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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 591254, 34 pages
http://dx.doi.org/10.1155/2011/591254
Research Article

Nonoscillation of Second-Order Dynamic Equations with Several Delays

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4
2Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 30 December 2010; Accepted 13 February 2011

Academic Editor: MiroslavaΒ Růžičková

Copyright Β© 2011 Elena Braverman and Başak Karpuz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Existence of nonoscillatory solutions for the second-order dynamic equation ( 𝐴 0 π‘₯ Ξ” ) Ξ” βˆ‘ ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) π‘₯ ( 𝛼 𝑖 ( 𝑑 ) ) = 0 for 𝑑 ∈ [ 𝑑 0 , ∞ ) 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 𝐴 0 ( 𝑑 ) ≑ 1 for 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ and for second-order nondelay difference equations ( 𝛼 𝑖 ( 𝑑 ) = 𝑑 + 1 for 𝑑 ∈ [ 𝑑 0 , ∞ ) β„• ). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary 𝐴 0 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 ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑓 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ( 1 . 1 ) where 𝑛 ∈ β„• , 𝑑 0 ∈ 𝕋 , 𝑓 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is the forcing term, 𝐴 0 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ + ) , and for all 𝑖 ∈ [ 1 , 𝑛 ] β„• , 𝐴 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is the coefficient corresponding to the function 𝛼 𝑖 , where 𝛼 𝑖 ≀ 𝜎 on [ 𝑑 0 , ∞ ) 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 ξ€· 𝐴 0 π‘₯ ξ…ž ξ€Έ ξ…ž  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ ℝ , ( 1 . 2 ) in the case 𝐴 0 ( 𝑑 ) ≑ 1 for 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ , they coincide with theorems in [8]. The case of a “quickly growing” function 𝐴 0 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 ξ€· 𝐴 0 π‘₯ ξ…ž ξ€Έ ξ…ž ( 𝑑 ) + 𝐴 1 ξ€Ί 𝑑 ( 𝑑 ) π‘₯ ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ ℝ , π‘₯ ( 1 . 3 ) ξ…ž ξ…ž ( 𝑑 ) + 𝐴 1 ( ξ€Ί 𝑑 𝑑 ) π‘₯ ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ ℝ , ( 1 . 4 ) where 𝐴 1 is nonnegative, which are particular cases of (1.2) with 𝑛 = 1 , 𝛼 1 ( 𝑑 ) = 𝑑 , and 𝐴 0 ( 𝑑 ) ≑ 1 for all 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ .

In [10], Leighton proved the following well-known oscillation test for (1.4); see [10, 11].

Theorem A (see [10]). Assume that ξ€œ ∞ 𝑑 0 1 𝐴 0 ξ€œ ( πœ‚ ) d πœ‚ = ∞ , ∞ 𝑑 0 𝐴 1 ( πœ‚ ) d πœ‚ = ∞ , ( 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 𝐴 1 .

In [13], Kneser proved the following result.

Theorem B (see [13]). Equation (1.4) is nonoscillatory if 𝑑 2 𝐴 1 ( 𝑑 ) ≀ 1 / 4 for all 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ , while oscillatory if 𝑑 2 𝐴 1 ( 𝑑 ) > πœ† 0 / 4 for all 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ and some πœ† 0 ∈ ( 1 , ∞ ) 𝕋 .

In [14], Hille proved the following result, which improves the one due to Kneser; see also [1416].

Theorem C (see [14]). Equation (1.4) is nonoscillatory if 𝑑 ξ€œ ∞ 𝑑 𝐴 1 1 ( πœ‚ ) d πœ‚ ≀ 4 ξ€Ί 𝑑 βˆ€ 𝑑 ∈ 0 ξ€Έ , ∞ ℝ , ( 1 . 6 ) while it is oscillatory if 𝑑 ξ€œ ∞ 𝑑 𝐴 1 πœ† ( πœ‚ ) d πœ‚ > 0 4 ξ€Ί 𝑑 βˆ€ 𝑑 ∈ 0 ξ€Έ , ∞ ℝ a n d s o m e πœ† 0 ∈ ( 1 , ∞ ) ℝ . ( 1 . 7 )

Another particular case of (1.1) is the second-order delay difference equation Ξ” ξ€· 𝐴 0 ξ€Έ (  Ξ” π‘₯ π‘˜ ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( ξ€· 𝛼 π‘˜ ) π‘₯ 𝑖 ( ξ€Έ ξ€Ί π‘˜ π‘˜ ) = 0 f o r ∈ 0 ξ€Έ , ∞ β„• , ( 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 equations Ξ” ξ€· 𝐴 0 ξ€Έ Ξ” π‘₯ ( π‘˜ ) + 𝐴 1 ξ€Ί π‘˜ ( π‘˜ ) π‘₯ ( π‘˜ + 1 ) = 0 f o r π‘˜ ∈ 0 ξ€Έ , ∞ β„• , Ξ” ( 1 . 9 ) 2 π‘₯ ( π‘˜ ) + 𝐴 1 ξ€Ί π‘˜ ( π‘˜ ) π‘₯ ( π‘˜ + 1 ) = 0 f o r π‘˜ ∈ 0 ξ€Έ , ∞ β„• ( 1 . 1 0 ) have been extensively studied in [1, 1822]; see also [23]. In particular, the following result is valid.

Theorem D. Assume that ∞  𝑗 = π‘˜ 0 𝐴 1 ( 𝑗 ) = ∞ , ( 1 . 1 1 ) 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 π‘˜ ( π‘˜ + 1 ) 𝐴 1 ( π‘˜ ) ≀ 1 / 4 for all π‘˜ ∈ [ π‘˜ 0 , ∞ ) β„• , 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 π‘˜ ∞  𝑗 = π‘˜ 𝐴 1 ( 1 𝑗 ) ≀ 4 ξ€Ί π‘˜ βˆ€ π‘˜ ∈ 0 ξ€Έ , ∞ β„• , ( 1 . 1 2 ) while is oscillatory if π‘˜ ∞  𝑗 = π‘˜ 𝐴 1 ( πœ† 𝑗 ) > 0 4 ξ€Ί π‘˜ βˆ€ π‘˜ ∈ 0 ξ€Έ , ∞ β„• a n d s o m e πœ† 0 ∈ ( 1 , ∞ ) ℝ . ( 1 . 1 3 )

In [23], Tang et al. studied nonoscillation and oscillation of the equation Ξ” 2 π‘₯ ( π‘˜ ) + 𝐴 1 ( ξ€Ί π‘˜ π‘˜ ) π‘₯ ( π‘˜ ) = 0 f o r π‘˜ ∈ 0 ξ€Έ , ∞ β„• , ( 1 . 1 4 ) where { 𝐴 1 ( π‘˜ ) } 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 π‘ž -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, 2529], 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: ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑓 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , π‘₯ ξ€· 𝑑 0 ξ€Έ = π‘₯ 1 , π‘₯ Ξ” ξ€· 𝑑 0 ξ€Έ = π‘₯ 2 ξ€Ί 𝑑 , π‘₯ ( 𝑑 ) = πœ‘ ( 𝑑 ) f o r 𝑑 ∈ βˆ’ 1 , 𝑑 0 ξ€Έ 𝕋 , ( 2 . 1 ) where 𝑛 ∈ β„• , 𝕋 is a time scale unbounded above, 𝑑 0 ∈ 𝕋 , π‘₯ 1 , π‘₯ 2 ∈ ℝ are the initial values, πœ‘ ∈ 𝐢 r d ( [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 , ℝ ) is the initial function, such that πœ‘ has a finite left-sided limit at the initial point 𝑑 0 provided that it is left dense, 𝑓 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is the forcing term, 𝐴 0 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ + ) , and for all 𝑖 ∈ [ 1 , 𝑛 ] β„• , 𝐴 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is the coefficient corresponding to the function 𝛼 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , 𝕋 ) , which satisfies 𝛼 𝑖 ( 𝑑 ) ≀ 𝜎 ( 𝑑 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 and l i m 𝑑 β†’ ∞ 𝛼 𝑖 ( 𝑑 ) = ∞ . Here, we denoted 𝛼 m i n ( 𝑑 ) ∢ = m i n 𝑖 ∈ [ 1 , 𝑛 ] β„• ξ€½ 𝛼 𝑖 ξ€Ύ ξ€Ί 𝑑 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , 𝑑 βˆ’ 1 ∢ = i n f 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 ξ€½ 𝛼 m i n ξ€Ύ , ( 𝑑 ) ( 2 . 2 ) then 𝑑 βˆ’ 1 is finite, since 𝛼 m i n asymptotically tends to infinity.

Definition 2.1. A function π‘₯ ∢ [ 𝑑 βˆ’ 1 , ∞ ) 𝕋 β†’ ℝ with π‘₯ ∈ 𝐢 1 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) and a derivative satisfying 𝐴 0 π‘₯ Ξ” ∈ 𝐢 1 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is called a solution of (2.1) if it satisfies the equation in the first line of (2.1) identically on [ 𝑑 0 , ∞ ) 𝕋 and also the initial conditions in the second line of (2.1).

For a given function πœ‘ ∈ 𝐢 r d ( [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 , ℝ ) with a finite left-sided limit at the initial point 𝑑 0 provided that it is left-dense and π‘₯ 1 , π‘₯ 2 ∈ ℝ , problem (2.1) admits a unique solution satisfying π‘₯ = πœ‘ on [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 with π‘₯ ( 𝑑 0 ) = π‘₯ 1 and π‘₯ Ξ” ( 𝑑 0 ) = π‘₯ 2 (see [30] and [31, Theorem 3.1]).

Definition 2.2. A solution of (2.1) is called eventually positive if there exists 𝑠 ∈ [ 𝑑 0 , ∞ ) 𝕋 such that π‘₯ > 0 on [ 𝑠 , ∞ ) 𝕋 , and if ( βˆ’ π‘₯ ) is eventually positive, then π‘₯ 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 β†’ ℝ be defined for some 𝐷 βŠ‚ ℝ , then it is always understood that 𝑓 ( 𝑑 ) = πœ’ 𝐷 ( 𝑑 ) 𝑓 ( 𝑑 ) for 𝑑 ∈ ℝ , where πœ’ 𝐷 is the characteristic function of the set 𝐷 βŠ‚ ℝ defined by πœ’ 𝐷 ( 𝑑 ) ≑ 1 for 𝑑 ∈ 𝐷 and πœ’ 𝐷 ( 𝑑 ) ≑ 0 for 𝑑 βˆ‰ 𝐷 .

Definition 2.3. Let 𝑠 ∈ 𝕋 and 𝑠 βˆ’ 1 ∢ = i n f 𝑑 ∈ [ 𝑠 , ∞ ) 𝕋 { 𝛼 m i n ( 𝑑 ) } . The solutions 𝒳 1 = 𝒳 1 ( β‹… , 𝑠 ) and 𝒳 2 = 𝒳 2 ( β‹… , 𝑠 ) of the problems ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ [ ( 𝑑 ) = 0 f o r 𝑑 ∈ 𝑠 , ∞ ) 𝕋 , π‘₯ Ξ” 1 ( 𝑠 ) = 𝐴 0 ξ€Ί 𝑠 ( 𝑠 ) , π‘₯ ( 𝑑 ) ≑ 0 f o r 𝑑 ∈ βˆ’ 1 ξ€» , 𝑠 𝕋 , ξ€· 𝐴 ( 2 . 3 ) 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ [ ) ( 𝑑 ) = 0 f o r 𝑑 ∈ 𝑠 , ∞ 𝕋 , π‘₯ Ξ” ( 𝑠 ) = 0 , π‘₯ ( 𝑑 ) = πœ’ { 𝑠 } ξ€Ί 𝑠 ( 𝑑 ) f o r 𝑑 ∈ βˆ’ 1 ξ€» , 𝑠 𝕋 , ( 2 . 4 ) which satisfy 𝒳 1 ( β‹… , 𝑠 ) , 𝒳 2 ( β‹… , 𝑠 ) ∈ 𝐢 1 r d ( [ 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 𝒳 1 and 𝒳 2 .

Lemma 2.4. Let π‘₯ be a solution of (2.1), then π‘₯ can be written in the following form: π‘₯ ( 𝑑 ) = π‘₯ 2 𝒳 1 ξ€· 𝑑 , 𝑑 0 ξ€Έ + π‘₯ 1 𝒳 2 ξ€· 𝑑 , 𝑑 0 ξ€Έ + ξ€œ 𝑑 𝑑 0 𝒳 1 ⎑ ⎒ ⎒ ⎣  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( πœ‚ ) Ξ” πœ‚ ( 2 . 5 ) for 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 .

Proof. For 𝑑 ∈ [ 𝑑 βˆ’ 1 , ∞ ) 𝕋 , let ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ€œ 𝑦 ( 𝑑 ) ∢ = 𝑑 𝑑 0 𝒳 1 ⎑ ⎒ ⎒ ⎣  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ξ€Ί 𝑑 ( πœ‚ ) Ξ” πœ‚ f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ξ€Ί 𝑑 πœ‘ ( 𝑑 ) f o r 𝑑 ∈ 1 , 𝑑 0 ξ€Έ 𝕋 . ( 2 . 6 ) We recall that 𝒳 1 ( β‹… , 𝑑 0 ) and 𝒳 2 ( β‹… , 𝑑 0 ) solve (2.3) and (2.4), respectively. To complete the proof, let us demonstrate that 𝑦 solves ξ€· 𝐴 0 𝑦 Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) 𝑦 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑓 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , 𝑦 ξ€· 𝑑 0 ξ€Έ = 0 , 𝑦 Ξ” ξ€· 𝑑 0 ξ€Έ ξ€Ί 𝑑 = 0 , 𝑦 ( 𝑑 ) = πœ‘ ( 𝑑 ) f o r 𝑑 ∈ βˆ’ 1 , 𝑑 0 ξ€Έ 𝕋 . ( 2 . 7 ) This will imply that the function 𝑧 defined by 𝑧 ∢ = π‘₯ 2 𝒳 1 ( β‹… , 𝑑 0 ) + π‘₯ 1 𝒳 2 ( β‹… , 𝑑 0 ) + 𝑦 on [ 𝑑 0 , ∞ ) 𝕋 is a solution of (2.1). Combining this with the uniqueness result in [31, Theorem 3.1] will complete the proof. For all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , we can compute that 𝑦 Ξ” ξ€œ ( 𝑑 ) = 𝑑 𝑑 0 𝒳 Ξ” 1 ⎑ ⎒ ⎒ ⎣  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( πœ‚ ) Ξ” πœ‚ + 𝒳 1 ⎑ ⎒ ⎒ ⎣  ( 𝜎 ( 𝑑 ) , 𝜎 ( 𝑑 ) ) 𝑓 ( 𝑑 ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ = ξ€œ ( 𝑑 ) 𝑑 𝑑 0 𝒳 Ξ” 1 ⎑ ⎒ ⎒ ⎣  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( πœ‚ ) Ξ” πœ‚ . ( 2 . 8 ) Therefore, 𝑦 ( 𝑑 0 ) = 0 , 𝑦 Ξ” ( 𝑑 0 ) = 0 , and 𝑦 = πœ‘ on [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 , that is, 𝑦 satisfies the initial conditions in (2.7). Differentiating 𝑦 Ξ” after multiplying by 𝐴 0 and using the properties of the first fundamental solution 𝒳 1 , we get ξ€· 𝐴 0 𝑦 Ξ” ξ€Έ Ξ” ξ€œ ( 𝑑 ) = 𝑑 𝑑 0 ξ€· 𝐴 0 𝒳 Ξ” 1 ξ€Έ ( β‹… , 𝜎 ( πœ‚ ) ) Ξ” ⎑ ⎒ ⎒ ⎣  ( 𝑑 ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( πœ‚ ) Ξ” πœ‚ + 𝐴 𝜎 0 ( 𝑑 ) 𝒳 Ξ” 1 ⎑ ⎒ ⎒ ⎣  ( 𝜎 ( 𝑑 ) , 𝜎 ( 𝑑 ) ) 𝑓 ( 𝑑 ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦  ( 𝑑 ) = βˆ’ 𝑗 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑗 ξ€œ ( 𝑑 ) 𝛼 𝑗 𝑑 ( 𝑑 ) 0 𝒳 1 ξ€· 𝛼 𝑗 ξ€Έ ⎑ ⎒ ⎒ ⎣  ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ βˆ’  ( πœ‚ ) Ξ” πœ‚ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) πœ‘ 𝑖 ξ€Έ ( 𝑑 ) + 𝑓 ( 𝑑 ) ( 2 . 9 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 . For 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , set 𝐼 ( 𝑑 ) = { 𝑖 ∈ [ 1 , 𝑛 ] β„• ∢ πœ’ [ 𝑑 0 , ∞ ) 𝕋 ( 𝛼 𝑖 ( 𝑑 ) ) = 1 } and 𝐽 ( 𝑑 ) ∢ = { 𝑖 ∈ [ 1 , 𝑛 ] β„• ∢ πœ’ [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 ( 𝛼 𝑖 ( 𝑑 ) ) = 1 } . Making some arrangements, for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , we find ξ€· 𝐴 0 𝑦 Ξ” ξ€Έ Ξ”  ( 𝑑 ) = βˆ’ 𝑗 ∈ 𝐼 ( 𝑑 ) 𝐴 𝑗 ξ€œ ( 𝑑 ) 𝛼 𝑗 𝑑 ( 𝑑 ) 0 𝒳 1 ξ€· 𝛼 𝑗 ξ€Έ ⎑ ⎒ ⎒ ⎣  ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ βˆ’  ( πœ‚ ) Ξ” πœ‚ 𝑗 ∈ 𝐽 ( 𝑑 ) 𝐴 𝑗 ξ€œ ( 𝑑 ) 𝛼 𝑗 𝑑 ( 𝑑 ) 0 𝒳 1 ξ€· 𝛼 𝑗 ξ€Έ ⎑ ⎒ ⎒ ⎣  ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑓 ( πœ‚ ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( πœ‚ ) πœ‘ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ βˆ’  ( πœ‚ ) Ξ” πœ‚ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) πœ‘ 𝑖 ξ€Έ ( 𝑑 ) + 𝑓 ( 𝑑 ) , ( 2 . 1 0 ) and thus ξ€· 𝐴 0 𝑦 Ξ” ξ€Έ Ξ”  ( 𝑑 ) = βˆ’ 𝑗 ∈ 𝐼 ( 𝑑 ) 𝐴 𝑗 ξ€œ ( 𝑑 ) 𝛼 𝑗 𝑑 ( 𝑑 ) 0 𝒳 1 ξ€· 𝛼 𝑗 ξ€Έ  ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑓 ( πœ‚ ) Ξ” πœ‚ βˆ’ 𝑗 ∈ 𝐽 ( 𝑑 ) 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) πœ‘ 𝑖 ξ€Έ  ( 𝑑 ) + 𝑓 ( 𝑑 ) = βˆ’ 𝑗 ∈ 𝐼 ( 𝑑 ) 𝐴 𝑗 ξ€· 𝛼 ( 𝑑 ) 𝑦 𝑗 ξ€Έ βˆ’  ( 𝑑 ) 𝑗 ∈ 𝐽 ( 𝑑 ) 𝐴 𝑗 ξ€· 𝛼 ( 𝑑 ) 𝑦 𝑗 ξ€Έ ( 𝑑 ) + 𝑓 ( 𝑑 ) , ( 2 . 1 1 ) which proves that 𝑦 satisfies (2.7) on [ 𝑑 0 , ∞ ) 𝕋 since 𝐼 ( 𝑑 ) ∩ 𝐽 ( 𝑑 ) = βˆ… and 𝐼 ( 𝑑 ) βˆͺ 𝐽 ( 𝑑 ) = [ 1 , 𝑛 ] β„• for each 𝑑 ∈ [ 𝑑 0 , ∞ ) 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 𝑑 0 ∈ 𝕋 and assume that 𝐾 is a nonnegative Ξ” -integrable function defined on { ( 𝑑 , 𝑠 ) ∈ 𝕋 Γ— 𝕋 ∢ 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , 𝑠 ∈ [ 𝑑 0 , 𝑑 ] 𝕋 } . If 𝑓 , 𝑔 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) satisfy ξ€œ 𝑓 ( 𝑑 ) = 𝑑 𝑑 0 ξ€Ί 𝑑 𝐾 ( 𝑑 , πœ‚ ) 𝑓 ( πœ‚ ) Ξ” πœ‚ + 𝑔 ( 𝑑 ) βˆ€ 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ( 2 . 1 2 ) then 𝑔 ( 𝑑 ) β‰₯ 0 for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 implies 𝑓 ( 𝑑 ) β‰₯ 0 for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 .

3. Nonoscillation Criteria

Consider the delay dynamic equation ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 ( 3 . 1 ) and its corresponding inequalities ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) ≀ 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ξ€· 𝐴 ( 3 . 2 ) 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) β‰₯ 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 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: ( 𝐴 1 ) 𝐴 0 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ + ) , ( 𝐴 2 ) for 𝑖 ∈ [ 1 , 𝑛 ] β„• , 𝐴 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ + 0 ) , ( 𝐴 3 ) for 𝑖 ∈ [ 1 , 𝑛 ] β„• , 𝛼 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , 𝕋 ) satisfies 𝛼 𝑖 ( 𝑑 ) ≀ 𝜎 ( 𝑑 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 and l i m 𝑑 β†’ ∞ 𝛼 𝑖 ( 𝑑 ) = ∞ , 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 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 and a function Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) with Ξ› / 𝐴 0 ∈ β„› + ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) satisfying the first-order dynamic Riccati inequality Ξ› Ξ” 1 ( 𝑑 ) + 𝐴 0 Ξ› ( 𝑑 ) 𝜎  ( 𝑑 ) Ξ› ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝑑 , 𝛼 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) ≀ 0 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 4 ) (iv)the first fundamental solution 𝒳 1 of (3.1) is eventually positive, that is, there exists a sufficiently large 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 such that 𝒳 1 ( 𝑑 , 𝑠 ) > 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 .

Proof. The proof follows the scheme: (i) β‡’ (ii) β‡’ (iii) β‡’ (iv) β‡’ (i).
(i) β‡’ (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) β‡’ (iii) Let π‘₯ be an eventually positive solution of (3.2), then there exists 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 such that π‘₯ ( 𝑑 ) > 0 for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . We may assume without loss of generality that π‘₯ ( 𝑑 1 ) = 1 (otherwise, we may proceed with the function π‘₯ / π‘₯ ( 𝑑 1 ) , which is also a solution since (3.2) is linear). Let Ξ› ( 𝑑 ) ∢ = 𝐴 0 ( π‘₯ 𝑑 ) Ξ” ( 𝑑 ) ξ€Ί 𝑑 π‘₯ ( 𝑑 ) f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 5 ) then evidently Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) and 1 + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝐴 0 π‘₯ ( 𝑑 ) = 1 + πœ‡ ( 𝑑 ) Ξ” ( 𝑑 ) = π‘₯ π‘₯ ( 𝑑 ) 𝜎 ( 𝑑 ) ξ€Ί 𝑑 π‘₯ ( 𝑑 ) > 0 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 6 ) which proves that Ξ› / 𝐴 0 ∈ β„› + ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) . This implies that the exponential function e Ξ› / 𝐴 0 ( β‹… , 𝑑 1 ) is well defined and is positive on the entire time scale [ 𝑑 1 , ∞ ) 𝕋 ; see [32, Theorem 2.48]. From (3.5), we see that Ξ› satisfies the ordinary dynamic equation π‘₯ Ξ” ( 𝑑 ) = Ξ› ( 𝑑 ) 𝐴 0 ( ξ€Ί 𝑑 𝑑 ) π‘₯ ( 𝑑 ) f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , π‘₯ ξ€· 𝑑 1 ξ€Έ = 1 , ( 3 . 7 ) whose unique solution is π‘₯ ( 𝑑 ) = e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ ξ€Ί 𝑑 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 8 ) see [32, Theorem 2.77]. Hence, using (3.8), for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we get π‘₯ Ξ” ( 𝑑 ) = Ξ› ( 𝑑 ) 𝐴 0 ( e 𝑑 ) Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ , ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ” ξ€· ( 𝑑 ) = Ξ› e Ξ› / 𝐴 0 ξ€· β‹… , 𝑑 1 ξ€Έ ξ€Έ Ξ” ( 𝑑 ) = Ξ› Ξ” ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ + Ξ› 𝜎 ( 𝑑 ) e Ξ” Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ = Ξ› Ξ” ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ + 1 𝐴 0 Ξ› ( 𝑑 ) 𝜎 ( 𝑑 ) Ξ› ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ , ( 3 . 9 ) which gives by substituting into (3.2) and using [32, Theorem 2.36] that 0 β‰₯ Ξ› Ξ” ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ + 1 𝐴 0 Ξ› ( 𝑑 ) 𝜎 ( 𝑑 ) Ξ› ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ +  𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ( 𝑑 ) , 𝑑 1 ξ€Έ = e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ ⎑ ⎒ ⎒ ⎣ Ξ› Ξ” 1 ( 𝑑 ) + 𝐴 0 ( Ξ› 𝑑 ) 𝜎  ( 𝑑 ) Ξ› ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 e ( 𝑑 ) Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ( 𝑑 ) , 𝑑 1 ξ€Έ e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ = e Ξ› / 𝐴 0 ξ€· 𝑑 , 𝑑 1 ξ€Έ ⎑ ⎒ ⎒ ⎣ Ξ› Ξ” 1 ( 𝑑 ) + 𝐴 0 Ξ› ( 𝑑 ) 𝜎  ( 𝑑 ) Ξ› ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝑑 , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( 𝑑 ) ( 3 . 1 0 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . Since the expression in the brackets is the same as the left-hand side of (3.4) and e Ξ› / 𝐴 0 ( β‹… , 𝑑 1 ) > 0 on [ 𝑑 1 , ∞ ) 𝕋 , the function Ξ› is a solution of (3.4).
(iii) β‡’ (iv) Consider the initial value problem ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑓 ( 𝑑 ) f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , π‘₯ Ξ” ξ€· 𝑑 1 ξ€Έ ξ€Ί 𝑑 = 0 , π‘₯ ( 𝑑 ) ≑ 0 f o r 𝑑 ∈ βˆ’ 1 , 𝑑 1 ξ€» 𝕋 . ( 3 . 1 1 ) Denote 𝑔 ( 𝑑 ) ∢ = 𝐴 0 ( 𝑑 ) π‘₯ Ξ” ( ξ€Ί 𝑑 𝑑 ) βˆ’ Ξ› ( 𝑑 ) π‘₯ ( 𝑑 ) f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 1 2 ) where π‘₯ is any solution of (3.11) and Ξ› is a solution of (3.4). From (3.12), we have π‘₯ Ξ” ( 𝑑 ) = Ξ› ( 𝑑 ) 𝐴 0 ( 𝑑 ) π‘₯ ( 𝑑 ) + 𝑔 ( 𝑑 ) 𝐴 0 ξ€Ί 𝑑 ( 𝑑 ) f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , π‘₯ ξ€· 𝑑 1 ξ€Έ = 0 , ( 3 . 1 3 ) whose unique solution is ξ€œ π‘₯ ( 𝑑 ) = 𝑑 𝑑 1 e Ξ› / 𝐴 0 ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) 𝐴 0 ξ€Ί 𝑑 ( πœ‚ ) Ξ” πœ‚ βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 1 4 ) see [32, Theorem 2.77]. Now, for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we compute that π‘₯ ( 𝑑 ) = e βŠ– ( Ξ› / 𝐴 0 ) ξ‚Έ ξ€œ ( 𝜎 ( 𝑑 ) , 𝑑 ) 𝑑 𝜎 ( 𝑑 ) 1 e Ξ› / 𝐴 0 ( 𝜎 ( 𝑑 ) , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) 𝐴 ( πœ‚ ) Ξ” πœ‚ βˆ’ πœ‡ ( 𝑑 ) e Ξ› / 𝐴 0 ( 𝜎 ( 𝑑 ) , 𝜎 ( 𝑑 ) ) 𝑔 ( 𝑑 ) 𝐴 0 ξ‚Ή = 𝐴 ( 𝑑 ) 0 ( 𝑑 ) 𝐴 0 ξ‚Έ π‘₯ ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝜎 ( 𝑑 ) βˆ’ πœ‡ ( 𝑑 ) 𝑔 ( 𝑑 ) 𝐴 0 ξ‚Ή = 1 ( 𝑑 ) 𝐴 0 ξ€Ί 𝐴 ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 0 ( 𝑑 ) π‘₯ 𝜎 ξ€» , ( 𝑑 ) βˆ’ πœ‡ ( 𝑑 ) 𝑔 ( 𝑑 ) ( 3 . 1 5 ) and similarly π‘₯ ξ€· 𝛼 𝑖 ξ€Έ ( 𝑑 ) = e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ Γ— ξ‚Έ ξ€œ ( 𝑑 ) 𝑑 𝜎 ( 𝑑 ) 1 e Ξ› / 𝐴 0 𝑔 ( 𝜎 ( 𝑑 ) , 𝜎 ( πœ‚ ) ) ( πœ‚ ) ξ€œ 𝐴 ( πœ‚ ) Ξ” πœ‚ βˆ’ 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 𝑔 ( 𝜎 ( 𝑑 ) , 𝜎 ( πœ‚ ) ) ( πœ‚ ) 𝐴 0 ( ξ‚Ή πœ‚ ) Ξ” πœ‚ = e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ ξ‚Έ π‘₯ ( 𝑑 ) 𝜎 ξ€œ ( 𝑑 ) βˆ’ 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ( 𝜎 ( 𝑑 ) , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) 𝐴 0 ξ‚Ή ( πœ‚ ) Ξ” πœ‚ = e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ π‘₯ ( 𝑑 ) 𝜎 ξ€œ ( 𝑑 ) βˆ’ 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ξ€Έ 𝑔 ( 𝑑 ) , 𝜎 ( πœ‚ ) ( πœ‚ ) 𝐴 0 ( πœ‚ ) Ξ” πœ‚ ( 3 . 1 6 ) for 𝑖 ∈ [ 1 , 𝑛 ] β„• . From (3.12) and (3.15), we have ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ” ( 𝑑 ) = ( Ξ› π‘₯ + 𝑔 ) Ξ” ( 𝑑 ) = Ξ› Ξ” ( 𝑑 ) π‘₯ 𝜎 ( 𝑑 ) + Ξ› ( 𝑑 ) π‘₯ Ξ” ( 𝑑 ) + 𝑔 Ξ” ( 𝑑 ) = Ξ› Ξ” ( 𝑑 ) π‘₯ 𝜎 Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 ( 𝑑 ) π‘₯ ( 𝑑 ) + Ξ› ( 𝑑 ) 𝐴 0 ( 𝑑 ) 𝑔 ( 𝑑 ) + 𝑔 Ξ” ( 𝑑 ) ( 3 . 1 7 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . We substitute (3.14), (3.15), (3.16), and (3.17) into (3.11) and find that ⎑ ⎒ ⎒ ⎣ Ξ› 𝑓 ( 𝑑 ) = Ξ” ( 𝑑 ) π‘₯ 𝜎 Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0  ( 𝑑 ) π‘₯ ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ + ( 𝑑 ) Ξ› ( 𝑑 ) 𝐴 0 ( 𝑑 ) 𝑔 ( 𝑑 ) + 𝑔 Ξ” = ⎑ ⎒ ⎒ ⎣ Ξ› ( 𝑑 ) Ξ” Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 ( +  𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ π‘₯ ( 𝑑 ) 𝜎 βˆ’ ⎑ ⎒ ⎒ ⎣ ( 𝑑 ) πœ‡ ( 𝑑 ) Ξ› 2 ( 𝑑 ) 𝐴 0 ( ξ€· 𝐴 𝑑 ) 0 ( ξ€Έ  𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑔 ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€œ ( 𝑑 ) 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ξ€Έ ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) 𝐴 0 ⎀ βŽ₯ βŽ₯ ⎦ + ( πœ‚ ) Ξ” πœ‚ Ξ› ( 𝑑 ) 𝐴 0 ( 𝑑 ) 𝑔 ( 𝑑 ) + 𝑔 Ξ” = ⎑ ⎒ ⎒ ⎣ Ξ› ( 𝑑 ) Ξ” Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 +  ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ Γ— ξ‚Έ ( 𝑑 ) 1 + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝐴 0 ( ξ‚Ή ξ€œ 𝑑 ) 𝑑 𝜎 ( 𝑑 ) 1 e Ξ› / 𝐴 0 ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) 𝐴 0 ( βˆ’  πœ‚ ) Ξ” πœ‚ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€œ ( 𝑑 ) 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ξ€Έ ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) 𝐴 0 + ( πœ‚ ) Ξ” πœ‚ Ξ› ( 𝑑 ) 𝐴 0 ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑔 ( 𝑑 ) + 𝑔 Ξ” ( 𝑑 ) ( 3 . 1 8 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . Then, (3.18) can be rewritten as 𝑔 Ξ” ( 𝑑 ) = βˆ’ Ξ› ( 𝑑 ) 𝐴 0 ξ€œ ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑔 ( 𝑑 ) + Ξ₯ ( 𝑑 ) 𝑑 𝜎 ( 𝑑 ) 1 e Ξ› / 𝐴 0 ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) 𝐴 0 +  ( πœ‚ ) Ξ” πœ‚ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( ξ€œ 𝑑 ) 𝛼 𝜎 ( 𝑑 ) 𝑖 ( 𝑑 ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ( ξ€Έ 𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) 𝐴 0 ( πœ‚ ) Ξ” πœ‚ + 𝑓 ( 𝑑 ) ( 3 . 1 9 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , where ξ‚Έ Ξ₯ ( 𝑑 ) ∢ = βˆ’ 1 + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝐴 0 ξ‚Ή ⎑ ⎒ ⎒ ⎣ Ξ› ( 𝑑 ) Ξ” Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 +  ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( 𝑑 ) ( 3 . 2 0 ) for 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . We now show that Ξ₯ β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 . Indeed, by using (3.4) and the simple useful formula (A.2), we get ⎑ ⎒ ⎒ ⎣ ξ‚΅ Ξ₯ ( 𝑑 ) = βˆ’ 1 + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝐴 0 ξ‚Ά Ξ› ( 𝑑 ) Ξ” 1 ( 𝑑 ) + 𝐴 0 Ξ› ( 𝑑 ) 2  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝑑 , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ⎑ ⎒ ⎒ ⎣ Ξ› ( 𝑑 ) = βˆ’ Ξ” 1 ( 𝑑 ) + 𝐴 0 ( Ξ› 𝑑 ) 𝜎  ( 𝑑 ) Ξ› ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝑑 , 𝛼 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( 𝑑 ) β‰₯ 0 ( 3 . 2 1 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 . On the other hand, from (3.11) and (3.12), we see that 𝑔 ( 𝑑 1 ) = 0 . Then, by [32, Theorem 2.77], we can write (3.19) in the equivalent form ξ€Ί 𝑑 𝑔 = β„‹ 𝑔 + β„Ž o n 1 ξ€Έ , ∞ 𝕋 , ( 3 . 2 2 ) where, for 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we have defined ξ€œ ( β„‹ 𝑔 ) ( 𝑑 ) ∢ = 𝑑 𝑑 1 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) ⎑ ⎒ ⎒ ⎣ ξ€œ ( 𝑑 , 𝜎 ( πœ‚ ) ) Ξ₯ ( πœ‚ ) 𝑑 𝜎 ( πœ‚ ) 1 e Ξ› / 𝐴 0 ( 𝜎 ( πœ‚ ) , 𝜎 ( 𝜁 ) ) 𝑔 ( 𝜁 ) 𝐴 0 +  ( 𝜁 ) Ξ” 𝜁 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€œ ( πœ‚ ) 𝛼 𝜎 ( πœ‚ ) 𝑖 ( πœ‚ ) e Ξ› / 𝐴 0 ξ€· 𝛼 𝑖 ξ€Έ 𝑔 ( πœ‚ ) , 𝜎 ( 𝜁 ) ( 𝜁 ) 𝐴 0 ( ⎀ βŽ₯ βŽ₯ ⎦ ξ€œ 𝜁 ) Ξ” 𝜁 Ξ” πœ‚ , ( 3 . 2 3 ) β„Ž ( 𝑑 ) ∢ = 𝑑 𝑑 1 e Ξ› / 𝐴 0 ( 𝑑 , 𝜎 ( 𝜁 ) ) 𝑓 ( πœ‚ ) Ξ” πœ‚ . ( 3 . 2 4 ) Note that Ξ› / 𝐴 0 ∈ β„› + ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) implies βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) ∈ β„› + ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) (indeed, we have 1 βˆ’ πœ‡ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) = 𝐴 0 / ( 𝐴 0 + πœ‡ Ξ› ) > 0 on [ 𝑑 1 , ∞ ) 𝕋 ), and thus the exponential function e βŠ– ( Ξ› / 𝐴 0 ) ( β‹… , 𝑑 1 ) is also well defined and positive on the entire time scale [ 𝑑 1 , ∞ ) 𝕋 , see [32, Exercise 2.28]. Thus, 𝑓 β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 implies β„Ž β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 . For simplicity of notation, for 𝑠 , 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we let 𝐾 1 ( 1 𝑑 , 𝑠 ) ∢ = 𝐴 0 ξ€œ ( 𝑠 ) 𝑑 𝑠 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) ( 𝑑 , 𝜎 ( πœ‚ ) ) Ξ₯ ( πœ‚ ) e Ξ› / 𝐴 0 ( 𝐾 𝜎 ( πœ‚ ) , 𝜎 ( 𝑠 ) ) Ξ” πœ‚ , 2 1 ( 𝑑 , 𝑠 ) ∢ = 𝐴 0 ( ξ€œ 𝑠 ) 𝑑 𝑠 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› )  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( πœ‚ ) πœ’ [ 𝛼 𝑖 ( πœ‚ ) , ∞ ) 𝕋 ( 𝑠 ) e Ξ› / 𝐴 0 ( 𝜎 ( πœ‚ ) , 𝜎 ( 𝑠 ) ) Ξ” πœ‚ . ( 3 . 2 5 ) Using the change of integration order formula in [33, Lemma 1], for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we obtain ξ€œ 𝑑 𝑑 1 ξ€œ 𝑑 𝜎 ( πœ‚ ) 1 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) ( 𝑑 , 𝜎 ( πœ‚ ) ) Ξ₯ ( πœ‚ ) e Ξ› / 𝐴 0 ( 𝜎 ( πœ‚ ) , 𝜎 ( 𝜁 ) ) 𝑔 ( 𝜁 ) 𝐴 0 = ξ€œ ( 𝜁 ) Ξ” 𝜁 Ξ” πœ‚ 𝑑 𝑑 1 ξ€œ 𝑑 𝜁 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› ) ( 𝑑 , 𝜎 ( πœ‚ ) ) Ξ₯ ( πœ‚ ) e Ξ› / 𝐴 0 ( 𝜎 ( πœ‚ ) , 𝜎 ( 𝜁 ) ) 𝑔 ( 𝜁 ) 𝐴 0 = ξ€œ ( 𝜁 ) Ξ” πœ‚ Ξ” 𝜁 𝑑 𝑑 1 𝐾 1 ( 𝑑 , 𝜁 ) 𝑔 ( 𝜁 ) Ξ” 𝜁 , ( 3 . 2 6 ) and similarly ξ€œ 𝑑 𝑑 1 ξ€œ 𝑑 𝜎 ( πœ‚ ) 1 e βˆ’ Ξ› / ( 𝐴 0 + πœ‡ Ξ› )  ( 𝑑 , 𝜎 ( πœ‚ ) ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( πœ‚ ) πœ’ [ 𝛼 𝑖 ( πœ‚ ) , ∞ ) 𝕋 ( 𝜁 ) e Ξ› / 𝐴 0 ( 𝜎 ( πœ‚ ) , 𝜎 ( 𝜁 ) ) 𝑔 ( 𝜁 ) 𝐴 0 = ξ€œ ( 𝜁 ) Ξ” 𝜁 Ξ” πœ‚ 𝑑 𝑑 1 𝐾 2 ( 𝑑 , 𝜁 ) 𝑔 ( 𝜁 ) Ξ” 𝜁 . ( 3 . 2 7 ) Therefore, we can rewrite (3.23) in the equivalent form of the integral operator ( ξ€œ β„‹ 𝑔 ) ( 𝑑 ) = 𝑑 𝑑 1 ξ€Ί 𝐾 1 ( 𝑑 , πœ‚ ) + 𝐾 2 ( ξ€» ξ€Ί 𝑑 𝑑 , πœ‚ ) 𝑔 ( πœ‚ ) Ξ” πœ‚ f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 2 8 ) whose kernel is nonnegative. Consequently, using (3.22), (3.24), and (3.28), we obtain that 𝑓 β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 implies β„Ž β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 ; this and Lemma 2.5 yield that 𝑔 β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 . Therefore, from (3.14), we infer that if 𝑓 β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 , then π‘₯ β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 too. On the other hand, by Lemma 2.4, π‘₯ has the following representation: ξ€œ π‘₯ ( 𝑑 ) = 𝑑 𝑑 1 𝒳 1 ( ξ€Ί 𝑑 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) Ξ” πœ‚ f o r 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 . ( 3 . 2 9 ) Since π‘₯ is eventually nonnegative for any eventually nonnegative function 𝑓 , we infer that the kernel 𝒳 1 of the integral on the right-hand side of (3.29) is eventually nonnegative. Indeed, assume to the contrary that π‘₯ β‰₯ 0 on [ 𝑑 1 , ∞ ) 𝕋 but 𝒳 1 is not nonnegative, then we may pick 𝑑 2 ∈ [ 𝑑 1 , ∞ ) 𝕋 and find 𝑠 ∈ [ 𝑑 1 , 𝑑 2 ) 𝕋 such that 𝒳 1 ( 𝑑 2 , 𝜎 ( 𝑠 ) ) < 0 . Then, letting 𝑓 ( 𝑑 ) ∢ = βˆ’ m i n { 𝒳 1 ( 𝑑 2 , 𝜎 ( 𝑑 ) ) , 0 } β‰₯ 0 for 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we are led to the contradiction π‘₯ ( 𝑑 2 ) < 0 , where π‘₯ is defined by (3.29). To prove that 𝒳 1 is eventually positive, set π‘₯ ( 𝑑 ) ∢ = 𝒳 1 ( 𝑑 , 𝑠 ) for 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , where 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 , to see that π‘₯ β‰₯ 0 and ( 𝐴 0 π‘₯ Ξ” ) Ξ” ≀ 0 on [ 𝑠 , ∞ ) 𝕋 , which implies 𝐴 0 π‘₯ Ξ” is nonincreasing on [ 𝑠 , ∞ ) 𝕋 . So that, we may let 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 so large that π‘₯ Ξ” (i.e., 𝐴 0 π‘₯ Ξ” ) is of fixed sign on [ 𝑠 , ∞ ) 𝕋 βŠ‚ [ 𝑑 1 , ∞ ) 𝕋 . The initial condition and (A1) together with π‘₯ Ξ” ( 𝑠 ) = 1 / 𝐴 0 ( 𝑠 ) > 0 imply that π‘₯ Ξ” > 0 on [ 𝑠 , ∞ ) 𝕋 . Consequently, we have π‘₯ ( 𝑑 ) = 𝒳 1 ( 𝑑 , 𝑠 ) > 𝒳 1 ( 𝑠 , 𝑠 ) = 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 βŠ‚ [ 𝑑 1 , ∞ ) 𝕋 .
(iv)(i) Clearly, 𝒳 1 ( β‹… , 𝑑 0 ) is an eventually positive solution of (3.1).
The proof is completed.

Let us introduce the following condition: ( 𝐴 4 ) 𝐴 0 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ + ) with ξ€œ ∞ 𝑑 0 1 𝐴 0 ( πœ‚ ) Ξ” πœ‚ = ∞ . ( 3 . 3 0 )

Remark 3.2. It is well known that (A4) ensures existence of 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 such that π‘₯ ( 𝑑 ) π‘₯ Ξ” ( 𝑑 ) β‰₯ 0 for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , for any nonoscillatory solution π‘₯ of (3.1). This fact follows from the formula π‘₯ ( 𝑑 ) = π‘₯ ( 𝑠 ) + 𝐴 0 ( 𝑠 ) π‘₯ Ξ” ξ€œ ( 𝑠 ) 𝑑 𝑠 1 𝐴 0 ξ€œ ( πœ‚ ) Ξ” πœ‚ βˆ’ 𝑑 𝑠 1 𝐴 0 ⎑ ⎒ ⎒ ⎣ ξ€œ ( πœ‚ ) πœ‚ 𝑠  𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ξ€· 𝛼 ( 𝜁 ) π‘₯ 𝑖 ξ€Έ ⎀ βŽ₯ βŽ₯ ⎦ ( 𝜁 ) Ξ” 𝜁 Ξ” πœ‚ ( 3 . 3 1 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 , obtained by integrating (3.1) twice, where 𝑠 ∈ [ 𝑑 0 , ∞ ) 𝕋 . In the case when (A4) holds, (iii) of Theorem 3.1 can be assumed to hold with Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ + 0 ) , which means that any positive (negative) solution is nondecreasing (nonincreasing).

Remark 3.3. Let (A4) hold and exist 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 and the function Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ + 0 ) satisfying inequality (3.4), then the assertions (i), (iii), and (iv) of Theorem 3.1 are also valid on [ 𝑑 1 , ∞ ) 𝕋 .

Remark 3.4. It should be noted that (3.4) is also equivalent to the inequality Ξ› Ξ” Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 +  ( 𝑑 ) + πœ‡ ( 𝑑 ) Ξ› ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝜎 ( 𝑑 ) , 𝛼 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) ≀ 0 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 3 . 3 2 ) see (3.20) and compare with [26, 28, 29, 34].

Example 3.5. For 𝕋 = ℝ , (3.4) has the form Ξ› ξ…ž ( 1 𝑑 ) + 𝐴 0 Ξ› ( 𝑑 ) 2 (  𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( ξ‚» βˆ’ ξ€œ 𝑑 ) e x p 𝑑 𝛼 𝑖 ( 𝑑 ) Ξ› ( πœ‚ ) 𝐴 0 ξ‚Ό ξ€Ί 𝑑 ( πœ‚ ) d πœ‚ ≀ 0 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ ℝ , ( 3 . 3 3 ) see [8] for the case 𝐴 0 ( 𝑑 ) ≑ 1 , 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ , and [35] for 𝑛 = 1 , 𝛼 1 ( 𝑑 ) = 𝑑 , 𝑑 ∈ [ 𝑑 0 , ∞ ) ℝ .

Example 3.6. For 𝕋 = β„• , (3.4) becomes Ξ› Ξ” Ξ› ( π‘˜ ) + 2 ( π‘˜ ) 𝐴 0 +  ( π‘˜ ) + Ξ› ( π‘˜ ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( π‘˜ ) π‘˜  𝑗 = 𝛼 𝑖 ( π‘˜ ) 𝐴 0 ( 𝑗 ) 𝐴 0 ξ€Ί π‘˜ ( 𝑗 ) + Ξ› ( 𝑗 ) ≀ 0 βˆ€ π‘˜ ∈ 1 ξ€Έ , ∞ β„• , ( 3 . 3 4 ) where the product over the empty set is assumed to be equal to one; see [1, 18] (or (1.10)) for 𝑛 = 1 , 𝛼 1 ( π‘˜ ) = π‘˜ + 1 , π‘˜ ∈ [ π‘˜ 0 , ∞ ) β„• , and [20] for 𝑛 = 1 , 𝐴 0 ( π‘˜ ) ≑ 1 , 𝛼 1 ( π‘˜ ) = π‘˜ + 1 , π‘˜ ∈ [ π‘˜ 0 , ∞ ) β„• . 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 = π‘ž β„€ with π‘ž ∈ ( 1 , ∞ ) ℝ , under the same assumption on the product as in the previous example, condition (3.4) reduces to the inequality 𝐷 π‘ž Ξ› Ξ› ( 𝑑 ) + 2 ( 𝑑 ) 𝐴 0 +  ( 𝑑 ) + ( π‘ž βˆ’ 1 ) 𝑑 Ξ› ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) l o g π‘ž ( 𝑑 )  πœ‚ = l o g π‘ž ( 𝛼 𝑖 ( 𝑑 ) ) 𝐴 0 ( π‘ž πœ‚ ) 𝐴 0 ( π‘ž πœ‚ ) + ( π‘ž βˆ’ 1 ) π‘ž πœ‚ Ξ› ( π‘ž πœ‚ ) ≀ 0 ( 3 . 3 5 ) for all 𝑑 ∈ [ 𝑑 1 , ∞ ) π‘ž β„€ .

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 ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ( 4 . 1 ) where 𝐡 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) for 𝑖 ∈ [ 1 , 𝑛 ] β„• .

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 ] β„• , 𝐡 𝑖 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) with 𝐴 𝑖 ( 𝑑 ) β‰₯ 𝐡 𝑖 ( 𝑑 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 . Assume further that (3.4) admits a solution Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ + 0 ) for some 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 , then the first fundamental solution 𝒴 1 of (4.1) satisfies 𝒴 1 ( 𝑑 , 𝑠 ) β‰₯ 𝒳 1 ( 𝑑 , 𝑠 ) > 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 , where 𝒳 1 denotes the first fundamental solution of (3.1).

Proof. We consider the initial value problem ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑓 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , π‘₯ Ξ” ξ€· 𝑑 0 ξ€Έ ξ€Ί 𝑑 = 0 , π‘₯ ( 𝑑 ) ≑ 0 f o r 𝑑 ∈ βˆ’ 1 , 𝑑 0 ξ€» 𝕋 , ( 4 . 2 ) where 𝑓 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) . Let 𝑔 ∈ 𝐢 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ ) , and define the function π‘₯ as ξ€œ π‘₯ ( 𝑑 ) = 𝑑 𝑑 1 𝒳 1 ( ξ€Ί 𝑑 𝑑 , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) Ξ” πœ‚ βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 . ( 4 . 3 ) By the Leibnitz rule (see [32, Theorem 1.117]), for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 , we have π‘₯ Ξ” ( ξ€œ 𝑑 ) = 𝑑 𝑑 1 𝒳 Ξ” 1 ( ξ€· 𝐴 𝑑 , 𝜎 ( πœ‚ ) ) 𝑔 ( πœ‚ ) Ξ” πœ‚ , ( 4 . 4 ) 0 π‘₯ Ξ” ξ€Έ Ξ” ξ€œ ( 𝑑 ) = 𝑑 𝑑 1 ξ€· 𝐴 0 𝒳 Ξ” 1 ξ€Έ ( β‹… , 𝜎 ( πœ‚ ) ) Ξ” ( 𝑑 ) 𝑔 ( πœ‚ ) Ξ” πœ‚ + 𝑔 ( 𝑑 ) . ( 4 . 5 ) Substituting (4.3) and (4.5) into (4.2), we get ξ€œ 𝑓 ( 𝑑 ) = 𝑑 𝑑 1 ξ€· 𝐴 0 𝒳 Ξ” 1 ξ€Έ ( β‹… , 𝜎 ( πœ‚ ) ) Ξ”  ( 𝑑 ) 𝑔 ( πœ‚ ) Ξ” πœ‚ + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ξ€œ ( 𝑑 ) 𝛼 𝑖 𝑑 ( 𝑑 ) 1 𝒳 1 ξ€· 𝛼 𝑖 ξ€Έ =  ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) Ξ” πœ‚ + 𝑔 ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• ξ€Ί 𝐡 𝑖 ( 𝑑 ) βˆ’ 𝐴 𝑖 ( ξ€» ξ€œ 𝑑 ) 𝛼 𝑖 𝑑 ( 𝑑 ) 1 𝒳 1 ξ€· 𝛼 𝑖 ( ξ€Έ =  𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) Ξ” πœ‚ + 𝑔 ( 𝑑 ) 𝑖 ∈ [ 1 , 𝑛 ] β„• ξ€Ί 𝐡 𝑖 ( 𝑑 ) βˆ’ 𝐴 𝑖 ξ€» ξ€œ ( 𝑑 ) 𝑑 𝑑 1 𝒳 1 ξ€· 𝛼 𝑖 ξ€Έ ( 𝑑 ) , 𝜎 ( πœ‚ ) 𝑔 ( πœ‚ ) Ξ” πœ‚ + 𝑔 ( 𝑑 ) , ( 4 . 6 ) where in the last step, we have used the fact that 𝒳 1 ( 𝑑 , 𝜎 ( 𝑠 ) ) ≑ 0 for all 𝑑 ∈ [ 𝑑 1 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 , ∞ ) 𝕋 . Therefore, we obtain the operator equation ξ€Ί 𝑑 𝑔 = β„‹ 𝑔 + 𝑓 o n 1 ξ€Έ , ∞ 𝕋 , ( 4 . 7 ) where ( ξ€œ β„‹ 𝑔 ) ( 𝑑 ) ∢ = 𝑑 𝑑 1  𝑖 ∈ [ 1 , 𝑛 ] β„• 𝒳 1 ξ€· 𝛼 𝑖 ( 𝐴 𝑑 ) , 𝜎 ( πœ‚ ) ξ€Έ ξ€Ί 𝑖 ( 𝑑 ) βˆ’ 𝐡 𝑖 ( ξ€» ξ€Ί 𝑑 𝑑 ) 𝑔 ( πœ‚ ) Ξ” πœ‚ f o r 𝑑 ∈ 1 ξ€Έ , ∞ 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 π‘₯ is nonnegative by (4.3). On the other hand, by Lemma 2.4, π‘₯ has the representation ξ€œ π‘₯ ( 𝑑 ) = 𝑑 𝑑 0 𝒴 1 ( ξ€Ί 𝑑 𝑑 , 𝜎 ( πœ‚ ) ) 𝑓 ( πœ‚ ) Ξ” πœ‚ βˆ€ 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 . ( 4 . 9 ) Proceeding as in the proof of the part (iii) β‡’ (iv) of Theorem 3.1, we conclude that the first fundamental solution 𝒴 1 of (4.1) satisfies 𝒴 1 ( 𝑑 , 𝑠 ) β‰₯ 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 . To complete the proof, we have to show that 𝒴 1 ( 𝑑 , 𝑠 ) β‰₯ 𝒳 1 ( 𝑑 , 𝑠 ) > 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 . Clearly, for any fixed 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 and all 𝑑 ∈ [ 𝑠 , ∞ ) 𝕋 , we have ξ€· 𝐴 0 𝒴 Ξ” 1 ξ€Έ ( β‹… , 𝑠 ) Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 𝑖 ( 𝑑 ) 𝒴 1 ξ€· 𝛼 𝑖 ξ€Έ =  ( 𝑑 ) , 𝑠 𝑖 ∈ [ 1 , 𝑛 ] β„• ξ€Ί 𝐴 𝑖 ( 𝑑 ) βˆ’ 𝐡 𝑖 ξ€» 𝒴 ( 𝑑 ) 1 ξ€· 𝛼 𝑖 ξ€Έ , ( 𝑑 ) , 𝑠 ( 4 . 1 0 ) which by the solution representation formula yields that 𝒴 1 ( 𝑑 , 𝑠 ) = 𝒳 1 ( ξ€œ 𝑑 , 𝑠 ) + 𝑑 𝑠 𝒳 1 (  𝑑 , 𝜎 ( πœ‚ ) ) 𝑖 ∈ [ 1 , 𝑛 ] β„• ξ€Ί 𝐴 𝑖 ( πœ‚ ) βˆ’ 𝐡 𝑖 ( ξ€» 𝒴 πœ‚ ) 1 ξ€· 𝛼 𝑖 ( ξ€Έ πœ‚ ) , 𝑠 Ξ” πœ‚ β‰₯ 𝒳 1 ( 𝑑 , 𝑠 ) ( 4 . 1 1 ) for all 𝑑 ∈ [ 𝑠 , ∞ ) 𝕋 . This completes the proof since the first fundamental solution 𝒳 1 satisfies 𝒳 1 ( 𝑑 , 𝑠 ) > 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 by Remark 3.3.

Corollary 4.2. Suppose that (A1), (A2), (A3), and (A5) hold, and (3.1) has a nonoscillatory solution on [ 𝑑 1 , ∞ ) 𝕋 βŠ‚ [ 𝑑 0 , ∞ ) 𝕋 , then (4.1) admits a nonoscillatory solution on [ 𝑑 2 , ∞ ) 𝕋 βŠ‚ [ 𝑑 1 , ∞ ) 𝕋 .

Corollary 4.3. Assume that (A2) and (A3) hold. (i)If (A1) holds and the dynamic inequality ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 + 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) ≀ 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , ( 4 . 1 2 ) where 𝐴 + 𝑖 ( 𝑑 ) ∢ = m a x { 𝐴 𝑖 ( 𝑑 ) , 0 } for 𝑑 ∈ [ 𝑑 0 , ∞ ) 𝕋 and 𝑖 ∈ [ 1 , 𝑛 ] β„• , has a positive solution on [ 𝑑 0 , ∞ ) 𝕋 , then (3.1) also admits a positive solution on [ 𝑑 1 , ∞ ) 𝕋 βŠ‚ [ 𝑑 0 , ∞ ) 𝕋 . (ii)If (A4) holds and there exist a sufficiently large 𝑑 1 ∈ [ 𝑑 0 , ∞ ) 𝕋 and a function Ξ› ∈ 𝐢 1 r d ( [ 𝑑 1 , ∞ ) 𝕋 , ℝ + 0 ) satisfying the inequality Ξ› Ξ” 1 ( 𝑑 ) + 𝐴 0 Ξ› ( 𝑑 ) 𝜎  ( 𝑑 ) Ξ› ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 + 𝑖 ( 𝑑 ) e βŠ– ( Ξ› / 𝐴 0 ) ξ€· 𝑑 , 𝛼 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) ≀ 0 βˆ€ 𝑑 ∈ 1 ξ€Έ , ∞ 𝕋 , ( 4 . 1 3 ) then the first fundamental solution 𝒳 1 of (3.1) satisfies 𝒳 1 ( 𝑑 , 𝑠 ) > 0 for all 𝑑 ∈ ( 𝑠 , ∞ ) 𝕋 and all 𝑠 ∈ [ 𝑑 1 , ∞ ) 𝕋 .

Proof. Consider the dynamic equation ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐴 + 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 0 f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 . ( 4 . 1 4 ) Theorem 3.1 implies that for this equation the assertions (i) and (ii) hold. Since for all 𝑖 ∈ [ 1 , 𝑛 ] β„• , we have 𝐴 𝑖 ( 𝑑 ) ≀ 𝐴 + 𝑖 ( 𝑑 ) for all 𝑑 ∈ [ 𝑑 0 , ∞ ) 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: ξ€· 𝐴 0 π‘₯ Ξ” ξ€Έ Ξ”  ( 𝑑 ) + 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ξ€· 𝛼 ( 𝑑 ) π‘₯ 𝑖 ξ€Έ ξ€Ί 𝑑 ( 𝑑 ) = 𝑔 ( 𝑑 ) f o r 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 , π‘₯ ξ€· 𝑑 0 ξ€Έ = 𝑦 1 , π‘₯ Ξ” ξ€· 𝑑 0 ξ€Έ = 𝑦 2 ξ€Ί 𝑑 , π‘₯ ( 𝑑 ) = πœ“ ( 𝑑 ) f o r 𝑑 ∈ βˆ’ 1 , 𝑑 0 ξ€Έ 𝕋 , ( 4 . 1 5 ) where 𝑦 1 , 𝑦 2 ∈ ℝ are the initial values, πœ“ ∈ 𝐢 r d ( [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 , ℝ ) is the initial function such that πœ“ has a finite left-sided limit at the initial point 𝑑 0 provided that it is left dense, 𝑔 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) is the forcing term.

Theorem 4.4. Suppose that (A2), (A3), (A4), (A5), and the following condition hold: ( 𝐴 6 ) 𝑓 , 𝑔 ∈ 𝐢 r d ( [ 𝑑 0 , ∞ ) 𝕋 , ℝ ) and πœ‘ , πœ“ ∈ 𝐢 r d ( [ 𝑑 βˆ’ 1 , 𝑑 0 ) 𝕋 , ℝ ) satisfy  𝑓 ( 𝑑 ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ( ξ€· 𝛼 𝑑 ) πœ‘ 𝑖 ( ξ€Έ  𝑑 ) ≀ 𝑔 ( 𝑑 ) βˆ’ 𝑖 ∈ [ 1 , 𝑛 ] β„• 𝐡 𝑖 ( ξ€· 𝛼 𝑑 ) πœ“ 𝑖 ( ξ€Έ ξ€Ί 𝑑 𝑑 ) βˆ€ 𝑑 ∈ 0 ξ€Έ , ∞ 𝕋 . ( 4 . 1 6 ) Moreover, let (2.1) have a positive solution π‘₯ on [ 𝑑 0 , ∞ ) 𝕋 , 𝑦 1 = π‘₯ 1 , and