Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

Abstract

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation (𝑝(𝑡)(𝜓∘𝑥)(𝑥^𝜎)^Δ)^Δ+𝑞(𝑡)𝑓∘𝑥^𝜎=0 on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the 𝑞-difference equations.

1. Introduction

In the recent decade there have been many literatures to study the oscillatory properties for second-order dynamic equations on time scales; see, for example, [1–11] and the references therein. In particular, the dynamic equation of the form𝑝(𝑡)𝑥^ΔΔ+𝑞(𝑡)𝑓∘𝑥^𝜎=0 (1.1) has been attracting one’s interesting; see, for example, [3, 5, 8]. Motivated by the papers mentioned as above, in this paper we consider the existence of nonoscillatory solutions for nonlinear dynamic equation𝑝(𝑡)𝜓∘𝑥𝑥^𝜎ΔΔ+𝑞(𝑡)𝑓∘𝑥^𝜎=0 (1.2) on a time scale 𝕋, where 𝑥^𝜎=𝑥∘𝜎.

Referring to [12, 13], a time scale 𝕋 can be defined as an arbitrary nonempty subset of the set ℝ of real numbers, with the properties that every Cauchy sequence in 𝕋 converges to a point of 𝕋 with the possible exception of Cauchy sequences which converge to a finite infimum or finite supremum of 𝕋. On any time scale 𝕋, the forward and backward jump operators are defined, respectively, by𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}, (1.3) where inf∅∶=sup𝕋 and sup∅∶=inf𝕋. A point 𝑡∈𝕋 is said to be right-scattered if 𝜎(𝑡)>𝑡, right-dense if 𝜎(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, and left-dense if 𝜌(𝑡)=𝑡. A derived set from 𝕋 is defined as follows: 𝕋^𝑘=𝕋−{𝑚} when 𝕋 has a left-scattered maximum 𝑚, otherwise 𝕋^𝑘=𝕋.

Definition 1.1. For a function 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝑘, we define the delta-derivative 𝑓^Δ(𝑡) of 𝑓(𝑡) to be the number (provided it exists) with the property that, for any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋 for some 𝛿) such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜀|𝜎(𝑡)−𝑠|∀𝑠∈𝑈. (1.4) We say that 𝑓 is delta-differentiable (or in short: differentiable) on 𝕋^𝑘 provided 𝑓^Δ(𝑡) exists for all 𝑡∈𝕋^𝑘.

For two differentiable functions 𝑓 and 𝑔 at 𝑡∈𝕋^𝑘 with 𝑔(𝑡)𝑔(𝜎(𝑡))≠0, it holds that𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡),𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝜎(𝑡))−𝑓(𝜎(𝑡))𝑔^Δ(𝑡)𝑔(𝑡)𝑔(𝜎(𝑡.)) (1.5)(1.6)

Definition 1.2. A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓 provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝕋^𝑘. By the antiderivative, the Cauchy integral of 𝑓 is defined as ∫^𝑏_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑏)−𝐹(𝑎), and ∫^∞_𝑎𝑓(𝑠)Δ𝑠=lim_𝑡→∞∫^𝑡_𝑎𝑓(𝑠)Δ𝑠.

Definition 1.3. Let 𝑓∶𝕋→ℝ be a function, where 𝑓 is called right-dense continuous (rd-continuous) if it is right continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋.

To distinguish from the traditional interval in ℝ, we define the interval in 𝕋 by[𝑎,∞)_𝕋∶={𝑡∈𝕋∶𝑎≤𝑡<∞}. (1.7) Let 𝐶_rd(𝕋) (or 𝐶_rd(𝕋,ℝ)) denote the set of all rd-continuous functions defined on 𝕋, and 𝐶¹_rd(𝕋) (or 𝐶¹_rd(𝕋,ℝ)) denote the set of all differentiable functions whose derivative is rd-continuous.

Since we are interested in the existence of nonoscillatory solutions of (1.2), we make the blanket assumption that inf𝕋=𝑡₀ and sup𝕋=∞. As defined in [1], by a solution of (1.2) we mean a nontrivial real function 𝑥(𝑡), with 𝑥(𝑡)∈𝐶¹_rd[𝑡₁,∞)_𝕋 and 𝑝(𝑡)𝜓(𝑥(𝑡))(𝑥^𝜎(𝑡))^Δ∈𝐶¹_rd[𝑡₁,∞)_𝕋 for some 𝑡₁≥𝑡₀, which satisfies (1.2) on [𝑡₁,∞)_𝕋. A solution of (1.2) is said to be nonoscillatory if it is eventually positive or eventually negative.

2. Preliminaries

Let us assume in (1.2) under consideration that(H1)𝜎∈𝐶¹_rd(𝕋,𝕋) with 𝜎^Δ(𝑡)>0 on 𝕋,(H2)𝑝,𝑞∈𝐶_rd(𝕋,ℝ) and there exists a 𝑡₁∈𝕋 such that 𝑝(𝑡)>0 and 𝑞(𝑡)≥0 on [𝑡₁,∞)_𝕋 and ∫^∞_𝑡1(Δ𝑠/𝑝(𝑠))=∞, ∫^∞_𝑡1𝑞(𝑠)Δ𝑠<∞; also, 𝑞 is not identically zero on [𝑡₁,∞)_𝕋,(H3)𝜓∈𝐶¹(ℝ,ℝ⁺) with 𝜓(𝑥)≤𝑚₀ on ℝ for some constant 𝑚₀>0 and, 𝜓 is nonincreasing on [0,∞) and nondecreasing on (−∞,0],(H4)𝑓∈𝐶¹(ℝ,ℝ) satisfies that 𝑥𝑓(𝑥)>0 for 𝑥≠0, 𝑓^′(𝑥)≥𝑚₁ on ℝ for some constant 𝑚₁>0, and 𝑓^′ is nondecreasing on [0,∞) and nonincreasing on (−∞,0].

Suppose that 𝕋={𝑥(𝑡)∈𝐶_rd(𝕋,ℝ)∶𝑡∈𝕋} is a time scale. By 𝜎 and 𝜌 we denote the forward and backward jump operators on 𝕋, respectively, and by Δ we denote the derivative on 𝕋. Let 𝜎²=𝜎∘𝜎. Then, 𝑓Δ(𝜎(𝑥))/𝜓(𝑥) is bounded below on 𝕋 by assumptions (H3)-(H4). Indeed, in case 𝜎(𝑥)=𝜎²(𝑥), we have 𝑓^′(𝜎(𝑥))/𝜓(𝑥)=𝑓Δ(𝜎(𝑥))/𝜓(𝑥) and the assertion holds. In case 𝜎(𝑥)<𝜎²(𝑥), by the definition of delta-derivative and the mean value theorem, there exits a constant 𝑐∈[𝜎(𝑥),𝜎²(𝑥)] such that𝑓Δ(𝜎(𝑥=𝑓𝜎))²(𝑥)−𝑓(𝜎(𝑥))𝜎²(𝑥)−𝜎(𝑥)=𝑓^′(𝑐), (2.1) and then𝑓Δ(𝜎(𝑥))𝜓(𝑥)=𝑓^′(𝑐)𝜓(𝑥)≥𝑚₁𝑚₀. (2.2)

Furthermore, 𝑓Δ(𝜎(𝑥)) is nondecreasing on [0,∞)_𝕋 by virtue of (H4). Indeed, for any 𝑥,𝑦∈[0,∞)_𝕋 with 𝑥≤𝑦, there are four cases to consider. In case 𝜎(𝑥)=𝜎²(𝑥) and 𝜎(𝑦)=𝜎²(𝑦), 𝑓Δ(𝜎(𝑥))=𝑓^′(𝜎(𝑥))≤𝑓^′(𝜎(𝑦))=𝑓Δ(𝜎(𝑦)). In case 𝜎(𝑥)=𝜎²(𝑥) and 𝜎(𝑦)<𝜎²(𝑦), we have 𝑓Δ(𝜎(𝑥))=𝑓^′(𝜎(𝑥)) and𝑓Δ𝜎𝑦=𝑓𝜎²𝑦−𝑓𝜎𝑦𝜎²𝑦−𝜎𝑦=𝑓^′𝑐_𝑦, (2.3) where 𝑐_𝑦∈[𝜎(𝑦),𝜎²(𝑦)]. Since 𝑐_𝑦≥𝜎(𝑥), it follows that 𝑓Δ(𝜎(𝑥))≤𝑓Δ(𝜎(𝑦)). The other cases can be shown likewise.

As a consequence, we see from assumption (H3) that 𝑓^Δ(𝜎(𝑥))/𝜓(𝑥) is nondecreasing on [0,∞)_𝕋. Similarly, we can show that 𝑓Δ(𝜎(𝑥))/𝜓(𝑥) is nonincreasing on (−∞,0]_𝕋 provided 𝜎(𝑥)∈(−∞,0]_𝕋.

As thus, we can extract the essences as above and obtain a result as follows.

Lemma 2.1. Suppose that 𝕋 is a time scale and 𝑓Δ(𝑥)/𝜓(𝑥) is defined on it. Then it follows that(i)𝑓Δ(𝜎(𝑥))/𝜓(𝑥) is positively bounded below and (ii)𝑓Δ(𝜎(𝑥))/𝜓(𝑥) is nondecreasing on [0,∞)_𝕋 and nonincreasing on (−∞,0]_𝕋.

For a given 𝑎∈𝕋 with 𝑎≠0, we introduce a function on 𝕋 as follows:Γ_𝑎(𝑥)=^𝑥_𝑎𝜓𝜌(𝑢)𝑓(𝑢)Δ𝑢for𝑥∈𝕋with𝑥≠0. (2.4) Then the function Γ_𝑎(𝑥) possesses the following properties. (i)If 𝑎>0, then Γ_𝑎(𝑥) is strictly increasing for 𝑥>0 and Γ_𝑎⁻¹(𝑦)≥𝑎 is strictly increasing for 𝑦≥0.(ii)If 𝑎<0, then Γ_𝑎(𝑥) is strictly decreasing for 𝑥<0 and Γ_𝑎⁻¹(𝑦)≤𝑎 is strictly decreasing for 𝑦≥0.(iii)If 𝑎≠0, then 𝐹(Γ_𝑎⁻¹(𝑦)) is nondecreasing for 𝑦≥0 (by Lemma 2.1 (ii)). Here 𝐹 is defined as in (2.6).

For the sake of convenience, we let𝑅(𝑡)=𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^Δ𝑓(𝑥^𝜎(𝑡,𝐹(𝑥)=𝑓Δ(𝜎(𝑥))))𝜓(𝑥) (2.5)(2.6) whenever they are defined.

Note that, if 𝑥(𝑡) is an eventually negative solution of (1.2), then 𝑦(𝑡)=−𝑥(𝑡) satisfies that𝑝(𝑡)𝜓∘𝑦𝑦^𝜎ΔΔ+𝑞(𝑡)𝑓∘𝑦^𝜎=0, (2.7)

where𝜓𝑦=𝜓−𝑦,𝑓𝑦=−𝑓−𝑦 (2.8) possess the same properties as 𝜓 and 𝑓, respectively. Therefore, in what follows we will restrict our attention to the eventually positive solutions of (1.2).

3. Main Results

Before entering our main discussions, we remark that 𝑥([𝑡₁,∞)_𝕋) is a time scale when 𝑥∈𝐶¹_rd[𝑡₁,∞)_𝕋 and 𝑥 is monotonic on [𝑡₁,∞)_𝕋, where 𝑡₁∈𝕋. In the following discussions, the notations Δ,𝜎 and 𝜌 will act upon the time scale 𝑥([𝑡₁,∞)_𝕋), while Δ, 𝜎, and 𝜌 will do on 𝕋. Then 𝜎∘𝑥=𝑥∘𝜎 when 𝑥 is strictly increasing on [𝑡₁,∞)_𝕋.

Lemma 3.1. Suppose that 𝑡₁∈𝕋 and 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋. Then it holds that 𝑅(𝑡)=^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝑅(𝑠)𝑅(𝜎(𝑠))𝐹(𝑥(𝑠))𝑝(𝑠)Δ𝑠 (3.1) for all 𝑡∈[𝑡₁,∞)_𝕋 and 𝑅(𝑡)=^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝑅(𝑠)𝑅(𝜎(𝑠))𝐹Γ_𝑥⁻¹_𝜎(𝑡1)∫_𝑡^𝜌(𝑠)₁𝑅(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠 (3.2) for all 𝑡∈[𝜎(𝑡₁),∞)_𝕋.

Proof. Without loss of generality, let 𝑝(𝑡)>0 and 𝑞(𝑡)≥0 on [𝑡₁,∞)_𝕋. Then it follows from (1.2) that 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^ΔΔ=−𝑞(𝑡)𝑓𝑥^𝜎(𝑡)≤0∀𝑡∈𝑡₁,∞_𝕋. (3.3) We assert that 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^Δ>0∀𝑡∈𝑡₁,∞_𝕋. (3.4) Otherwise, note from assumption (H2) that 𝑞 is not identically zero on [𝑡₁,∞)_𝕋, (3.3) implies that there exits a 𝑡₂∈[𝑡₁,∞)_𝕋 and a constant 𝑚>0 such that 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^Δ≤−𝑚∀𝑡∈𝑡₂,∞_𝕋. (3.5) Consequently we find that 𝑥^𝜎(𝑡)≤𝑥^𝜎𝑡₂−𝑚𝑚₀^𝑡_𝑡2Δ𝑠𝑝(𝑠)⟶−∞as𝑡⟶∞, (3.6) which contradicts 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋. Since (3.4) holds, it is clear that 𝑥^𝜎(𝑡)^Δ>0,𝑅(𝑡)>0∀𝑡∈𝑡₁,∞_𝕋. (3.7)
On the other hand, we see from (3.3) that 𝑅(𝑡) is nonincreasing on [𝑡₁,∞)_𝕋, which, associated with (3.7), means that lim_𝑡→∞𝑅(𝑡) exists as a finite number.
Next we will show that ^∞_𝑡1𝑅(𝑠)𝑅(𝜎(𝑠))𝐹(𝑥(𝑠))𝑝(𝑠)Δ𝑠<∞. (3.8) Note that 𝑥^𝜎 is strictly increasing due to assumption (H1) and (3.7); by (1.2), (1.5) and (1.6) and the chain rule [13, Theorem 1.93] we have 𝑅^Δ(𝑡)=𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^ΔΔ𝑓𝑥^𝜎2(𝑡)𝑓(𝑥^𝜎(𝑡))𝑓𝑥^𝜎2(𝑡)−𝑝(𝜎(𝑡))𝜓𝑥^𝜎(𝑡)𝑥^𝜎2(𝑡)^Δ𝑓Δ𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)^Δ𝑓(𝑥^𝜎(𝑡))𝑓𝑥^𝜎2(𝑡)(𝑡)−𝑝(𝜎(𝑡𝜓𝑥=−𝑞))^𝜎(𝑡)𝑥^𝜎2(𝑡)^Δ𝑓𝑥^𝜎2(𝑡)⋅𝑓Δ𝑥^𝜎(𝑡)𝑝𝜓∘𝑥𝑥^𝜎Δ(𝑡)𝑝(𝑡)𝜓(𝑥(𝑡))𝑓(𝑥^𝜎(𝑡(𝑡)−𝑅(𝜎(𝑡)𝑅(𝑡𝐹(𝑥(𝑡))=−𝑞))))𝑝(𝑡). (3.9) Taking Δ-integral on (3.9) from 𝑡₁ to 𝑡, we obtain that 𝑅(𝑡)−𝑅𝑡₁=−^𝑡_𝑡1𝑞(𝑠)Δ𝑠−^𝑡_𝑡1𝑅(𝑠)𝑅(𝜎(𝑠))𝐹(𝑥(𝑠))𝑝(𝑠)Δ𝑠. (3.10)
Now that the limit of 𝑅(𝑡) exists as 𝑡→∞, by assumption (H2) and (3.10) we see that (3.8) holds. Note that 𝐹(𝑥(𝑡)) is positive bounded below (see Lemma 2.1(i)), (3.8) infers that lim_𝑡→∞𝑅(𝑡)=0. To sum up, it is easy to see that (3.10) yields (3.1).
Next we prove (3.2). Again note that 𝑥^𝜎 is strictly increasing on [𝑡₁,∞)_𝕋, by the substitution theorem [13, Theorem 1.98], it follows that ^𝑡_𝑡1𝑅(𝑡)Δ𝑠𝑝(𝑡)=^𝑡_𝑡1𝜓(𝑥(𝑠))𝑥^𝜎(𝑠)^ΔΔ𝑠𝑓(𝑥^𝜎(𝑠=))^𝑥𝜎_𝑥^(𝑡)𝜎_(𝑡1)𝜓𝜌(𝑢)Δ𝑢𝑓(𝑢), (3.11) which, together with the definition of Γ_𝑎, induces 𝑥(𝑡)=Γ_𝑥⁻¹_{𝜎(𝑡1)𝑡}^𝜌(𝑡)₁𝑅(𝑠)Δ𝑠𝑝(𝑠)∀𝑡∈𝜎𝑡₁,∞_𝕋, (3.12) where we have used 𝜌(𝜎(𝑡))=𝑡 (see assumption (H1)). Now let us substitute 𝑥(𝑡) in (3.12) into (3.1) and then we obtain (3.2). The proof is complete.

Theorem 3.2. Suppose that 𝑡₁∈𝕋. Then 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋 (or, on [𝜎(𝑡₁),∞)_𝕋) if and only if there exists a constant 𝛼>0 and a function 𝜑∈𝐶_rd([𝑡₁,∞)_𝕋,ℝ⁺) such that 𝜑(𝑡)≥^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝜑(𝑠)𝜑(𝜎(𝑠))𝑝(𝑠)𝐹Γ_𝛼⁻¹_𝑡^𝜌(𝑠)₁𝜑(𝑢)Δ𝑢𝑝(𝑢)Δ𝑠 (3.13) for all 𝑡∈[𝜎(𝑡₁),∞)_𝕋.

Proof. Suppose that 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋. Then, by Lemma 3.1, 𝑅 defined as in (2.5) satisfies 𝑅∈𝐶_rd([𝑡₁,∞)_𝕋,ℝ⁺) and (3.13) holds, where 𝛼=𝑥^𝜎(𝑡₁).
Conversely, suppose that there exists a constant 𝛼>0 and a function 𝜑∈𝐶_rd(𝕋,ℝ⁺) such that (3.13) holds. Let 𝑦₀(𝑡)≡0 and define a sequence of functions {𝑦_𝑛(𝑡)}^∞_𝑛=0 on [𝜎(𝑡₁),∞)_𝕋 as follows: 𝑦_𝑛+1(𝑡)=^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝑦_𝑛(𝑠)𝑦_𝑛(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑦_𝑛(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠. (3.14) It is clear that 0≤𝑦₁(𝑡)≤𝜑(𝑡) on [𝜎(𝑡₁),∞)_𝕋. Suppose that 𝑦_𝑛−1(𝑡)≤𝑦_𝑛(𝑡)≤𝜑(𝑡) on [𝜎(𝑡₁),∞)_𝕋. Then, by the monotone of 𝐹(Γ_𝑎⁻¹(𝑦)) for 𝑦≥0, we learn that 𝑦_𝑛+1(𝑡)≥^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝑦_𝑛−1(𝑠)𝑦_𝑛−1(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑦_𝑛−1(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠=𝑦_𝑛(𝑡) (3.15) as well as 𝑦_𝑛+1≤𝜑(𝑡). (3.16) So by the mathematical induction we obtain that 0≤𝑦_𝑛(𝑡)≤𝑦_𝑛+1(𝑡)≤𝜑(𝑡)∀𝜎𝑡₁,∞_𝕋, (3.17) which means that there exists a function 𝑦 such that lim_𝑛→∞𝑦_𝑛(𝑡)=𝑦(𝑡)≤𝜑(𝑡)∀𝑡∈𝜎𝑡₁,∞_𝕋. (3.18)
Now by Lebesgue's domination convergence theorem on time scales [14, Chapter 5], we may deduce from (3.14) that𝑦(𝑡)=^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝑦(𝑠)𝑦(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑦(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠. (3.19) Let 𝑥(𝑡)=Γ_𝛼⁻¹_𝑡^𝜌(𝑡)₁𝑦(𝑠)Δ𝑠𝑝(𝑠). (3.20) Then 𝑥(𝑡)≥𝛼 on [𝜎(𝑡₁),∞)_𝕋 and 𝑥^𝜎(𝑡) is strictly increasing on [𝑡₁,∞)_𝕋. Moreover, 𝑥([𝜎(𝑡₁),∞)_𝕋) is another time scale with 𝜎∘𝑥=𝑥∘𝜎. Therefore, by the theorem on derivative of the inverse [13, Theorem 1.97] we learn that 𝑥^𝜎(𝑡)^Δ=𝑓𝑥^𝜎(𝑡)𝑦(𝑡)𝜓(𝑥(𝑡𝑝(𝑡)𝑡))for𝑡∈₁,∞_𝕋. (3.21) Now from (3.20) we have Γ_𝛼𝑥^𝜎(𝑡)=^𝑡_𝑡1𝑦(𝑠)Δ𝑠𝑝(𝑠)for𝑡∈𝑡₁,∞_𝕋, (3.22) which implies that 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^Δ=𝑦(𝑡)𝑓𝑥^𝜎(𝑡), (3.23) and this results in 𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^ΔΔ=𝑦^𝜎(𝑡)𝑓Δ𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)^Δ+𝑓𝑥^𝜎(𝑡)𝑦^Δ(𝑡)=𝑦^𝜎(𝑡)𝑓Δ𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)^Δ+𝑓𝑥^𝜎(𝑡(𝑡)−𝑦(𝑡)𝑦(𝜎(𝑡𝐹(𝑥(𝑡)−𝑞))))𝑝(𝑡)=𝑦^𝜎(𝑡)𝑓Δ𝑥^𝜎(𝑡)𝑥^𝜎(𝑡)^Δ−𝑞(𝑡)𝑓𝑥^𝜎(𝑡)−𝑝(𝑡)𝜓(𝑥(𝑡))𝑥^𝜎(𝑡)^Δ𝑦^𝜎(𝑡)𝐹(𝑥(𝑡))𝑝(𝑡)=−𝑞(𝑡)𝑓𝑥^𝜎(𝑡), (3.24) where we have imposed formulas (3.19) and (3.20) for the second equal sign and (3.23) for the third, respectively.
Now we see from (3.24) that 𝑥(𝑡) defined by (3.20) is a positive solution of (1.2). The proof is complete.

In the remainder of this section, we define formally a sequence of functions {𝑄_𝑛(𝑡)}^∞_𝑛=0 as follows. Let 𝑡₁∈𝕋, 𝛼>0, and𝑄₀(𝑡)=^∞_𝑡𝑞(𝑠)𝑄Δ𝑠,₁(𝑡)=^∞_𝑡𝑄₀(𝑠)𝑄₀(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑄₀(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠 (3.25)(3.26) as well as𝑄_𝑛+1(𝑡)=^∞_𝑡𝑄₀(𝑠)+𝑄_𝑛(𝑠)𝑄₀(𝜎(𝑠))+𝑄_𝑛(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑄₀(𝑢)+𝑄_𝑛(𝑢)/𝑝(𝑢)Δ𝑢𝑝(𝑠)Δ𝑠. (3.27) Then 𝑄₁(𝑡)≤𝑄₂(𝑡). By induction, it follows that0<𝑄_𝑛(𝑡)≤𝑄_𝑛+1(𝑡)for𝑛=1,2,3,…. (3.28)

If 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋 for some 𝑡₁∈𝕋, then (3.1) holds by Lemma 3.1 and hence𝑄₀(𝑡)≤𝑅(𝑡)∀𝑡∈𝜎𝑡₁,∞_𝕋, (3.29) which, together with (3.26), results in𝑄₁(𝑡)≤^∞_𝑡𝑅(𝑠)𝑅(𝜎(𝑠))𝐹Γ_𝑎⁻¹∫_𝑡^𝜌(𝑠)₁𝑅(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠∀𝑡∈𝜎𝑡₁,∞_𝕋. (3.30) Let us now define 𝑅₀(𝑡) by𝑅₀(𝑡)=^∞_𝑡𝑅(𝑠)𝑅(𝜎(𝑠))𝐹Γ_𝑎⁻¹∫_𝑡^𝜌(𝑠)₁𝑅(𝑢)Δ𝑢/𝑝(𝑢)𝑝(𝑠)Δ𝑠,𝑡∈𝜎𝑡₁,∞_𝕋. (3.31) Then, in view of (3.2) and (3.30), we have𝑄₀(𝑡)+𝑄₁(𝑡)≤𝑄₀(𝑡)+𝑅₀(𝑡)=𝑅(𝑡)∀𝑡∈𝜎𝑡₁,∞_𝕋. (3.32) By the mathematical induction again we educe𝑄₀(𝑡)+𝑄_𝑛(𝑡)≤𝑅(𝑡)∀𝑡∈𝜎𝑡₁,∞_𝕋,𝑛=1,2,3,…. (3.33)

Now we learn from (3.28) and (3.33) that {𝑄_𝑛(𝑡)}^∞₀ is well defined and converges to some function 𝑄(𝑡) when 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋.

Conversely, suppose that {𝑄_𝑛(𝑡)}^∞_𝑛=0 is well defined andlim_𝑛→∞𝑄_𝑛(𝑡)=𝑄(𝑡)<∞on𝜎𝑡₁,∞_𝕋. (3.34) Then, by (3.28), we have 𝑄_𝑛(𝑡)≤𝑄(𝑡) for all 𝑛≥1. Hence, by Lebesgue's domination convergence theorem on time scales, we may obtain from (3.27) that𝑄(𝑡)=^∞_𝑡𝑄₀(𝑠)+𝑄(𝑠)𝑄₀(𝜎(𝑠))+𝑄(𝜎(𝑠))𝐹Γ_𝛼⁻¹∫_𝑡^𝜌(𝑠)₁𝑄₀(𝑢)+𝑄(𝑢)/𝑝(𝑢)Δ𝑢𝑝(𝑠)Δ𝑠. (3.35)

Let 𝜑(𝑡)=𝑄₀(𝑡)+𝑄(𝑡). Then, from (3.35) we see that (3.13) holds. Furthermore, by means of Theorem 3.2, (1.2) has a solution 𝑥(𝑡)>0 on [𝜎(𝑡₁),∞)_𝕋.

To sum up, we obtain our last result as follows.

Corollary 3.3. Suppose that 𝑡₁∈𝕋. Then 𝑥(𝑡) is a solution of (1.2) with 𝑥(𝑡)>0 on [𝑡₁,∞)_𝕋 (or, on [𝜎(𝑡₁),∞)_𝕋) if and only if there exists a constant 𝛼>0 such that the sequence of functions {𝑄_𝑛(𝑡)}^∞_𝑛=0 defined as in (3.25)–(3.27) is well defined and lim_𝑛→∞𝑄_𝑛(𝑡)=𝑄(𝑡)<∞on𝜎𝑡₁,∞_𝕋. (3.36)

Example 3.4. Let 𝕋=[0,∞). Suppose in (1.2) that 𝑝(𝑡)=𝑡+1, 𝑞(𝑡)=1/(4𝑡²), 𝜓(𝑥)=1/2(1+|𝑥|), and 𝑓(𝑥)=𝑥. Let 𝜑(𝑡)=1/(2𝑡). It is easy to verify that Γ₁^−1𝑠₁𝜑(𝑠)d𝑠𝑝(𝑠)=𝑠,𝐹Γ₁^−1𝑠₁𝜑(𝑢)d𝑢𝑝(𝑢)=12(1+|𝑠|)for𝑠≥1 (3.37) as well as ^∞_𝑡𝑞(𝑠)d𝑠+^∞_𝑡𝜑²(𝑠)𝑝(𝑠)𝐹Γ₁^−1𝑠₁𝜑(𝑢)d𝑢𝑝(𝑢)3𝑑𝑠=(𝑡)8𝑡≤𝜑,𝑡≥1. (3.38) By Theorem 3.2, (1.2) has a solution 𝑥(𝑡)>0 on [1,∞).

Example 3.5. Let ℎ>0 and 𝑞₀>1, and let ℕ₀ be a set of nonnegative integers. Suppose in (1.2) that 𝜓(𝑥)=1 and 𝑓(𝑥)=𝑥.
In case 𝕋=ℎℕ₀, let𝑝(𝑡)(𝑡)=1=1,𝑞().2𝑡+2ℎ)(2𝑡+ℎ (3.39) Then, for a given 𝜑(𝑡)=1/(2𝑡+ℎ) and any given 𝛼>0, we have ^∞_𝑡𝑞(𝑠)Δ𝑠≤^∞_𝑡Δ𝑡𝜎()=12𝑠+ℎ)(2𝑠+ℎ2(),2𝑡+ℎ_∞𝑡𝜑(𝑠)𝜑(𝜎(𝑠))𝑝(𝑠)𝐹Γ_𝛼⁻¹₀^𝜌(𝑠)𝜑(𝑢)Δ𝑢𝑝(𝑢)Δ𝑠=^∞_𝑡Δ𝑠((𝑠))≤12𝑠+ℎ)(2𝜎+ℎ2(2𝑡+ℎ) (3.40) and hence (3.13) holds. By means of Theorem 3.2, (1.2) has a solution 𝑥(𝑡)>0 on 𝕋=ℎℕ₀.
In case 𝕋={𝑞^𝑘₀∶𝑘∈ℕ₀}, let𝑝(𝑡)(𝑡)=1=2,𝜑𝑡(𝑡)=1,𝑞(𝑡)𝑡.2𝜎 (3.41) Then, for any given 𝛼>0, we have ^∞_𝑡𝑞(𝑠)Δ𝑠+^∞_𝑡𝜑(𝑠)𝜑(𝜎(𝑠))𝑝(𝑠)𝐹Γ_𝛼⁻¹₁^𝜌(𝑠)𝜑(𝑢)Δ𝑢𝑝(𝑢)Δ𝑠=𝜑(𝑡), (3.42) and hence Theorem 3.2 implies that (1.2) has a solution 𝑥(𝑡)>0 on 𝕋={𝑞^𝑘₀∶𝑘∈ℕ₀}.