ℕ,(1.12) while is oscillatory if 𝑘^∞_𝑗=𝑘𝐴₁(𝑗)>𝜆⁰4∀𝑘∈𝑘⁰,∞_ℕandsome𝜆₀∈(1,∞)_ℝ.(1.13)

In [23], Tang et al. studied nonoscillation and oscillation of the equationΔ²𝑥(𝑘)+𝐴₁(𝑘)𝑥(𝑘)=0for𝑘∈𝑘₀,∞_ℕ,(1.14) 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 the 𝑞-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 Section 2, 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:𝑥𝐴₀𝑡𝑥^Δ₀^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₀,∞_𝕋,=𝑥₁,𝑥^Δ𝑡₀=𝑥₂,𝑥(𝑡)=𝜑(𝑡)for𝑡∈𝑡₋₁,𝑡₀_𝕋,(2.1) where 𝑛∈ℕ, 𝕋 is a time scale unbounded above, 𝑡₀∈𝕋, 𝑥₁,𝑥₂∈ℝ are the initial values, 𝜑∈𝐶_rd([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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) is the forcing term, 𝐴₀∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺), and for all 𝑖∈[1,𝑛]_ℕ, 𝐴_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) is the coefficient corresponding to the function 𝛼_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,𝕋), which satisfies 𝛼_𝑖(𝑡)≤𝜎(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋 and lim_𝑡→∞𝛼_𝑖(𝑡)=∞. Here, we denoted𝛼_min(𝑡)∶=min_{𝑖∈[1,𝑛]ℕ}𝛼_𝑖(𝑡)for𝑡∈𝑡₀,∞_𝕋,𝑡₋₁∶=inf_{𝑡∈[𝑡0,∞)𝕋}𝛼_min(𝑡),(2.2) then 𝑡₋₁ is finite, since 𝛼_min asymptotically tends to infinity.

Definition 2.1. A function 𝑥∶[𝑡₋₁,∞)_𝕋→ℝ with 𝑥∈𝐶^1rd([𝑡₀,∞)_𝕋,ℝ) and a derivative satisfying 𝐴₀𝑥^Δ∈𝐶^1rd([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 function 𝜑∈𝐶_rd([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 𝑥^Δ(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, let 𝑓∶𝐷→ℝ be defined for some 𝐷⊂ℝ, then it is always understood that 𝑓(𝑡)=𝜒_𝐷(𝑡)𝑓(𝑡) for 𝑡∈ℝ, where 𝜒_𝐷 is the characteristic function of the set 𝐷⊂ℝ defined by 𝜒_𝐷(𝑡)≡1 for 𝑡∈𝐷 and 𝜒_𝐷(𝑡)≡0 for 𝑡∉𝐷.

Definition 2.3. Let 𝑠∈𝕋 and 𝑠₋₁∶=inf_{𝑡∈[𝑠,∞)𝕋}{𝛼_min(𝑡)}. The solutions 𝒳₁=𝒳₁(⋅,𝑠) and 𝒳₂=𝒳₂(⋅,𝑠) of the problems 𝐴₀𝑥^Δ𝑥^Δ_Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈[𝑠,∞)_𝕋,(𝑠)=1𝐴₀(𝑠),𝑥(𝑡)≡0for𝑡∈𝑠⁻¹𝐴,𝑠_𝕋,(2.3)₀𝑥^Δ𝑥^Δ_Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈[𝑠,∞)_𝕋,(𝑠)=0,𝑥(𝑡)=𝜒_{𝑠}(𝑡)for𝑡∈𝑠₋₁,𝑠_𝕋,(2.4) which satisfy 𝒳₁(⋅,𝑠),𝒳₂(⋅,𝑠)∈𝐶^1rd([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. Let 𝑥 be a solution of (2.1), then 𝑥 can be written in the following form: 𝑥(𝑡)=𝑥₂𝒳₁𝑡,𝑡₀+𝑥₁𝒳₂𝑡,𝑡₀+^𝑡𝑡₀𝒳₁(𝑡,𝜎(𝜂))⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂(2.5) for 𝑡∈[𝑡₀,∞)_𝕋.

Proof. For 𝑡∈[𝑡₋₁,∞)_𝕋, let 𝑦(𝑡)∶=⎧⎪⎨⎪⎩^𝑡𝑡₀𝜑(𝑡𝒳)₁(𝑡,𝜎(𝜂))⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂ffoorr𝑡𝑡∈∈𝑡𝑡₀₁,∞_𝕋,,𝑡₀_𝕋.(2.6) 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 𝐴𝑦₀𝑦𝑡^Δ₀^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑦𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₀,∞_𝕋,=0,𝑦^Δ𝑡₀=0,𝑦(𝑡)=𝜑(𝑡)for𝑡∈𝑡₋₁,𝑡₀_𝕋.(2.7) 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 𝑦^Δ(𝑡)=+^𝑡𝑡₀𝒳𝒳^Δ1₁(𝑡,𝜎(𝜂))⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂=(𝜎(𝑡),𝜎(𝑡))⎡⎢⎢⎣𝑓(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)_𝑡𝑡0⎤⎥⎥⎦𝒳^Δ1(𝑡,𝜎(𝜂))⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂.(2.8) Therefore, 𝑦(𝑡₀)=0, 𝑦^Δ(𝑡₀)=0, and 𝑦=𝜑 on [𝑡₋₁,𝑡₀)_𝕋, that is, 𝑦 satisfies the initial conditions in (2.7). Differentiating 𝑦^Δ after multiplying by 𝐴₀ and using the properties of the first fundamental solution 𝒳₁, we get 𝐴₀𝑦^Δ^Δ(𝑡)=+^𝑡𝑡₀𝐴_𝜎0𝐴₀𝒳^Δ1(⋅,𝜎(𝜂))^Δ(𝑡)⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂=−(𝑡)_{𝑗∈[1,𝑛]ℕ}𝒳^Δ1(𝜎(𝑡),𝜎(𝑡))⎡⎢⎢⎣𝑓(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)⎤⎥⎥⎦−𝐴_𝑗(𝑡)^{𝛼𝑗𝑡(𝑡)}₀𝒳₁𝛼_𝑗(𝑡),𝜎(𝜂)⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)+𝑓(𝑡)(2.9) for all 𝑡∈[𝑡₀,∞)_𝕋. For 𝑡∈[𝑡₀,∞)_𝕋, set 𝐼(𝑡)={𝑖∈[1,𝑛]_ℕ∶𝜒_{[𝑡0,∞)𝕋}(𝛼_𝑖(𝑡))=1} and 𝐽(𝑡)∶={𝑖∈[1,𝑛]_ℕ∶𝜒_{[𝑡−1,𝑡0)𝕋}(𝛼_𝑖(𝑡))=1}. Making some arrangements, for all 𝑡∈[𝑡₀,∞)_𝕋, we find 𝐴₀𝑦^Δ^Δ(𝑡)=−−_{𝑗∈𝐼(𝑡)}𝐴_𝑗(𝑡)^{𝛼𝑗𝑡(𝑡)}₀𝒳₁𝛼_𝑗(𝑡),𝜎(𝜂)⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂_{𝑗∈𝐽(𝑡)}−𝐴_𝑗(𝑡)^{𝛼𝑗𝑡(𝑡)}₀𝒳₁𝛼_𝑗(𝑡),𝜎(𝜂)⎡⎢⎢⎣𝑓(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)+𝑓(𝑡),(2.10) and thus 𝐴₀𝑦^Δ^Δ(𝑡)==−−_{𝑗∈𝐼(𝑡)𝑗∈𝐼(𝑡)}𝐴_𝑗(𝑡)^{𝛼𝑗𝑡(𝑡)}₀𝒳₁𝛼_𝑗(𝑡),𝜎(𝜂)𝑓(𝜂)Δ𝜂−_{𝑗∈𝐽(𝑡)}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)+𝑓(𝑡)𝐴_𝑗(𝑡)𝑦𝛼_𝑗(𝑡)−_{𝑗∈𝐽(𝑡)}𝐴_𝑗(𝑡)𝑦𝛼_𝑗(𝑡)+𝑓(𝑡),(2.11) 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]). Let 𝑡₀∈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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) satisfy 𝑓(𝑡)=^𝑡𝑡₀𝐾(𝑡,𝜂)𝑓(𝜂)Δ𝜂+𝑔(𝑡)∀𝑡∈𝑡₀,∞_𝕋,(2.12) 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𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝕋(3.1) and its corresponding inequalities𝐴𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)≤0for𝑡∈𝑡₀,∞_𝕋,(3.2)₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)≥0for𝑡∈𝑡₀,∞_𝕋.(3.3)

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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺), (𝐴2)for 𝑖∈[1,𝑛]_ℕ, 𝐴_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰), (𝐴3)for 𝑖∈[1,𝑛]_ℕ, 𝛼_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,𝕋) satisfies 𝛼_𝑖(𝑡)≤𝜎(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋 and lim_𝑡→∞𝛼_𝑖(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 large 𝑡₁∈[𝑡₀,∞)_𝕋 and a function Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ) with Λ/𝐴₀∈ℛ⁺([𝑡₁,∞)_𝕋,ℝ) satisfying the first-order dynamic Riccati inequality Λ^Δ(𝑡)+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)≤0∀𝑡∈𝑡₁,∞_𝕋,(3.4)(iv)the first fundamental solution 𝒳₁ 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 Λ(𝑡)∶=𝐴₀(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)for𝑡∈𝑡¹,∞_𝕋,(3.5) then evidently Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ) and 1+𝜇(𝑡)Λ(𝑡)𝐴₀(𝑡)=1+𝜇(𝑡)𝑥^Δ(𝑡)𝑥(𝑡)=𝑥^𝜎(𝑡)𝑥(𝑡)>0∀𝑡∈𝑡¹,∞_𝕋,(3.6) which proves that Λ/𝐴₀∈ℛ⁺([𝑡₁,∞)_𝕋,ℝ). This implies that the exponential function e_Λ/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 𝑥^Δ(𝑡)=Λ(𝑡)𝐴₀(𝑡)𝑥(𝑡)for𝑡∈𝑡¹𝑥𝑡₁,∞_𝕋,=1,(3.7) whose unique solution is 𝑥(𝑡)=e_Λ/𝐴0𝑡,𝑡₁∀𝑡∈𝑡₁,∞_𝕋,(3.8) see [32, Theorem 2.77]. Hence, using (3.8), for all 𝑡∈[𝑡₁,∞)_𝕋, we get 𝑥^Δ(𝑡)=Λ(𝑡)𝐴₀𝐴₀(𝑡)e^Λ/𝐴0𝑡,𝑡₁,𝑥^Δ^Δ(𝑡)==ΛΛ_Δe_Λ/𝐴0⋅,𝑡₁^Δ(𝑡)=Λ^Δ(𝑡)e_Λ/𝐴0𝑡,𝑡₁+Λ^𝜎(𝑡)e^ΔΛ/𝐴₀𝑡,𝑡₁(𝑡)e_Λ/𝐴0𝑡,𝑡₁+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)e_Λ/𝐴0𝑡,𝑡₁,(3.9) which gives by substituting into (3.2) and using [32, Theorem 2.36] that 0≥Λ^Δ(𝑡)e_Λ/𝐴0𝑡,𝑡₁+1𝐴₀(𝑡)Λ^𝜎=e_Λ/𝐴0(𝑡)Λ(𝑡)e_Λ/𝐴0𝑡,𝑡₁+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_Λ/𝐴0𝛼_𝑖(𝑡),𝑡₁𝑡,𝑡₁⎡⎢⎢⎣Λ^Δ(𝑡)+𝐴₀1(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e^Λ/𝐴0𝛼_𝑖(𝑡),𝑡₁=e_Λ/𝐴0e_Λ/𝐴0𝑡,𝑡₁⎤⎥⎥⎦𝑡,𝑡₁⎡⎢⎢⎣Λ^Δ(𝑡)+𝐴₀1(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)⎤⎥⎥⎦(3.10) for all 𝑡∈[𝑡₁,∞)_𝕋. Since the expression in the brackets is the same as the left-hand side of (3.4) and e_Λ/𝐴0(⋅,𝑡₁)>0 on [𝑡₁,∞)_𝕋, the function Λ is a solution of (3.4).
(iii)⇒(iv) Consider the initial value problem 𝐴₀𝑥^Δ^Δ(𝑥𝑡)_Δ+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₁,∞_𝕋,𝑡₁=0,𝑥(𝑡)≡0for𝑡∈𝑡₋₁,𝑡₁_𝕋.(3.11) Denote 𝑔(𝑡)∶=𝐴₀(𝑡)𝑥^Δ(𝑡)−Λ(𝑡)𝑥(𝑡)for𝑡∈𝑡₁,∞_𝕋,(3.12) where 𝑥 is any solution of (3.11) and Λ is a solution of (3.4). From (3.12), we have 𝑥^Δ(𝑡)=Λ(𝑡)𝐴₀(𝑡)𝑥(𝑡)+𝐴𝑔₀((𝑡𝑡))for𝑡∈𝑡¹𝑥𝑡₁,∞_𝕋,=0,(3.13) whose unique solution is 𝑥(𝑡)=^𝑡𝑡₁e_Λ/𝐴0(𝑡,𝜎(𝜂))𝑔(𝜂)𝐴₀(𝜂)Δ𝜂∀𝑡∈𝑡¹,∞_𝕋,(3.14) see [32, Theorem 2.77]. Now, for all 𝑡∈[𝑡₁,∞)_𝕋, we compute that 𝑥(𝑡)=e_{⊖(Λ/𝐴0)}(𝜎(𝑡),𝑡)^{𝑡𝜎(𝑡)}₁e_Λ/𝐴0(𝜎(𝑡),𝜎(𝜂))𝑔(𝜂)𝐴(𝜂)Δ𝜂−𝜇(𝑡)e^Λ/𝐴0(𝜎(𝑡),𝜎(𝑡))𝑔(𝑡)=𝐴𝐴₀(𝑡)₀(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)𝑥^𝜎(𝑡)−𝜇(𝑡)𝑔(𝑡)=1𝐴₀(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)𝐴⁰(𝑡)𝑥^𝜎(𝑡)−𝜇(𝑡)𝑔(𝑡),(3.15) and similarly 𝑥𝛼_𝑖(𝑡)=×e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)_{𝑡𝜎(𝑡)1}e_Λ/𝐴0(𝜎(𝑡),𝜎(𝜂))𝑔(𝜂)𝐴(𝜂)Δ𝜂−^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0(𝜎(𝑡),𝜎(𝜂))𝑔(𝜂)=e_{⊖(Λ/𝐴0)}𝐴₀(𝜂)Δ𝜂=e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)𝑥^𝜎(𝑡)−^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0(𝜎(𝑡),𝜎(𝜂))𝐴𝑔₀((𝜂𝜂))Δ𝜂𝜎(𝑡),𝛼_𝑖(𝑡)𝑥^𝜎(𝑡)−^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0𝛼_𝑖(𝑡),𝜎(𝜂)𝐴𝑔₀((𝜂𝜂))Δ𝜂(3.16) for 𝑖∈[1,𝑛]_ℕ. From (3.12) and (3.15), we have 𝐴₀𝑥^Δ^Δ(𝑡)==Λ(Λ_Δ𝑥+𝑔)^Δ(𝑡)=Λ^Δ(𝑡)𝑥^𝜎(𝑡)+Λ(𝑡)𝑥^Δ(𝑡)+𝑔^Δ(𝑡)(𝑡)𝑥^𝜎(𝑡)+Λ²(𝑡)𝐴₀(𝑡)𝑥(𝑡)+𝐴Λ₀((𝑡𝑡))𝑔(𝑡)+𝑔^Δ(𝑡)(3.17) for all 𝑡∈[𝑡₁,∞)_𝕋. We substitute (3.14), (3.15), (3.16), and (3.17) into (3.11) and find that 𝑓(𝑡)=⎡⎢⎢⎣Λ^Δ(𝑡)𝑥^𝜎(𝑡)+Λ²(𝑡)𝐴₀(𝑡)𝑥(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)⎤⎥⎥⎦+Λ(𝑡)𝐴₀(𝑡)𝑔(𝑡)+𝑔^Δ=⎡⎢⎢⎣Λ(𝑡)_Δ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}−⎡⎢⎢⎣𝜇(𝑡)Λ₂𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)⎤⎥⎥⎦𝑥^𝜎(𝑡)(𝑡)𝐴₀(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)𝑔(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)+Λ(𝑡)𝐴₀(𝜂)Δ𝜂⎤⎥⎥⎦𝐴₀(𝑡)𝑔(𝑡)+𝑔^Δ=⎡⎢⎢⎣Λ(𝑡)_Δ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}×1+𝜇(𝑡)Λ(𝑡)𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)⎤⎥⎥⎦𝐴₀(𝑡)^{𝑡𝜎(𝑡)}₁e_Λ/𝐴0(𝑡,𝜎(𝜂))𝑔(𝜂)−𝐴₀(𝜂)Δ𝜂_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)+Λ(𝑡)𝐴₀(𝜂)Δ𝜂𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)𝑔(𝑡)+𝑔^Δ(𝑡)(3.18) for all 𝑡∈[𝑡₁,∞)_𝕋. Then, (3.18) can be rewritten as 𝑔^Δ(𝑡)=−Λ(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)𝑔(𝑡)+Υ(𝑡)^{𝑡𝜎(𝑡)}₁e_Λ/𝐴0(𝑡,𝜎(𝜂))𝑔(𝜂)+𝐴₀(𝜂)Δ𝜂_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)^{𝛼𝜎(𝑡)}_𝑖(𝑡)e_Λ/𝐴0𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)𝐴₀(𝜂)Δ𝜂+𝑓(𝑡)(3.19) for all 𝑡∈[𝑡₁,∞)_𝕋, where Υ(𝑡)∶=−1+𝜇(𝑡)Λ(𝑡)𝐴₀(𝑡)⎡⎢⎢⎣Λ^Δ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)⎤⎥⎥⎦(3.20) 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 Υ(𝑡)=−⎡⎢⎢⎣1+𝜇(𝑡)Λ(𝑡)𝐴₀(𝑡)Λ^Δ(𝑡)+1𝐴₀(𝑡)Λ²=−⎡⎢⎢⎣Λ_Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)⎤⎥⎥⎦(𝑡)+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)⎤⎥⎥⎦≥0(3.21) 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 𝑔=ℋ𝑔+ℎon𝑡₁,∞_𝕋,(3.22) where, for 𝑡∈[𝑡₁,∞)_𝕋, we have defined (ℋ𝑔)(𝑡)∶=^𝑡𝑡₁e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))⎡⎢⎢⎣Υ(𝜂)^{𝑡𝜎(𝜂)}₁e_Λ/𝐴0(𝜎(𝜂),𝜎(𝜁))𝑔(𝜁)+𝐴₀(𝜁)Δ𝜁_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)^{𝛼𝜎(𝜂)}_𝑖(𝜂)e_Λ/𝐴0𝛼_𝑖(𝜂),𝜎(𝜁)𝑔(𝜁)ℎ(𝑡)∶=_𝑡𝑡1𝐴₀(𝜁)Δ𝜁⎤⎥⎥⎦Δ𝜂,(3.23)e_Λ/𝐴0(𝑡,𝜎(𝜁))𝑓(𝜂)Δ𝜂.(3.24) 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 𝐾₁(𝑡,𝑠)∶=1𝐴₀(𝑠)^𝑡𝑠𝐾e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))Υ(𝜂)e_Λ/𝐴0(𝜎(𝜂),𝜎(𝑠))Δ𝜂,₂(𝑡,𝑠)∶=1𝐴₀(𝑠)^𝑡𝑠e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜒_{[𝛼𝑖(𝜂),∞)𝕋}(𝑠)e_Λ/𝐴0(𝜎(𝜂),𝜎(𝑠))Δ𝜂.(3.25) Using the change of integration order formula in [33, Lemma 1], for all 𝑡∈[𝑡₁,∞)_𝕋, we obtain ^𝑡𝑡₁^{𝑡𝜎(𝜂)}₁e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))Υ(𝜂)e_Λ/𝐴0(𝜎(𝜂),𝜎(𝜁))𝑔(𝜁)=𝐴₀(𝜁)Δ𝜁Δ𝜂_𝑡𝑡1^𝑡𝜁e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))Υ(𝜂)e_Λ/𝐴0(𝜎(𝜂),𝜎(𝜁))𝑔(𝜁)=𝐴₀(𝜁)Δ𝜂Δ𝜁_𝑡𝑡1𝐾₁(𝑡,𝜁)𝑔(𝜁)Δ𝜁,(3.26) and similarly ^𝑡𝑡₁^{𝑡𝜎(𝜂)}₁e_{−Λ/(𝐴0+𝜇Λ)}(𝑡,𝜎(𝜂))_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)𝜒_{[𝛼𝑖(𝜂),∞)𝕋}(𝜁)e_Λ/𝐴0(𝜎(𝜂),𝜎(𝜁))𝑔(𝜁)=𝐴₀(𝜁)Δ𝜁Δ𝜂_𝑡𝑡1𝐾₂(𝑡,𝜁)𝑔(𝜁)Δ𝜁.(3.27) Therefore, we can rewrite (3.23) in the equivalent form of the integral operator (ℋ𝑔)(𝑡)=^𝑡𝑡₁𝐾₁(𝑡,𝜂)+𝐾₂(𝑡,𝜂)𝑔(𝜂)Δ𝜂for𝑡∈𝑡₁,∞_𝕋,(3.28) 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: 𝑥(𝑡)=^𝑡𝑡₁𝒳₁(𝑡,𝜎(𝜂))𝑓(𝜂)Δ𝜂for𝑡∈𝑡₁,∞_𝕋.(3.29) 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 (𝐴₀𝑥^Δ)^Δ≤0 on [𝑠,∞)_𝕋, which implies 𝐴₀𝑥^Δ is nonincreasing on [𝑠,∞)_𝕋. So that, we may let 𝑡₁∈[𝑡₀,∞)_𝕋 so large that 𝑥^Δ (i.e., 𝐴₀𝑥^Δ) is of fixed sign on [𝑠,∞)_𝕋⊂[𝑡₁,∞)_𝕋. The initial condition and (A1) together with 𝑥^Δ(𝑠)=1/𝐴₀(𝑠)>0 imply that 𝑥^Δ>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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺) with^∞𝑡₀1𝐴₀(𝜂)Δ𝜂=∞.(3.30)

Remark 3.2. It is well known that (A4) ensures existence of 𝑡₁∈[𝑡₀,∞)_𝕋 such that 𝑥(𝑡)𝑥^Δ(𝑡)≥0 for all 𝑡∈[𝑡₁,∞)_𝕋, for any nonoscillatory solution 𝑥 of (3.1). This fact follows from the formula 𝑥(𝑡)=𝑥(𝑠)+𝐴₀(𝑠)𝑥^Δ(𝑠)^𝑡𝑠1𝐴₀(𝜂)Δ𝜂−^𝑡𝑠1𝐴₀(𝜂)⎡⎢⎢⎣^𝜂𝑠_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜁)𝑥𝛼_𝑖(𝜁)Δ𝜁⎤⎥⎥⎦Δ𝜂(3.31) 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∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰), which means that any positive (negative) solution is nondecreasing (nonincreasing).

Remark 3.3. Let (A4) hold and exist 𝑡₁∈[𝑡₀,∞)_𝕋 and the function Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰) 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 Λ^Δ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)≤0∀𝑡∈𝑡₁,∞_𝕋,(3.32) see (3.20) and compare with [26, 28, 29, 34].

Example 3.5. For 𝕋=ℝ, (3.4) has the form Λ^(𝑡)+1𝐴₀(𝑡)Λ²(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)exp−^𝑡𝛼_𝑖(𝑡)Λ(𝜂)𝐴₀(𝜂)d𝜂≤0∀𝑡∈𝑡¹,∞_ℝ,(3.33) 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. For 𝕋=ℕ, (3.4) becomes ΔΛ(𝑘)+Λ²(𝑘)𝐴₀(𝑘)+Λ(𝑘)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑘)^𝑘_𝑗=_{𝛼𝑖(𝑘)}𝐴₀(𝑗)𝐴₀(𝑗)+Λ(𝑗)≤0∀𝑘∈𝑘¹,∞_ℕ,(3.34) 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. For 𝕋=𝑞^ℤ with 𝑞∈(1,∞)_ℝ, under the same assumption on the product as in the previous example, condition (3.4) reduces to the inequality 𝐷_𝑞Λ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+(𝑞−1)𝑡Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)^{log𝑞(𝑡)}_𝜂=lo_{g𝑞(𝛼𝑖(𝑡))}𝐴₀(𝑞^𝜂)𝐴₀(𝑞^𝜂)+(𝑞−1)𝑞^𝜂Λ(𝑞^𝜂)≤0(3.35) for all 𝑡∈[𝑡₁,∞)_𝑞ℤ.

4. Comparison Theorems

Theorem 3.1 can be employed to obtain comparison nonoscillation results. To this end, together with (3.1), we consider the second-order dynamic equation𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝕋,(4.1) where 𝐵_𝑖∈𝐶_rd([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)for 𝑖∈[1,𝑛]_ℕ, 𝐵_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) with 𝐴_𝑖(𝑡)≥𝐵_𝑖(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋. Assume further that (3.4) admits a solution Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰) 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 𝐴₀𝑥^Δ^Δ(𝑥𝑡)_Δ+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₀,∞_𝕋,𝑡₀=0,𝑥(𝑡)≡0for𝑡∈𝑡₋₁,𝑡₀_𝕋,(4.2) where 𝑓∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ). Let 𝑔∈𝐶_rd([𝑡₁,∞)_𝕋,ℝ), and define the function 𝑥 as 𝑥(𝑡)=^𝑡𝑡₁𝒳₁(𝑡,𝜎(𝜂))𝑔(𝜂)Δ𝜂∀𝑡∈𝑡₁,∞_𝕋.(4.3) By the Leibnitz rule (see [32, Theorem 1.117]), for all 𝑡∈[𝑡₁,∞)_𝕋, we have 𝐴𝑥^Δ(𝑡)=^𝑡𝑡₁𝒳^Δ1(𝑡,𝜎(𝜂))𝑔(𝜂)Δ𝜂,(4.4)₀𝑥^Δ^Δ(𝑡)=^𝑡𝑡₁𝐴₀𝒳^Δ1(⋅,𝜎(𝜂))^Δ(𝑡)𝑔(𝜂)Δ𝜂+𝑔(𝑡).(4.5) Substituting (4.3) and (4.5) into (4.2), we get 𝑓(𝑡)==^𝑡𝑡₁𝐴₀𝒳^Δ1(⋅,𝜎(𝜂))^Δ(𝑡)𝑔(𝜂)Δ𝜂+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)^{𝛼𝑖𝑡(𝑡)}₁𝒳₁𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)Δ𝜂+𝑔(𝑡)_{𝑖∈[1,𝑛]ℕ}=𝐵_𝑖(𝑡)−𝐴_𝑖(𝑡)^{𝛼𝑖𝑡(𝑡)}₁𝒳₁𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)Δ𝜂+𝑔(𝑡)_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)−𝐴_𝑖(𝑡)^𝑡𝑡₁𝒳₁𝛼_𝑖(𝑡),𝜎(𝜂)𝑔(𝜂)Δ𝜂+𝑔(𝑡),(4.6) 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 𝑔=ℋ𝑔+𝑓on𝑡₁,∞_𝕋,(4.7) where (ℋ𝑔)(𝑡)∶=^𝑡𝑡_1𝑖∈_[1,𝑛]ℕ𝒳₁𝛼_𝑖(𝑡),𝜎(𝜂)𝐴_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑔(𝜂)Δ𝜂for𝑡∈𝑡₁,∞_𝕋,(4.8) 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 𝑥(𝑡)=^𝑡𝑡₀𝒴₁(𝑡,𝜎(𝜂))𝑓(𝜂)Δ𝜂∀𝑡∈𝑡₀,∞_𝕋.(4.9) 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 𝐴₀𝒴^Δ1(⋅,𝑠)^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝒴₁𝛼_𝑖(𝑡),𝑠=_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)−𝐵_𝑖(𝑡)𝒴₁𝛼_𝑖(𝑡),𝑠,(4.10) which by the solution representation formula yields that 𝒴₁(𝑡,𝑠)=𝒳₁(𝑡,𝑠)+^𝑡𝑠𝒳₁(𝑡,𝜎(𝜂))_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)−𝐵_𝑖(𝜂)𝒴₁𝛼_𝑖(𝜂),𝑠Δ𝜂≥𝒳₁(𝑡,𝑠)(4.11) 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 𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴^+𝑖(𝑡)𝑥𝛼_𝑖(𝑡)≤0for𝑡∈𝑡₀,∞_𝕋,(4.12) where 𝐴^+𝑖(𝑡)∶=max{𝐴_𝑖(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 large 𝑡₁∈[𝑡₀,∞)_𝕋 and a function Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰) satisfying the inequality Λ^Δ(𝑡)+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴^+𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)≤0∀𝑡∈𝑡₁,∞_𝕋,(4.13) 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 𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴^+𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝕋.(4.14) Theorem 3.1 implies that for this equation the assertions (i) and (ii) hold. Since for all 𝑖∈[1,𝑛]_ℕ, we have 𝐴_𝑖(𝑡)≤𝐴^+𝑖(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:𝑥𝐴₀𝑡𝑥^Δ₀^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑔(𝑡)for𝑡∈𝑡₀,∞_𝕋,=𝑦₁,𝑥^Δ𝑡₀=𝑦₂,𝑥(𝑡)=𝜓(𝑡)for𝑡∈𝑡₋₁,𝑡₀_𝕋,(4.15) where 𝑦₁,𝑦₂∈ℝ are the initial values, 𝜓∈𝐶_rd([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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) is the forcing term.

Theorem 4.4. Suppose that (A2), (A3), (A4), (A5), and the following condition hold: (𝐴6)𝑓,𝑔∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) and 𝜑,𝜓∈𝐶_rd([𝑡₋₁,𝑡₀)_𝕋,ℝ) satisfy 𝑓(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)≤𝑔(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝜓𝛼_𝑖(𝑡)∀𝑡∈𝑡₀,∞_𝕋.(4.16) 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 Theorem 3.1 and Remark 3.3, we can assume that Λ∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰) 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 Λ^Δ(𝑡)+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)≤0∀𝑡∈𝑡₀,∞_𝕋,(4.17) 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 𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑥𝛼_𝑖(𝑡),𝑡∈𝑡₀,∞_𝕋𝑥𝑡₀=𝑥₁,𝑥^Δ𝑡₀=𝑥₂,𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₋₁,𝑡₀_𝕋,(4.18) applying Lemma 2.4, and using (𝐴6), we have 𝑥(𝑡)=×𝑥₂⎡⎢⎢⎣𝒴𝑓₁(𝜂𝑡),−𝑡₀+𝑥_{1𝑖∈[1,𝑛]ℕ}𝒴₂𝑡,𝑡₀+^𝑡𝑡₀𝒴₁(𝑡,𝜎(𝜂))≤𝑦₂𝐴_𝑖(𝜂)−𝐵_𝑖(𝜂)𝜒_{[𝑡0,∞)𝕋}𝛼_𝑖(𝜂)𝑥𝛼_𝑖(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝜂)𝜑𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂=𝑦(𝒴𝑡)₁𝑡,𝑡₀+𝑦₁𝒴₂𝑡,𝑡₀+^𝑡𝑡₀𝒴₁(𝑡,𝜎(𝜂))⎡⎢⎢⎣𝑔(𝜂)−_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝜂)𝜓𝛼_𝑖(𝜂)⎤⎥⎥⎦Δ𝜂(4.19) for all 𝑡∈[𝑡₀,∞)_𝕋. This completes the proof.

Remark 4.5. If 𝐵_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰) 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 Theorem 4.4 for the quantum time scale 𝕋=2^ℤ.

Example 4.6. Let 2^ℤ∶={2^𝑘∶𝑘∈ℤ}∪{0}, and consider the following initial value problems: 𝐷₂Id₂^ℤ𝐷₂𝑥(𝑡)+2𝑡⁴𝑥𝑡4=−𝑡1⁴𝐷₂for𝑡∈[1,∞)_2ℤ,𝑥(1)=1,𝑥(𝑡)≡1for𝑡∈14,1₂^ℤ,(4.20) where Id_2ℤ is the identity function on 2^ℤ, that is, Id_2ℤ(𝑡)=𝑡 for 𝑡∈2^ℤ, and 𝐷₂𝑥(𝑡)=1𝑡(𝑥(2𝑡)−𝑥(𝑡))for𝑡∈𝐷2^ℤ,(4.21)₂I𝐷d₂^ℤ₂𝐷₂𝑥(𝑡)+𝑡1₄𝑥4𝑡=𝑡1⁴for𝑡∈[1,∞)_2ℤ,𝑥(1)=1,𝑥(𝑡)≡1for𝑡∈14,1₂^ℤ.(4.22) 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 Theorem 4.4, we obtain the following corollary.

Corollary 4.7. Suppose that (A1), (A2), and (A3) hold and that (3.1) is nonoscillatory, then, for 𝑓∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰), the dynamic equation 𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑓(𝑡)for𝑡∈𝑡₀,∞_𝕋(4.23) is also nonoscillatory.

We now consider the following dynamic equation:𝑥𝐴₀𝑡𝑥^Δ₀^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=𝑔(𝑡)for𝑡∈𝑡₀,∞_𝕋,=𝑦₁,𝑥^Δ𝑡₀=𝑦₂,𝑥(𝑡)=𝜓(𝑡)for𝑡∈𝑡₋₁,𝑡₀_𝕋,(4.24) 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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ) and 𝜑,𝜓∈𝐶_rd([𝑡₋₁,𝑡₀)_𝕋,ℝ) satisfy 𝑓(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜑𝛼_𝑖(𝑡)≤𝑔(𝑡)−_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝜓𝛼_𝑖(𝑡)∀𝑡∈𝑡₀,∞_𝕋.(4.25) 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. Corollary 4.3 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 Theorem 4.4 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𝐴₀𝑥^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵_𝑖(𝑡)𝑥𝛽_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝕋.(4.26)

Theorem 4.10. Suppose that (A2), (A4), (A5), and the following condition hold: (𝐴8)for 𝑖∈[1,𝑛]_ℕ, 𝛽_𝑖∈𝐶_rd([𝑡₀,∞)_𝕋,𝕋) satisfies 𝛽_𝑖(𝑡)≤𝛼_𝑖(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋 and lim_𝑡→∞𝛽_𝑖(𝑡)=∞.
Assume further that the first-order dynamic Riccati inequality (3.4) has a solution Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰) 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 𝐴_𝑖(𝑡)≥𝐵^+𝑖(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋 and 𝑖∈[1,𝑛]_ℕ, then we have 0≥Λ^Δ(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}≥Λ_Δ𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛼_𝑖(𝑡)(𝑡)+Λ²(𝑡)𝐴₀(𝑡)+𝜇(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐵^+𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝜎(𝑡),𝛽_𝑖(𝑡)(4.27) 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 function𝛼_max(𝑡)∶=max_{𝑖∈[1,𝑛]ℕ}𝛼_𝑖(𝑡)for𝑡∈𝑡₀,∞_𝕋.(4.28)

Corollary 4.12. Suppose that (A1), (A2), (A3), and (A5) hold. If 𝐴₀𝑥^Δ^Δ(𝑡)+⎛⎜⎜⎝_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)⎞⎟⎟⎠𝑥𝛼_max(𝑡)=0for𝑡∈𝑡₀,∞_𝕋(4.29) is nonoscillatory, then (4.1) is also nonoscillatory.

Remark 4.13. The claim of Corollary 4.12 is also true when 𝛼_max is replaced by 𝜎.

5. Explicit Nonoscillation and Oscillation Results

Theorem 5.1. Suppose that (A1), (A2), and (A3) hold and that 𝜎(𝑡)2𝑡𝐴₀(𝑡)+𝜇(𝑡)+2𝑡𝜎(𝑡)_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(1/(2Id𝕋𝐴0))}𝜎(𝑡),𝛼_𝑖(𝑡)≤1∀𝑡∈𝑡₁,∞_𝕋,(5.1) where 𝑡₁∈[𝑡₀,∞)_𝕋 and Id_𝕋 is the identity function on 𝕋, then (3.1) is nonoscillatory.

Proof. The statement of the theorem yields that Λ(𝑡)=1/(2𝑡) for 𝑡∈[𝑡₀,∞)_𝕋⁺ is a positive solution of the Riccati inequality (3.32).

Next, let us apply Theorem 5.1 to delay differential equations.

Corollary 5.2. Let 𝐴₀∈C([𝑡₀,∞)_ℝ,ℝ⁺), for 𝑖∈[1,𝑛]_ℕ, 𝐴_𝑖∈C([𝑡₀,∞)_ℝ,ℝ⁺⁰), and 𝛼_𝑖∈C([𝑡₀,∞)_ℝ,ℝ) such that 𝛼_𝑖(𝑡)≤𝑡 for all 𝑡∈[𝑡₀,∞)_ℝ and lim_𝑡→∞𝛼_𝑖(𝑡)=∞. If 12𝐴₀(𝑡)+2𝑡²_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)exp−^𝑡𝛼_𝑖(𝑡)12𝜂𝐴₀(𝜂)d𝜂≤1∀𝑡∈𝑡¹,∞_ℝ(5.2) 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 {𝐴_𝑖(𝑘)} be a nonnegative sequence, and let {𝛼_𝑖(𝑘)} be a divergent sequence such that 𝛼_𝑖(𝑘)≤𝑘+1 for all 𝑘∈[𝑘₀,∞)_ℕ. If 𝑘+12𝑘𝐴₀(𝑘)+1+2𝑘(𝑘+1)_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑘)^𝑘_𝑗=_{𝛼𝑖(𝑘)}2𝑗𝐴₀(𝑗)2𝑗𝐴₀(𝑗)+1≤1∀𝑘∈𝑘¹,∞_ℕ(5.3) for some 𝑘₁∈[𝑘₀,∞)_ℕ, then (1.8) is nonoscillatory.

Let us introduce the function𝐴(𝑡,𝑠)∶=^𝑡𝑠1𝐴₀(𝜂)Δ𝜂for𝑠,𝑡∈𝑡⁰,∞_𝕋.(5.4)

Theorem 5.4. Suppose that (A1), (A2), and (A3) hold, and for every 𝑡₁∈[𝑡₀,∞)_𝕋, the dynamic equation 𝐴₀𝑥^Δ^Δ(𝑡)+1𝐴𝛼_max(𝑡),𝑡₁⎛⎜⎜⎝_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝐴𝛼_𝑖(𝑡),𝑡₁⎞⎟⎟⎠𝑥𝛼_max(𝑡)=0,𝑡∈𝑡₂,∞_𝕋(5.5) is oscillatory, where 𝑡₂∈[𝑡₁,∞)_𝕋 satisfies 𝛼_min(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, (𝐴₀𝑥^Δ)^Δ≤0 on [𝑡₁,∞)_𝕋⊂[𝑡₀,∞)_𝕋. This implies that 𝐴₀𝑥^Δ is nonincreasing on [𝑡₁,∞)_𝕋, then it follows that 𝑥(𝑡)≥𝑥(𝑡)−𝑥𝑡₁=^𝑡𝑡₁1𝐴₀(𝜂)𝐴⁰(𝜂)𝑥^Δ(𝜂)Δ𝜂≥𝐴𝑡,𝑡₁𝐴₀(𝑡)𝑥^Δ(𝑡)∀𝑡∈𝑡₁,∞_𝕋,(5.6) or simply by using (5.4), 𝑥(𝑡)−𝐴𝑡,𝑡₁𝐴₀(𝑡)𝑥^Δ(𝑡)≥0∀𝑡∈𝑡₁,∞_𝕋.(5.7) Now, let 𝜓(𝑡)∶=𝑥(𝑡)𝐴𝑡,𝑡₁for𝑡∈𝑡¹,∞_𝕋.(5.8) By the quotient rule, (5.4) and (5.7), we have 𝜓^Δ(𝑡)=𝐴𝑡,𝑡¹𝐴₀(𝑡)𝑥^Δ(𝑡)−𝑥(𝑡)𝐴𝜎(𝑡),𝑡₁𝐴𝑡,𝑡₁𝐴₀(𝑡)≤0∀𝑡∈𝑡¹,∞_𝕋,(5.9) proving that 𝜓 is nonincreasing on (𝑡₁,∞)_𝕋. Therefore, for all 𝑖∈[1,𝑛]_ℕ, we obtain 𝑥𝛼_max(𝑡)𝐴𝛼_max(𝑡),𝑡₁=𝜓𝛼^max(𝑡)≤𝜓𝛼_𝑖(𝑡)=𝑥𝛼^𝑖(𝑡)𝐴𝛼_𝑖(𝑡),𝑡₁∀𝑡∈𝑡²,∞_𝕋,(5.10) where 𝑡₂∈[𝑡₁,∞)_𝕋 satisfies 𝛼_min(𝑡)>𝑡₁ for all 𝑡∈[𝑡₂,∞)_𝕋. Using (5.10) in (3.1), we see that 𝑥 solves 𝐴₀𝑥^Δ^Δ(𝑡)+1𝐴𝛼_max(𝑡),𝑡₁⎛⎜⎜⎝_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝐴𝛼_𝑖(𝑡),𝑡₁⎞⎟⎟⎠𝑥𝛼_max(𝑡)≤0∀𝑡∈𝑡₂,∞_𝕋,(5.11) 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 ^∞𝑡₂_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝜂)e_{⊖(1/(𝐴0𝐴(⋅,𝑡1)))}𝜎(𝜂),𝛼_𝑖(𝜂)Δ𝜂=∞,(5.12) 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∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰). Using (3.5) and (5.7), we see that Λ(𝑡)≤1𝐴𝑡,𝑡₁∀𝑡∈𝑡²,∞_𝕋,(5.13) which together with (3.4) implies that Λ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(1/(𝐴0𝐴(⋅,𝑡1)))}𝜎(𝑡),𝛼_𝑖(𝑡)≤0∀𝑡∈𝑡₂,∞_𝕋.(5.14) Integrating the last inequality, we get Λ(𝑡)−Λ𝑡₂+^𝑡𝑡_2𝑖∈_[1,𝑛]ℕ𝐴_𝑖(𝜂)e_{⊖(1/(𝐴0𝐴(⋅,𝑡1)))}𝜎(𝜂),𝛼_𝑖(𝜂)Δ𝜂≤0∀𝑡∈𝑡₂,∞_𝕋,(5.15) which is in a contradiction with (5.12). This completes the proof.

We conclude this section with applications of Theorem 5.5 to delay differential equations and difference equations.

Corollary 5.6. Let 𝐴₀∈C([𝑡₀,∞)_ℝ,ℝ⁺), for 𝑖∈[1,𝑛]_ℕ, 𝐴_𝑖∈C([𝑡₀,∞)_ℝ,ℝ⁺⁰), and 𝛼_𝑖∈C([𝑡₀,∞)_ℝ,ℝ) such that 𝛼_𝑖(𝑡)≤𝑡 for all 𝑡∈[𝑡₀,∞)_ℝ and lim_𝑡→∞𝛼_𝑖(𝑡)=∞. If lim_𝑡→∞𝐴𝑡,𝑡₀=∞,^∞𝑡_0𝑖∈_[1,𝑛]ℕ𝐴_𝑖(𝜂)𝐴𝛼^𝑖(𝜂),𝑡₀𝐴𝜂,𝑡₀d𝜂=∞,(5.16) where 𝐴(𝑡,𝑠)∶=^𝑡𝑠1𝐴₀(𝜂)d𝜂for𝑠,𝑡∈𝑡⁰,∞_ℝ,(5.17) then (1.2) is oscillatory.

Corollary 5.7. Let {𝐴₀(𝑘)} be a positive sequence, for 𝑖∈[1,𝑛]_ℕ, let {𝐴_𝑖(𝑘)} be a nonnegative sequence and let {𝛼_𝑖(𝑘)} be a divergent sequence such that 𝛼_𝑖(𝑘)≤𝑘+1 for all 𝑘∈[𝑘₀,∞)_ℕ. If lim_𝑘→∞𝐴𝑘,𝑘₀=∞,^∞_𝑗_=𝑘0_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑗)^𝑗_ℓ_{=𝛼𝑖(𝑗)}𝐴₀(ℓ)𝐴ℓ,𝑘₀𝐴₀(ℓ)𝐴ℓ,𝑘₀+1=∞,(5.18) where 𝐴(𝑘,𝑙)∶=^𝑘−1_𝑗=𝑙1𝐴₀(𝑗)for𝑙,𝑘∈𝑘⁰,∞_ℕ,(5.19) then (1.8) is oscillatory.

6. Existence of a Positive Solution

Theorem 6.1. Suppose that (A2), (A3), and (A4) hold, 𝑓∈𝐶_rd([𝑡₀,∞)_𝕋,ℝ⁺⁰), and the first-order dynamic Riccati inequality (3.4) has a solution Λ∈𝐶^1rd([𝑡₀,∞)_𝕋,ℝ⁺⁰). 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 that 𝑦 is the solution of the following initial value problem: 𝐴₀𝑦𝑦^Δ_Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑦𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝕋,𝑡₀=Λ𝑡⁰𝐴₀𝑡₀𝑥¹,𝑦(𝑡)≡𝑥₁for𝑡∈𝑡₋₁,𝑡₀_𝕋.(6.1) Denote 𝑧(𝑡)∶=𝑥¹𝑥₁e_Λ/𝐴0𝑡,𝑡₀for𝑡∈𝑡₀,∞_𝕋,for𝑡∈𝑡₋₁,𝑡₀_𝕋,(6.2) then, by following similar arguments to those in the proof of the part (ii)⇒(iii) of Theorem 3.1, we obtain 𝑔(𝑡)∶==𝑥₁𝐴₀𝑧^Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑧𝛼_𝑖(𝑡)e_Λ/𝐴0𝑡,𝑡₀⎡⎢⎢⎣Λ^Δ(𝑡)+1𝐴₀(𝑡)Λ^𝜎(𝑡)Λ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)e_{⊖(Λ/𝐴0)}𝑡,𝛼_𝑖(𝑡)⎤⎥⎥⎦≤0(6.3) for all 𝑡∈[𝑡₀,∞)_𝕋. So 𝑧 is a solution to 𝐴₀𝑧𝑧^Δ_Δ^Δ(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑧𝛼_𝑖(𝑡)=𝑔(𝑡)for𝑡∈𝑡₀,∞_𝕋,𝑡₀=Λ𝑡⁰𝐴₀𝑡₀𝑥¹,𝑧(𝑡)≡𝑥₁for𝑡∈𝑡₋₁,𝑡₀_𝕋.(6.4) 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 where 𝑓≢0 on [𝑡₀,∞)_𝕋 is also a consequence of Theorem 4.4.

Let us illustrate the result of Theorem 6.1 with the following example.

Example 6.2. Let √ℕ₀∶={√𝑘∶𝑘∈ℕ₀}, and consider the following delay dynamic equation: Id^√ℕ₀𝑥^Δ^Δ(𝑡)+18𝑡√𝑡²+1𝑥(𝑡)+21𝑥√𝑡²−1=1𝑡√𝑡²+1,𝑡∈[1,∞)^√ℕ0𝑥_Δ,(1)=2,𝑥(𝑡)≡2for𝑡∈[0,1]_√ℕ0,(6.5) then (5.1) takes the form Φ(𝑡)≤1 for all 𝑡∈[1,∞)_√ℕ0, where the function Φ is defined by Φ(𝑡)∶=12𝑡²+√𝑡²+1−𝑡⎛⎜⎜⎝√𝑡²+1+𝑡²2⎛⎜⎜⎝1+𝑡²−12𝑡²−1+𝑡−√𝑡²−1⎞⎟⎟⎠⎞⎟⎟⎠for𝑡∈[1,∞)^ℝ(6.6) 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. Corollary 5.6 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 satisfyingΔΛ(𝑘)+Λ²(𝑘)𝐴₀(𝑘)+Λ(𝑘)+𝐴¹(𝑘)≤0∀𝑘∈𝑘₁,∞_ℕ.(7.1) Since this result is a necessary and sufficient condition, the conclusion of Theorem F could be deduced fromΔΛ(𝑘)+Λ²(𝑘)1+Λ(𝑘)+𝐴¹(𝑘)≤0∀𝑘∈𝑘₁,∞_ℕ,(7.2) 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 lettingΛ(𝑘)∶=14(𝑘−1)+^∞_𝑗=𝑘𝐴₁(𝑗)for𝑘∈𝑘₁,∞_ℕ⊂[2,∞)_ℕ,(7.3) we get14(𝑘−1)+41𝑘≥Λ(𝑘)≥4(𝑘1−1)∀𝑘∈𝑘¹,∞_ℕ,(7.4) and this yieldsΔΛ(𝑘)+Λ²(𝑘)1+Λ(𝑘)+𝐴¹(𝑘)≤−14𝑘²(4𝑘−3)<0∀𝑘∈𝑘¹,∞_ℕ.(7.5) 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 Theorem 3.1 with Λ(𝑡)∶=𝜆/𝑡 for 𝑡∈[𝑡₀,∞)_𝑞ℤ+ and 𝜆∈ℝ⁺ implies the following result for quantum scales.

Example 7.1. Let 𝕋=𝑞^ℤ∶={𝑞^𝑘∶𝑘∈ℤ}∪{0} with 𝑞∈(1,∞)_ℝ. If there exist 𝜆∈ℝ⁺⁰ and 𝑡₁∈[𝑡₀,∞)_𝑞ℤ+ such that 𝜆²𝐴₀(𝑡)+(𝑞−1)𝜆+𝑡²_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)^{log𝑞(𝑡)}_𝜂=lo_{g𝑞(𝛼𝑖(𝑡))}𝐴₀(𝑞^𝜂)𝐴₀(𝑞^𝜂)+(𝑞−1)𝜆≤𝜆𝑞,𝑡∈𝑡¹,∞_𝑞ℤ,(7.6) then the delay 𝑞-difference equation 𝐷_𝑞𝐴₀𝐷_𝑞𝑥(𝑡)+_{𝑖∈[1,𝑛]ℕ}𝐴_𝑖(𝑡)𝑥𝛼_𝑖(𝑡)=0for𝑡∈𝑡₀,∞_𝑞ℤ(7.7) is nonoscillatory.

In [36], Bohner and Ünal studied nonoscillation and oscillation of the 𝑞-difference equation𝐷^2𝑞𝑥(𝑡)+𝑎𝑞𝑡²𝑥(𝑞𝑡)=0for𝑡∈𝑡₀,∞_𝑞ℤ,(7.8) where 𝑎∈ℝ⁺⁰, and proved that (7.7) is nonoscillatory if and only if𝑎≤1√𝑞+1².(7.9) For the above 𝑞-difference equation, (7.6) reduces to the algebraic inequality𝜆²1+(𝑞−1)𝜆+𝑎𝑞≤𝜆𝑞or𝜆²−(1−(𝑞−1)𝑎)𝜆+𝑎≤0,(7.10) whose discriminant is (1−(𝑞−1)𝑎)²−4𝑎=(𝑞−1)²𝑎²−(𝑞+1)𝑎+1. The discriminant is nonnegative if and only if𝑎≥𝑞+2√𝑞+1𝑞²−2𝑞+1=1√𝑞−1²or𝑎≤𝑞−2√𝑞+1𝑞²−2𝑞+1=1√𝑞+1².(7.11) If the latter one holds, then the inequality (7.6) holds with an equality for the value𝜆∶=121−(𝑞−1)𝑎+(1−(𝑞−1)𝑎)²−4𝑎.(7.12) 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 Theorem 4.10 if we remark that a solution Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ) of the inequality (3.32) satisfying Λ/𝐴₀∈ℛ⁺([𝑡₁,∞)_𝕋,ℝ) is also a solution of (3.32) with 𝐵_𝑖 instead of 𝐴_𝑖 for 𝑖=0,1.

Proposition 7.2 (see [28, 32, 36]). Suppose that 𝐵₀(𝑡)≥𝐴₀(𝑡)>0, 𝐴₁(𝑡)≥0, and 𝐴₁(𝑡)≥𝐵₁(𝑡) for all 𝑡∈[𝑡₀,∞)_𝕋, then nonoscillation of 𝐴₀𝑥^Δ^Δ(𝑡)+𝐴₁(𝑡)𝑥^𝜎(𝑡)=0for𝑡∈𝑡₀,∞_𝕋(7.13) implies nonoscillation of 𝐵₀𝑥^Δ^Δ(𝑡)+𝐵₁(𝑡)𝑥^𝜎(𝑡)=0for𝑡∈𝑡₀,∞_𝕋.(7.14)

The following result can also be regarded as another generalization of the Sturm-Picone comparison theorem. It is easily deduced that there is a solution Λ∈𝐶^1rd([𝑡₁,∞)_𝕋,ℝ⁺⁰) of the inequality (3.4).

Proposition 7.3. Suppose that (A4) and the conditions of Proposition 7.2 are fulfilled, then nonoscillation of 𝐴₀𝑥^Δ^Δ(𝑡)+𝐴₁(𝑡)𝑥(𝑡)=0for𝑡∈𝑡₀,∞_𝕋(7.15) implies the same for 𝐵₀𝑥^Δ^Δ(𝑡)+𝐵₁(𝑡)𝑥(𝑡)=0for𝑡∈𝑡₀,∞_𝕋.(7.16)

Finally, let us present some open problems. To this end, we will need the following definition.

Definition 7.4. A solution 𝑥 of (3.1) is said to be slowly oscillating if for every 𝑡₁∈[𝑡₀,∞)_𝕋 there exist 𝑡₂∈(𝑡₁,∞)_𝕋 with 𝛼_min(𝑡)≥𝑡₁ for all 𝑡∈[𝑡₂,∞)_𝕋 and 𝑡₃∈[𝑡₂,∞)_𝕋 such that 𝑥(𝑡₁)𝑥^𝜎(𝑡₁)≤0, 𝑥(𝑡₂)𝑥^𝜎(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?

Appendix

Time Scales Essentials

A time scale, which inherits the standard topology on ℝ, is a nonempty closed subset of reals. Here, and later throughout this paper, a time scale will be denoted by the symbol 𝕋, and the intervals with a subscript 𝕋 are used to denote the intersection of the usual interval with 𝕋. For 𝑡∈𝕋, we define the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf(𝑡,∞)_𝕋 while the backward jump operator 𝜌∶𝕋→𝕋 is defined by 𝜌(𝑡)∶=sup(−∞,𝑡)_𝕋, and the graininess function 𝜇∶𝕋→ℝ⁺⁰ is defined to be 𝜇(𝑡)∶=𝜎(𝑡)−𝑡. A point 𝑡∈𝕋 is called right dense if 𝜎(𝑡)=𝑡 and/or equivalently 𝜇(𝑡)=0 holds; otherwise, it is called right scattered, and similarly left dense and left scattered points are defined with respect to the backward jump operator. For 𝑓∶𝕋→ℝ and 𝑡∈𝕋, the Δ-derivative 𝑓^Δ(𝑡) of 𝑓 at the point 𝑡 is defined to be the number, provided it exists, with the property that, for any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 such that||𝑓^𝜎(𝑡)−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]||≤𝜀||𝜎(𝑡)−𝑠||∀𝑠∈𝑈,(A.1) where 𝑓^𝜎∶=𝑓∘𝜎 on 𝕋. We mean the Δ-derivative of a function when we only say derivative unless otherwise is specified. A function 𝑓 is called rd-continuous provided that it is continuous at right-dense points in 𝕋 and has a finite limit at left-dense points, and the set of rd-continuous functions is denoted by 𝐶_rd(𝕋,ℝ). The set of functions 𝐶^1rd(𝕋,ℝ) includes the functions whose derivative is in 𝐶_rd(𝕋,ℝ) too. For a function 𝑓∈𝐶^1rd(𝕋,ℝ), the so-called simple useful formula holds𝑓^𝜎(𝑡)=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡)∀𝑡∈𝕋^𝜅,(A.2) where 𝕋^𝜅∶=𝕋⧵{sup𝕋} if sup𝕋=max𝕋 and satisfies 𝜌(max𝕋)≠max𝕋; otherwise, 𝕋^𝜅∶=𝕋. For 𝑠,𝑡∈𝕋 and a function 𝑓∈𝐶_rd(𝕋,ℝ), the Δ-integral of 𝑓 is defined by ^𝑡𝑠𝑓(𝜂)Δ𝜂=𝐹(𝑡)−𝐹(𝑠)for𝑠,𝑡∈𝕋,(A.3) where 𝐹∈𝐶^1rd(𝕋,ℝ) is an antiderivative of 𝑓, that is, 𝐹^Δ=𝑓 on 𝕋^𝜅. Table 1 gives the explicit forms of the forward jump, graininess, Δ-derivative, and Δ-integral on the well-known time scales of reals, integers, and the quantum set, respectively.

Table 1 

Forward jump, Δ-derivative, and Δ-integral.

A function 𝑓∈𝐶_rd(𝕋,ℝ) is called regressive if 1+𝜇𝑓≠0 on 𝕋^𝜅, and positively regressive if 1+𝜇𝑓>0 on 𝕋^𝜅. The set of regressive functions and the set of positively regressive functions are denoted by ℛ(𝕋,ℝ) and ℛ⁺(𝕋,ℝ), respectively, and ℛ⁻(𝕋,ℝ) is defined similarly.

Let 𝑓∈ℛ(𝕋,ℝ), then the exponential function e_𝑓(⋅,𝑠) on a time scale 𝕋 is defined to be the unique solution of the initial value problem𝑥^Δ(𝑡)=𝑓(𝑥𝑡)(𝑥𝑠)(𝑡=)1for𝑡∈𝕋^𝜅,(A.4) for some fixed 𝑠∈𝕋. For ℎ∈ℝ⁺, set ℂ_ℎ∶={𝑧∈ℂ∶𝑧≠−1/ℎ}, ℤ_ℎ∶={𝑧∈ℂ∶−𝜋/ℎ<Im(𝑧)≤𝜋/ℎ}, and ℂ₀∶=ℤ₀∶=ℂ. For ℎ∈ℝ⁺⁰, we define the cylinder transformation 𝜉_ℎ∶ℂ_ℎ→ℤ_ℎ by𝜉_ℎ(𝑧)∶=⎧⎪⎨⎪⎩𝑧1,ℎ=0,ℎLog(1+ℎ𝑧),ℎ>0(A.5) for 𝑧∈ℂ_ℎ, then the exponential function can also be written in the forme_𝑓(𝑡,𝑠)∶=exp^𝑡𝑠𝜉_𝜇(𝜂)(𝑓(𝜂))Δ𝜂for𝑠,𝑡∈𝕋.(A.6) Table 2 illustrates the explicit forms of the exponential function on some well-known time scales.

Table 2 

The exponential function.

The exponential function e_𝑓(⋅,𝑠) is strictly positive on [𝑠,∞)_𝕋 if 𝑓∈ℛ⁺([𝑠,∞)_𝕋,ℝ), while e_𝑓(⋅,𝑠) alternates in sign at right-scattered points of the interval [𝑠,∞)_𝕋 provided that 𝑓∈ℛ⁻([𝑠,∞)_𝕋,ℝ). For ℎ∈ℝ⁺⁰, let 𝑧,𝑤∈ℂ_ℎ, the circle plus ⊕_ℎ and the circle minus ⊖_ℎ are defined by 𝑧⊕_ℎ𝑤∶=𝑧+𝑤+ℎ𝑧𝑤 and 𝑧⊖_ℎ𝑤∶=(𝑧−𝑤)/(1+ℎ𝑤), respectively. Further throughout the paper, we will abbreviate the operations ⊕_𝜇 and ⊖_𝜇 simply by ⊕ and ⊖, respectively. It is also known that ℛ⁺(𝕋,ℝ) is a subgroup of ℛ(𝕋,ℝ), that is, 0∈ℛ⁺(𝕋,ℝ), 𝑓,𝑔∈ℛ⁺(𝕋,ℝ) implies 𝑓⊕_𝜇𝑔∈ℛ⁺(𝕋,ℝ) and ⊖_𝜇𝑓∈ℛ⁺(𝕋,ℝ), where ⊖_𝜇𝑓∶=0⊖_𝜇𝑓 on 𝕋.

The readers are referred to [32] for further interesting details in the time scale theory.

Acknowledgment

E. Braverman is partially supported by NSERC Research grant. This work is completed while B. Karpuz is visiting the Department of Statistics and Mathematics, University of Calgary, Canada, in the framework of Doctoral Research Scholarship of the Council of Higher Education of Turkey.