New Inequalities of Opial's Type on Time Scales and Some of Their Applications

Abstract

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

1. Introduction

In 1960 Opial [1] proved that if 𝑦 is an absolutely continuous function on [𝑎,𝑏] with 𝑦(𝑎)=𝑦(𝑏)=0,then^𝑏_𝑎𝑦(𝑡)𝑦^′(𝑡())𝑑𝑡≤𝑏−𝑎4^𝑏_𝑎𝑦^′(𝑡)²𝑑𝑡. (1.1) In further simplifying the proof of the Opial inequality which had already been simplified by Olech [2], Beesack [3], Levinson [4], Mallows [5], and Pederson [6], it is proved that if 𝑦 is real absolutely continuous on (0,𝑏) and with 𝑦(0)=0, then^𝑏₀𝑦(𝑡)𝑦^′(𝑡𝑏)𝑑𝑡≤2^𝑏₀𝑦^′(𝑡)²𝑑𝑡. (1.2) Since the discovery of Opial’s inequality much work has been done, and many papers which deal with new proofs, various generalizations, extensions, and their discrete analogues have appeared in the literature. The discrete analogy of the (1.1) has been proved in [7] and the discrete analogy of (1.2) has been proved in [8, Theorem  5.2.2]. It is worth to mention here that many results concerning differential inequalities carry over quite easily to corresponding results for difference inequalities, while other results seem to be completely different from their continuous counterparts.

In recent years, there has been much research activity concerning the qualitative theory of dynamic equations on time scales. It has been created in [9] in order to unify the study of differential and difference equations, and it also extends these classical cases to cases “in between,” for example, to the so-called 𝑞-difference equations. The general idea is to prove a result for a dynamic equation or a dynamic inequality where the domain of the unknown function is a so-called time scale 𝕋, which may be an arbitrary closed subset of the real numbers ℝ. A cover story article in New Scientist [10] discusses several possible applications of time scales. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [11]), that is, when 𝕋=ℝ,𝕋=ℕ and 𝕋=𝑞^ℕ0={𝑞^𝑡∶𝑡∈ℕ₀} where 𝑞>1.

One of the main subjects of the qualitative analysis on time scales is to prove some new dynamic inequalities. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of dynamic equations on time scales. In the following, we recall some results obtained for dynamic inequalities on time scales that serve and motivate the contents of this paper.

Bohner and Kaymakçalan in [12] established the time scale analogy of (1.2) and proved that if 𝑦∶[0,𝑏]∩𝕋→ℝ is delta differentiable with 𝑦(0)=0, then^ℎ₀𝑦(𝑡)+𝑦^𝜎(𝑡)𝑦^Δ(𝑡)Δ𝑡≤ℎ^ℎ₀𝑦^Δ(𝑡)²Δ𝑡. (1.3) Also in [12] the authors proved that if 𝑟 and 𝑞 are positive rd-continuous functions on [0,𝑏]_𝕋,∫^𝑏₀(Δ𝑡/𝑟(𝑡))<∞,𝑞 nonincreasing and 𝑦∶[0,𝑏]∩𝕋→ℝ is delta differentiable with 𝑦(0)=0, then^𝑏₀𝑞^𝜎(𝑡)𝑦(𝑡)+𝑦^𝜎(𝑡)𝑦^Δ(𝑡)Δ𝑡≤^𝑏₀Δ𝑡𝑟(𝑡)^𝑏₀𝑟(𝑡)𝑞(𝑡)𝑦^Δ(𝑡)²Δ𝑡. (1.4) Since the discovery of the inequalities (1.3) and (1.4) some papers which deal with new proofs, various generalizations, and extensions of (1.3) and (1.4) have appeared in the literature, we refer to the results in [13–15] and the references cited therein. Karpuz et al. [13] proved an inequality similar to the inequality (1.4) of the form^𝑏_𝑎𝑞(𝑡)𝑦(𝑡)+𝑦^𝜎(𝑡)𝑦^Δ(𝑡)Δ𝑡≤𝐾_𝑞(𝑎,𝑏)^𝑏_𝑎𝑦^Δ(𝑡)²Δ𝑡, (1.5) where 𝑞 is a positive rd-continuous function on [𝑎,𝑏], and 𝑦∶[𝑎,𝑏]∩𝕋→ℝ is delta differentiable with 𝑦(𝑎)=0, and𝐾_𝑞(𝑎,𝑏)=2^𝑏_𝑎𝑞²(𝑢)(𝜎(𝑢)−𝑎)Δ𝑢^1/2. (1.6) Wong et al. [14] and Sirvastava et al. [15] proved that if 𝑟 is a positive rd-continuous function on [𝑎,𝑏], we have^𝑏_𝑎𝑟(𝑡)𝑦(𝑡)^𝑝𝑦^Δ(𝑡)^𝑞𝑞Δ𝑡≤()𝑝+𝑞𝑏−𝑎^𝑝𝑏_𝑎𝑟(𝑡)𝑦^Δ(𝑡)^𝑝+𝑞Δ𝑡, (1.7) where 𝑦∶[𝑎,𝑏]∩𝕋→ℝ is delta differentiable with 𝑦(𝑎)=0.

In [16] the author proved that if 𝑦∶[𝑎,𝑋]∩𝕋→ℝ is delta differentiable with 𝑦(𝑎)=0, then^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾₁𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (1.8) where 𝑝,𝑞 are positive real numbers such that 𝑝+𝑞>1, and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑎,𝑋)_𝕋 such that ∫^𝑋_𝑎𝑟^{−1/(𝑝+𝑞−1)}(𝑡)Δ𝑡<∞ and𝐾₁=𝑞𝑎,𝑋,𝑝,𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑋_𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{−𝑞/𝑝𝑥}_𝑎(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}. (1.9) If 𝑦(𝑎) is replaced by 𝑦(𝑏), then^𝑏_𝑋𝑠(𝑥)𝑦(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾₂𝑋,𝑏,𝑝,𝑞^𝑏_𝑋𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (1.10) where𝐾₂=𝑞𝑋,𝑏,𝑝,𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑏_𝑋(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{−𝑞/𝑝𝑏}_𝑥(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}. (1.11) For contributions of different types of inequalities on time scales, we also refer the reader to the papers [16–24] and the references cited therein.

The paper is organized as follows: in Section 2, we will prove some new dynamic inequalities of Opial’s type on time scales of the form^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (1.12) where 𝑝,𝑞 are positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and 𝐾(𝑎,𝑋,𝑝,𝑞) is the coefficient of the inequality. As special cases, we derive some differential and discrete inequalities on continuous and discrete time scales. In Section 3, we will apply the obtained inequalities in Section 2 on the second-order half-linear dynamic equation𝑟(𝑡)𝑦^Δ(𝑡)^𝛾Δ+𝑞(𝑡)𝑦^𝜎(𝑡)^𝛾=0,on[𝑎,𝑏]_𝕋, (1.13) where 𝕋 is an arbitrary time scale, 0<𝛾≤1 is a quotient of odd positive integers, 𝑟 and 𝑞 are real rd-continuous functions defined on 𝕋 with 𝑟(𝑡)>0. In particular, we will prove several results related to the problems:(i)obtain lower bounds for the spacing 𝛽−𝛼 where 𝑦 is a solution of (1.13) and satisfies 𝑦(𝛼)=𝑦^Δ(𝛽)=0, or 𝑦^Δ(𝛼)=𝑦(𝛽)=0,(ii)obtain lower bounds for the spacing between consecutive zeros of solutions of (1.13).

Our motivation comes from that fact that the inequalities obtained in the literature cannot be applied on the half-linear dynamic equation (1.13) to prove results related to the problems (i)-(ii).

2. Main Results

The main inequalities will be proved in this section by making use of the Hölder inequality (see [25, Theorem 6.13])^ℎ_𝑎𝑓(𝑡)𝑔(𝑡)Δ𝑡≤^ℎ_𝑎𝑓(𝑡)^𝛾Δ𝑡^1/𝛾ℎ_𝑎𝑔(𝑡)^𝜈Δ𝑡^1/𝜈, (2.1) where 𝑎,ℎ∈𝕋 and 𝑓;𝑔∈𝐶_rd(𝕀,ℝ),𝛾>1 and 1/𝜈+1/𝛾=1, and the inequality (see [8, page 51])2^𝑟−1𝑎^𝑟+𝑏^𝑟≤(𝑎+𝑏)^𝑟≤𝑎^𝑟+𝑏^𝑟,where𝑎,𝑏≥0,0≤𝑟≤1. (2.2) For completeness, we recall the following concepts related to the notion of time scales. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. We assume throughout that 𝕋 has the topology that it inherits from the standard topology on the real numbers ℝ. The forward jump operator and the backward jump operator are defined by𝜎(𝑡)∶=inf{s∈𝕋∶𝑠>𝑡},𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}, (2.3) where sup∅=inf𝕋. A point 𝑡∈𝕋 is said to be left dense if 𝜌(𝑡)=𝑡 and 𝑡>inf𝕋, is right dense if 𝜎(𝑡)=𝑡, is left scattered if 𝜌(𝑡)<𝑡 and right scattered if 𝜎(𝑡)>𝑡.

A function 𝑔∶𝕋→ℝ is said to be right dense continuous (rd-continuous) provided 𝑔 is continuous at right dense points and at left dense points in 𝕋, left hand limits exist and are finite. The set of all such rd-continuous functions is denoted by 𝐶_rd(𝕋).

The graininess function 𝜇 for a time scale 𝕋 is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡, and for any function 𝑓∶𝕋→ℝ the notation 𝑓^𝜎(𝑡) denotes 𝑓(𝜎(𝑡)). We will assume that sup𝕋=∞ and define the time scale interval [𝑎,𝑏]_𝕋 by [𝑎,𝑏]_𝕋∶=[𝑎,𝑏]∩𝕋.

Fix 𝑡∈𝕋 and let 𝑥∶𝕋→ℝ. Define 𝑥^Δ(𝑡) to be the number (if it exists) with the property that given any 𝜖>0 there is a neighborhood 𝑈 of 𝑡 with[𝑥(𝜎(𝑡))−𝑥(𝑠)]−𝑥^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜖|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (2.4) In this case, we say 𝑥^Δ(𝑡) is the (delta) derivative of 𝑥 at 𝑡 and that 𝑥 is (delta) differentiable at 𝑡.

We will frequently use the results in the following theorem which is due to Hilger [9].

Assume that 𝑔∶𝕋→ℝ and let 𝑡∈𝕋.(i)If 𝑔 is differentiable at 𝑡, then 𝑔 is continuous at 𝑡.(ii)If 𝑔 is continuous at 𝑡 and 𝑡 is right scattered, then 𝑔 is differentiable at 𝑡 with 𝑔^Δ(𝑡)=𝑔(𝜎(𝑡))−𝑔(𝑡)𝜇(𝑡). (2.5)(iii)If 𝑔 is differentiable and 𝑡 is right-dense, then 𝑔^Δ(𝑡)=lim_𝑠→𝑡𝑔(𝑡)−𝑔(𝑠).𝑡−𝑠 (2.6)(iv)If 𝑔 is differentiable at 𝑡, then 𝑔(𝜎(𝑡))=𝑔(𝑡)+𝜇(𝑡)𝑔^Δ(𝑡).

In this paper, we will refer to the (delta) integral which we can define as follows: if 𝐺^Δ(𝑡)=𝑔(𝑡), then the Cauchy (delta) integral of 𝑔 is defined by^𝑡_𝑎𝑔(𝑠)Δ𝑠∶=𝐺(𝑡)−𝐺(𝑎). (2.7) We will make use of the following product and quotient rules for the derivative of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔^𝜎≠0, here 𝑔^𝜎=𝑔∘𝜎) of two differentiable functions 𝑓 and 𝑔𝑓𝑔^Δ=𝑓^Δ𝑔+𝑓^𝜎𝑔^Δ=𝑓𝑔^Δ+𝑓^Δ𝑔^𝜎,𝑓𝑔^Δ=𝑓^Δ𝑔−𝑓𝑔^Δ𝑔𝑔^𝜎. (2.8) We say that a function 𝑝∶𝕋→ℝ is regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0,𝑡∈𝕋. The integration by parts formula is given by^𝑏_𝑎𝑓(𝑡)𝑔^Δ(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑡)𝑔^𝜎(𝑡)Δ𝑡. (2.9) It can be shown (see [25]) that if 𝑔∈𝐶_rd(𝕋), then the Cauchy integral 𝐺(𝑡)∶=∫^𝑡_𝑡0𝑔(𝑠)Δ𝑠 exists, 𝑡₀∈𝕋, and satisfies 𝐺^Δ(𝑡)=𝑔(𝑡),𝑡∈𝕋. The integration on discrete time scales is defined by^𝑏_𝑎𝑓(𝑡)Δ𝑡=_{[)𝑡∈𝑎,𝑏}𝜇(𝑡)𝑓(𝑡). (2.10) To prove the main results, we need the formula𝑥^𝛾(𝑡)^Δ=𝛾¹₀ℎ𝑥^𝜎+(1−ℎ)𝑥^𝛾−1𝑑ℎ𝑥^Δ(𝑡),𝛾>0, (2.11) which is a simple consequence of Keller’s chain rule [25, Theorem 1.90]. The books on the subject of time scales by Bohner and Peterson [25, 26] summarize and organize much of time scale calculus and contain some results for dynamic equations on time scales. To admit functions such that 𝑦 and 𝑦^Δ may change sign on (𝑎,𝑏)_𝕋, we note that if𝑦(𝑥)=^𝑥_𝑎𝑦^Δ(𝑥)Δ𝑥,then𝑣(𝑥)=^𝑥_𝑎𝑦^Δ(𝑥)Δ𝑥≥𝑦(𝑥), (2.12) with equality if and only if 𝑦^Δ does not change sign. (The same result holds if 𝑦=−∫^𝑏_𝑥𝑦^Δ(𝑥)Δ𝑥 and 𝑣(𝑥)=∫^𝑏_𝑥|𝑦^Δ(𝑥)|Δ𝑥.) Now, we are ready to state and prove the main results.

Theorem 2.1. Let 𝕋 be a time scale with 𝑎,𝑋∈𝕋 and 𝑝,𝑞 be positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑎,𝑋)_𝕋 such that ∫^𝑋_𝑎𝑟^{−1/(𝑝+𝑞−1)}(𝑡)Δ𝑡<∞. If 𝑦∶[𝑎,𝑋]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑎)=0, then one has ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾₁𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.13) where 𝐾₁𝑎,𝑋,𝑝,q=sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝(𝑥)𝑠(𝑥)𝑟(𝑥)+2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑋_𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{𝑞/𝑝𝑥}_𝑎𝑟^{−1/(𝑝+𝑞−1)}(𝑡)Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}. (2.14)

Proof. Since 𝑦(𝑥)≤^𝑥_𝑎𝑦^Δ(𝑡)Δ𝑡,for𝑥∈[𝑎,𝑋]_𝕋, (2.15) and 𝑟 is nonnegative on (𝑎,𝑋)_𝕋, then it follows from the Hölder inequality (2.1) with 𝑓(𝑡)=1(𝑟(𝑡))^{1/(𝑝+𝑞)},𝑔(𝑡)=(𝑟(𝑡))^{1/(𝑝+𝑞)}𝑦^Δ(𝑡),𝛾=𝑝+𝑞𝑝+𝑞−1,𝜈=𝑝+𝑞, (2.16) that ^𝑥_𝑎𝑦^Δ(𝑡)Δ𝑡≤^𝑥_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)/(𝑝+𝑞)𝑥}_𝑎𝑟(𝑡)𝑦^Δ(𝑡)^𝑝+𝑞Δ𝑡^{1/(𝑝+𝑞)}. (2.17) Then, for 𝑎≤𝑥≤𝑋, we get (noting that 𝑦(𝑎)=0) that 𝑦(𝑥)^𝑝≤^𝑥_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{𝑝((𝑝+𝑞−1)/(𝑝+𝑞))𝑥}_𝑎𝑟(𝑡)𝑦^Δ(𝑡)^𝑝+𝑞Δ𝑡^{𝑝/(𝑝+𝑞)}. (2.18) Since 𝑦^𝜎=𝑦+𝜇𝑦^Δ, we have 𝑦(𝑥)+𝑦^𝜎(𝑥)=2𝑦(𝑥)+𝜇𝑦^Δ(𝑥). (2.19) Applying the inequality (2.2), we get (where 𝑝≤1) that 𝑦+𝑦^𝜎𝑝=2𝑦(𝑥)+𝜇𝑦^Δ(𝑥)^𝑝≤2^𝑝𝑦^𝑝+𝜇^𝑝𝑦^Δ𝑝. (2.20) Setting 𝑧(𝑥)∶=^𝑥_𝑎𝑟(𝑡)𝑦^Δ(𝑡)^𝑝+𝑞Δ𝑡, (2.21) we see that 𝑧(𝑎)=0, and 𝑧^Δ(𝑥)=𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞>0. (2.22) From this, we get that 𝑦^Δ(𝑥)^𝑝+𝑞=𝑧^Δ(𝑥)𝑟(𝑥),𝑦^Δ(𝑥)^𝑞=𝑧^Δ(𝑥)𝑟(𝑥)^{𝑞/(𝑝+𝑞)}. (2.23) Thus, since 𝑠 is nonnegative on (𝑎,𝑋)_𝕋, we have from (2.20) and (2.23) that 𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞≤2^𝑝𝑠(𝑥)𝑦(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞+𝜇^𝑝(𝑥)𝑠(𝑥)𝑦^Δ𝑝+𝑞≤2^𝑝𝑠(𝑥)1𝑟(𝑥)^{𝑞/(𝑝+𝑞)}×^𝑥_𝑎1𝑟^{1/(𝑝+𝑞−1)}(𝑡)Δ𝑡^{𝑝((𝑝+𝑞−1)/(𝑝+𝑞))}×(𝑧(𝑥))^{𝑝/(𝑝+𝑞)}𝑧^Δ(𝑥)^{𝑞/(𝑝+𝑞)}+𝜇^𝑝(𝑥)𝑠(𝑥)𝑧^Δ(𝑥)𝑟(𝑥). (2.24) This implies that ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤2^𝑝𝑋_𝑎𝑠(𝑥)1𝑟(𝑥)^{𝑞/(𝑝+𝑞)}×^𝑥_𝑎1𝑟^{1/(𝑝+𝑞−1)}(𝑡)Δ𝑡^{𝑝((𝑝+𝑞−1)/(𝑝+𝑞))}×(𝑧(𝑥))^{𝑝/(𝑝+𝑞)}𝑧^Δ(𝑥)^{𝑞/(𝑝+𝑞)}Δ𝑥+^𝑋_𝑎𝜇^𝑝𝑠(𝑥)𝑟(𝑥)𝑧^Δ(𝑥)Δ𝑥≤2^𝑝𝑋_𝑎𝑠(𝑥)1𝑟(𝑥)^{𝑞/(𝑝+𝑞)}×^𝑥_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{𝑝((𝑝+𝑞−1)/(𝑝+𝑞))}×(𝑧(𝑥))^{𝑝/(𝑝+𝑞)}𝑧^Δ(𝑥)^{𝑞/(𝑝+𝑞)}Δ𝑥+sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝𝑠(𝑥)𝑟(𝑥)^𝑋_𝑎𝑧^Δ(𝑥)Δ𝑥. (2.25) Supposing that the integrals in (2.25) exist and again applying the Hölder inequality (2.1) with indices 𝑝+𝑞/𝑝 and 𝑝+𝑞/𝑞 on the first integral on the right-hand side of (2.25), we have ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤2^𝑝𝑋_𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}1𝑟(𝑥)^{𝑞/𝑝𝑥}_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}×^𝑋_𝑎(𝑧(𝑥))^𝑝/𝑞𝑧^Δ(𝑥)Δ𝑥^{𝑞/(𝑝+𝑞)}+sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝𝑠(𝑥)𝑟(𝑥)^𝑋_𝑎𝑧^Δ(𝑥)Δ𝑥. (2.26) From (2.22), and the chain rule (2.11), we obtain 𝑧^𝑝/𝑞(𝑥)𝑧^Δ(𝑥)≤𝑞𝑧𝑝+𝑞^{(𝑝+𝑞)/𝑞}(𝑥)^Δ. (2.27) Substituting (2.27) into (2.26) and using the fact that 𝑧(𝑎)=0, we have that ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤2^𝑝𝑋_𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}1𝑟(𝑥)^{𝑞/𝑝𝑥}_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}×𝑝𝑝+𝑞^{𝑞/(𝑝+𝑞)𝑋}_𝑎(𝑧(𝑡))^{(𝑝+𝑞)/𝑞Δ}Δ𝑡^{𝑞/(𝑝+𝑞)}+𝜇^𝑝𝑠(𝑥)𝑟(𝑥)^𝑋_𝑎𝑧^Δ(𝑥)=Δ𝑥_𝑋𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}1𝑟(𝑥)^{𝑞/𝑝𝑥}_𝑎1(𝑟(𝑡))^{1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}×2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}𝑧(𝑋)+sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝𝑠(𝑥)𝑟(𝑥)𝑧(𝑋). (2.28) Using (2.21), we have from the last inequality that ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾₁𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.29) which is the desired inequality (2.13). The proof is complete.

Here, we only state the following theorem, since its proof is the same as that of Theorem 2.1, with [𝑎,𝑋] replaced by [𝑏,𝑋] and |𝑦(𝑥)|=∫^𝑏_𝑥|𝑦^Δ(𝑡)|Δ𝑡.

Theorem 2.2. Let 𝕋 be a time scale with 𝑋,𝑏∈𝕋 and let 𝑝,𝑞 be positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑋,𝑏)_𝕋 such that ∫^𝑏_𝑋𝑟^{−1/(𝑝+𝑞−1)}(𝑡)Δ𝑡<∞. If 𝑦∶[𝑋,𝑏]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑏)=0, then one has ^𝑏_𝑋𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾₂𝑋,𝑏,𝑝,𝑞^𝑏_𝑋𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.30) where 𝐾₂𝑋,𝑏,𝑝,𝑞=sup_{𝑋≤𝑥≤𝑏}𝜇^𝑝(𝑥)𝑠(𝑥)𝑟(𝑥)+2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑏_𝑋(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{𝑞/𝑝𝑏}_𝑥(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}. (2.31)

Note that when 𝕋=ℝ, we have 𝑦^𝜎=𝑦 and 𝜇(𝑥)=0. Then from Theorems 2.1 and 2.2 we have the following differential inequalities.

Corollary 2.3. Assume that 𝑝,𝑞 are positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟,𝑠 be nonnegative continuous functions on  (𝑎,𝑋)_ℝ such that ∫^𝑋_𝑎(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}𝑑𝑡<∞. If 𝑦∶[𝑎,𝑋]∩ℝ→ℝ⁺ is differentiable with 𝑦(𝑎)=0, then one has ^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)^𝑝𝑦^′(𝑥)^𝑞𝑑𝑥≤𝐶₁𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^′(𝑥)^𝑝+𝑞𝑑𝑥, (2.32) where 𝐶₁𝑎,𝑋,𝑝,𝑞=2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑋_𝑎(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{𝑞/𝑝𝑥}_𝑎(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}𝑑𝑡^{(𝑝+𝑞−1)}𝑑𝑥^{𝑝/(𝑝+𝑞)}. (2.33)

Corollary 2.4. Assume that 𝑝,𝑞 are positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟,𝑠 be nonnegative continuous functions on(𝑋,𝑏)_ℝ such that ∫^𝑏_𝑋(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}𝑑𝑡<∞. If 𝑦∶[𝑋,𝑏]∩ℝ→ℝ⁺ is delta differentiable with 𝑦(𝑏)=0, then one has ^𝑏_𝑋𝑠(𝑥)𝑦(𝑥)^𝑝𝑦^′(𝑥)^𝑞𝑑𝑥≤𝐶₂𝑋,𝑏,𝑝,𝑞^𝑏_𝑋𝑟(𝑥)𝑦^′(𝑥)^𝑝+𝑞𝑑𝑥, (2.34) where 𝐶₂𝑋,𝑏,𝑝,𝑞=2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑏_𝑋(𝑠(𝑥))^{(𝑝+𝑞)/𝑝}(𝑟(𝑥))^{𝑞/𝑝𝑏}_𝑥(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}𝑑𝑡^{(𝑝+𝑞−1)}𝑑𝑥^{𝑝/(𝑝+𝑞)}. (2.35)

In the following, we assume that there exists ℎ∈(𝑎,𝑏) which is the unique solution of the equation𝐾𝑝,𝑞=𝐾₁𝑎,ℎ,𝑝,𝑞=𝐾₂ℎ,𝑏,𝑝,𝑞<∞, (2.36) where 𝐾₁(𝑎,ℎ,𝑝,𝑞) and 𝐾₂(ℎ,𝑏,𝑝,𝑞) are defined as in Theorems 2.1 and 2.2.

Theorem 2.5. Let 𝕋 be a time scale with 𝑎,𝑏∈𝕋 and let 𝑝,𝑞 be positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑎,𝑏)_𝕋 such that ∫^𝑏_𝑎(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}Δ𝑡<∞. If 𝑦∶[𝑎,𝑏]∩𝕋→ℝ⁺ is delta differentiable with y(𝑎)=0=𝑦(𝑏), then one has ^𝑏_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾𝑝,𝑞^𝑏_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥. (2.37)

Proof. Since ^𝑏_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥=^𝑋_𝑎𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞+Δ𝑥_𝑏𝑋𝑠(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥, (2.38) then the rest of the proof will be a combination of Theorems 2.1 and 2.2 and hence is omitted. The proof is complete.

As a special case if 𝑟=𝑠 in Theorem 2.1, then we obtain the following result.

Corollary 2.6. Let 𝕋 be a time scale with 𝑎,𝑋∈𝕋 and let 𝑝,𝑞 be positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and let 𝑟 be a nonnegative rd-continuous function on (𝑎,𝑋)_𝕋 such that ∫^𝑋_𝑎(𝑟(𝑡))^{−1/(𝑝+𝑞−1)}Δ𝑡<∞. If 𝑦∶[𝑎,𝑋]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑎)=0, then one has ^𝑋_𝑎𝑟(𝑥)𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐾^∗₁𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑟(𝑥)𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.39) where 𝐾^∗₁𝑎,𝑋,𝑝,𝑞=sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝(𝑥)+2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑋_𝑎𝑟(𝑥)^𝑥_𝑎𝑟^{−1/(𝑝+𝑞−1)}(𝑡)Δ𝑡^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}. (2.40)

From Theorems 2.2 and 2.5 one can derive similar results by setting 𝑟=𝑠. The details are left to the reader.

On a time scale 𝕋, we note from the chain rule (2.11) that(𝑡−𝑎)^𝑝+𝑞Δ=𝑝+𝑞¹₀[ℎ(𝜎(𝑡)−𝑎)+(1−ℎ)(𝑡−𝑎)]^{𝑝+𝑞−1}≥𝑑ℎ𝑝+𝑞¹₀[ℎ(𝑡−𝑎)+(1−ℎ)(𝑡−𝑎)]^{𝑝+𝑞−1}=()𝑑ℎ𝑝+𝑞𝑡−𝑎^{𝑝+𝑞−1}. (2.41) This implies that^𝑋_𝑎(𝑥−𝑎)^{(𝑝+𝑞−1)}Δ𝑥≤^𝑋_𝑎1()𝑝+𝑞𝑥−𝑎^𝑝+𝑞Δ()Δ𝑥=𝑋−𝑎^𝑝+𝑞.𝑝+𝑞 (2.42) From this and (2.40) (by putting 𝑟(𝑡)=1), we get that𝐾^∗₁𝑎,𝑋,𝑝,𝑞=2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}×^𝑋_𝑎(𝑥−𝑎)^{(𝑝+𝑞−1)}Δ𝑥^{𝑝/(𝑝+𝑞)}≤2^𝑝𝑞𝑝+𝑞^{𝑞/(𝑝+𝑞)}(𝑋−𝑎)^𝑝+𝑞𝑝+𝑞^{𝑝/(𝑝+𝑞)}+max_{𝑎≤𝑥≤𝑋}𝜇^𝑝(𝑥)=max_{𝑎≤𝑥≤𝑋}𝜇^𝑝(𝑥)+2^𝑝𝑞^{𝑞/(𝑝+𝑞)}()𝑝+𝑞𝑋−𝑎^𝑝. (2.43) So setting 𝑟=1 in (2.39) and using (2.43), we have the following result.

Corollary 2.7. Let 𝕋 be a time scale with 𝑎,𝑋∈𝕋 and let 𝑝,𝑞 be positive real numbers such that 𝑝≤1 and 𝑝+𝑞>1. If 𝑦∶[𝑎,𝑋]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑎)=0, then one has ^𝑋_𝑎𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐿𝑎,𝑋,𝑝,𝑞^𝑋_𝑎𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.44) where 𝐿𝑎,𝑋,𝑝,𝑞∶=2^𝑝𝑞^{𝑞/(𝑝+𝑞)}×()𝑝+𝑞𝑋−𝑎^𝑝+sup_{𝑎≤𝑥≤𝑋}𝜇^𝑝(𝑥). (2.45)

Remark 2.8. Note that when 𝕋=ℝ, we have 𝑦^𝜎=𝑦, 𝜇(𝑥)=0 and then the inequality (2.44) becomes ^𝑋_𝑎𝑦(𝑥)^𝑝𝑦^′(𝑥)^𝑞𝑞𝑑𝑥≤^{𝑞/(𝑝+𝑞)}×()𝑝+𝑞𝑋−𝑎^𝑝𝑋_𝑎𝑦^′(𝑥)^𝑝+𝑞𝑑𝑥. (2.46) Note also that when 𝑝=1 and 𝑞=1, then the inequality (2.46) becomes ^𝑋_𝑎𝑦(𝑥)𝑦^′(𝑥())𝑑𝑥≤𝑋−𝑎2^𝑋_𝑎𝑦^′(𝑥)²𝑑𝑥, (2.47) which is the Opial inequality (1.2).

When 𝕋=ℕ, we have form (2.44) the following discrete Opial’s type inequality.

Corollary 2.9. Assume that 𝑝,𝑞 are positive real numbers such that 𝑝≤1,𝑝+𝑞>1 and {𝑟_𝑖}_{0≤𝑖≤𝑁} are a nonnegative real sequence. If {𝑦_𝑖}_{0≤𝑖≤𝑁} is a sequence of positive real numbers with 𝑦(0)=0, then ^𝑁−1_𝑛=1𝑟(𝑛)𝑦(𝑛)+𝑦(𝑛+1)^𝑝Δ𝑦(𝑛)^𝑞≤2^𝑝𝑞^{𝑞/(𝑝+𝑞)}(𝑁−𝑎)^𝑝𝑝+𝑞+1^𝑁−1_𝑛=0𝑟(𝑛)Δ𝑦(𝑛)^𝑝+𝑞. (2.48)

The inequality (2.44) has an immediate application to the case where 𝑦(𝑎)=𝑦(𝑏)=0. Choose 𝑋=(𝑎+𝑏)/2 and apply (2.40) to [𝑎,𝑋] and [𝑋,𝑏] and adding we obtain the following inequality.

Corollary 2.10. Let 𝕋 be a time scale with 𝑎,𝑏∈𝕋 and let 𝑝,𝑞 be positive real numbers such that 𝑝≤1 and 𝑝+𝑞>1. If 𝑦∶[𝑎,𝑏]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑎)=0=𝑦(𝑏), then one has ^𝑏_𝑎𝑦(𝑥)+𝑦^𝜎(𝑥)^𝑝𝑦^Δ(𝑥)^𝑞Δ𝑥≤𝐹𝑎,𝑏,𝑝,𝑞^𝑏_𝑎𝑦^Δ(𝑥)^𝑝+𝑞Δ𝑥, (2.49) where 𝐹𝑞𝑎,𝑏,𝑝,𝑞∶=^{𝑞/(𝑝+𝑞)}()𝑝+𝑞𝑏−𝑎^𝑝+sup_{𝑎≤𝑥≤𝑏}𝜇^𝑝(𝑥). (2.50)

From this inequality, we have the following discrete Opial type inequality.

Corollary 2.11. Assume that 𝑝,𝑞 are positive real numbers such that 𝑝≤1 and 𝑝+𝑞>1. If {𝑦_𝑖}_{0≤𝑖≤𝑁} is a sequence of real numbers with 𝑦(0)=0=𝑦(𝑁), then ^𝑁−1_𝑛=1𝑟(𝑛)𝑦(𝑛)+𝑦(𝑛+1)^𝑝Δ𝑦(𝑛)^𝑞≤𝑞^{𝑞/(𝑝+𝑞)}()𝑝+𝑞𝑁−𝑎^𝑝+1^𝑁−1_𝑛=0𝑟(𝑛)Δ𝑦(𝑛)^𝑝+𝑞. (2.51)

By setting 𝑝=𝑞=1 in (2.49) we have the following Opial type inequality on a time scale.

Corollary 2.12. Let 𝕋 be a time scale with 𝑎,𝑏∈𝕋. If 𝑦∶[𝑎,𝑋]∩𝕋→ℝ⁺ is delta differentiable with 𝑦(𝑎)=0=𝑦(𝑏), then one has ^𝑏_𝑎𝑦(𝑥)+𝑦^𝜎(𝑥)𝑦^Δ(𝑥())Δ𝑥≤𝑏−𝑎2+sup_{𝑎≤𝑥≤𝑏}𝜇(𝑥)^𝑏_𝑎𝑦^Δ(𝑥)²Δ𝑥. (2.52)

As special cases from (2.52) on the continuous and discrete spaces, that is, when 𝕋=ℝ and 𝕋=ℕ, one has the following inequalities.

Corollary 2.13. If 𝑦∶[𝑎,𝑏]∩𝕋→ℝ is differentiable with 𝑦(𝑎)=0=𝑦(𝑏), then one has the Opial inequality ^𝑏_𝑎𝑦(𝑥)𝑦^′(𝑥())𝑑𝑥≤𝑏−𝑎4^𝑏_𝑎𝑦^′(𝑥)²𝑑𝑥. (2.53)

Corollary 2.14. If {y_𝑖}_{0≤𝑖≤𝑁} is a sequence of real numbers with 𝑦(0)=0=𝑦(𝑁), then ^𝑁−1_𝑛=1𝑦(𝑛)((𝑛≤𝑁+𝑦𝑛+1)Δ𝑦)2+1^𝑁−1_𝑛=0Δ𝑦(𝑛)². (2.54)

3. Applications

Our aim in this section, is to apply the dynamic inequalities of Opial’s type proved in Section 2 to prove several results related to the problems (i)-(ii) for the second-order half-linear dynamic equation𝑟(𝑡)𝑦^Δ(𝑡)^𝛾Δ+𝑞(𝑡)𝑦^𝜎(𝑡)^𝛾=0,on[𝑎,𝑏]_𝕋, (3.1) on an arbitrary time scale 𝕋, where 0<𝛾≤1 is a quotient of odd positive integers, 𝑟 and 𝑞 are real rd-continuous functions defined on 𝕋 with 𝑟(𝑡)>0. The terminology half-linear arises because of the fact that the space of all solutions of (3.1) is homogeneous, but not generally additive. Thus, it has just “half” of the properties of a linear space. It is easily seen that if 𝑦(𝑡) is a solution of (3.1), then so also is 𝑐𝑦(𝑡).

By a solution of (3.1) on an interval 𝕀, we mean a nontrivial real-valued function 𝑦∈𝐶_rd(𝕀), which has the property that 𝑟(𝑡)𝑦^Δ(𝑡)∈𝐶¹_rd(𝕀) and satisfies (3.1) on 𝕀. We say that a solution 𝑦 of (3.1) has a generalized zero at 𝑡 if 𝑦(𝑡)=0 and has a generalized zero in (𝑡,𝜎(𝑡)) in case 𝑦(𝑡)𝑦^𝜎(𝑡)<0 and 𝜇(𝑡)>0. Equation (3.1) is disconjugate on the interval [𝑡₀,𝑏]_𝕋, if there is no nontrivial solution of (3.1) with two (or more) generalized zeros in [𝑡₀,𝑏]_𝕋.

Equation (3.1) is said to be nonoscillatory on [𝑡₀,∞)_𝕋 if there exists 𝑐∈[𝑡₀,∞)_𝕋 such that this equation is disconjugate on [𝑐,𝑑]_𝕋 for every 𝑑>𝑐. In the opposite case (3.1) is said to be oscillatory on [𝑡₀,∞)_𝕋. The oscillation of solutions of (3.1) may equivalently be defined as follows: a nontrivial solution 𝑦(𝑡) of (3.1) is called oscillatory if it has infinitely many (isolated) generalized zeros in [𝑡₀,∞)_𝕋; otherwise it is called nonoscillatory. So that the solution 𝑦(𝑡) of (3.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. This means that the property of oscillation or nonoscillation is the behavior in the neighborhood of the infinite points.

We say that (3.1) is right disfocal (left disfocal) on [𝛼,𝛽]_𝕋 if the solutions of (3.1) such that 𝑦^Δ(𝛼)=0(𝑦^Δ(𝛽)=0) have no generalized zeros in [𝛼,𝛽]_𝕋.

Note that (3.1) in its general form involves some different types of differential and difference equations depending on the choice of the time scale 𝕋. For example when 𝕋=ℝ, (3.1) becomes a second-order half-linear differential equation and 𝜎(𝑡)=𝑡. When 𝕋=ℤ, (3.1) becomes a second-order half-linear difference equation and 𝜎(𝑡)=𝑡+1. When 𝕋=ℎℕ, (3.1) becomes a generalized difference equation and 𝜎(𝑡)=𝑡+ℎ. When 𝕋={𝑡∶𝑡=𝜚^𝑘,𝑘∈ℕ₀,𝜚>1}, (3.1) becomes a quantum difference equation (see [11]) and 𝜎(𝑡)=𝜚𝑡. Note also that the results in this paper can be applied on the time scales 𝕋=ℕ²={𝑡²∶𝑡∈ℕ},𝕋₂={√𝑛∶𝑛∈ℕ₀},𝕋₃={√³𝑛∶𝑛∈ℕ₀}, and when 𝕋=𝕋_𝑛={𝑡_𝑛∶𝑛∈ℕ₀} where {𝑡_𝑛} is the set of harmonic numbers. In these cases we see that when 𝕋=ℕ²₀={𝑡²∶𝑡∈ℕ₀}, we have 𝜎(𝑡)=(√𝑡+1)² and when 𝕋=𝕋_𝑛={𝑡_𝑛∶𝑛∈ℕ} where (𝑡_𝑛} is the harmonic numbers that are defined by 𝑡₀=0 and 𝑡_𝑛=∑^𝑛_𝑘=11/𝑘,𝑛∈ℕ₀, we have 𝜎(𝑡_𝑛)=𝑡_𝑛+1. When 𝕋₂={√𝑛∶𝑛∈ℕ₀}, we have 𝜎(𝑡)=√𝑡²+1, and when 𝕋₃={√³𝑛∶𝑛∈ℕ₀}, we have 𝜎(𝑡)=√³𝑡³+1.

Perhaps the best known existence results of types (i)-(ii) for a special case of (3.1) (when 𝛾=1 and 𝑟(𝑡)=1) is due to Bohner et al. [27], where they extended the Lyapunov inequality obtained for differential equations in [28]. In particular the authors in [27] considered the dynamic equation𝑦^ΔΔ(𝑡)+𝑞(𝑡)𝑦^𝜎(𝑡)=0, (3.2) where 𝑞(𝑡) is a positive rd-continuous function defined on 𝕋 and proved that if 𝑦(𝑡) is a solution of (3.2) with𝑦(𝑎)=𝑦(𝑏)=0(𝑎<𝑏), then^𝑏_𝑎𝑝(𝑡)4Δ𝑡>.𝑏−𝑎 (3.3) Karpuz et al. [13] proved that if 𝑞∈𝐶_rd([𝑎,𝑏]_𝕋,ℝ) and 𝑦(𝑡) is a solution of (3.2) with 𝑦(𝑎)=𝑦^Δ𝜎(𝑏)=0, then2_𝑎^𝜎(𝑏)𝑄²(𝑢)(𝜎(𝑢)−𝑎)Δ𝑢^1/2≥1,where𝑄(𝑢)=_𝑢^𝜎(𝑏)𝑞(𝑡)Δ𝑡. (3.4) Saker [22] considered the second-order half-linear dynamic equation𝑟(𝑡)𝜑𝑥^ΔΔ+𝑞(𝑡)𝜑𝑥^𝜎(𝑡)=0, (3.5) on an arbitrary time scale 𝕋, where 𝜑(𝑢)=|𝑢|^𝛾−1𝑢,𝛾≥1 is a positive constant, 𝑟 and 𝑞 are real rd-continuous positive functions defined on 𝕋 and proved that if 𝑥(𝑡) is a positive solution of (3.1) which satisfies 𝑥(𝑎)=𝑥(𝑏)=0,𝑥(𝑡)≠0 for 𝑡∈(𝑎,𝑏) and 𝑥(𝑡) has a maximum at a point 𝑐∈(𝑎,𝑏), then^𝑏_𝑎Δ𝑡𝑟^𝛾(𝑡)^𝛾𝑏_𝑎𝑞(𝑡)Δ𝑡≥2^𝛾+1. (3.6) Of particular interest in this paper is when 𝑞 is oscillatory which is different from the conditions imposed on 𝑞 in [12, 22, 27]. The results also yield conditions for disfocality for (3.1) on time scales. As special cases, the results include some results obtained for differential equations and give new results for difference equations on discrete time scales.

Now, we are ready to state and prove the main results in this section. To simplify the presentation of the results, we define𝑀𝛽∶=sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)|𝑄(𝑡)|𝑟(𝑡),where𝑄(𝑡)=^𝛽_𝑡𝑞(𝑠)𝑀(𝛼)Δ𝑠,∶=sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)|𝑄(𝑡)|𝑟(𝑡),where𝑄(𝑡)=^𝑡_𝛼𝑞(𝑠)Δ𝑠. (3.7) Note that when 𝕋=ℝ, we have 𝑀(𝛼)=0=𝑀(𝛽), and when 𝕋=ℤ, we have𝑀𝛽=sup_{𝛼≤𝑡≤𝛽}∑^𝛽−1_𝑠=𝑡𝑞(𝑠)𝑟(𝑡),𝑀(𝛼)=sup_{𝛼≤𝑡≤𝛽}∑^𝑡−1_𝑠=𝛼𝑞(𝑠)𝑟(𝑡). (3.8)

Theorem 3.1. Suppose that 𝑦 is a nontrivial solution of (3.1). If 𝑦(𝛼)=𝑦^Δ(𝛽)=0, then 2𝛾+1^1/(𝛾+1)×^𝛽_𝛼|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^1/𝛾(𝑥)^𝑥_𝛼Δ𝑡𝑟^1/𝛾(𝑡)^𝛾Δ𝑥^{𝛾/(𝛾+1)}+2^1−𝛾𝑀𝛽≥1, (3.9) where 𝑄(𝑡)=∫^𝛽_𝑡𝑞(𝑠)Δ𝑠. If 𝑦^Δ(𝛼)=𝑦(𝛽)=0, then 2𝛾+1^{1/(𝛾+1)𝛽𝛼}|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^1/𝛾(𝑥)^𝛽_𝑥Δ𝑡𝑟^1/𝛾(𝑡)^𝛾Δ𝑥^{𝛾/(𝛾+1)}+2^1−𝛾𝑀(𝛼)≥1, (3.10) where𝑄(𝑡)=∫^𝑡_𝛼𝑞(𝑠)Δ𝑠.

Proof. We prove (3.9). Without loss of generality we may assume that 𝑦(𝑡)>0 in [𝛼,𝛽]_𝕋. Multiplying (3.1) by 𝑦^𝜎 and integrating by parts, we have 𝑟(𝑡)𝑦^𝛽_𝛼^Δ(𝑡)^𝛾Δ𝑦^𝜎(𝑡)Δ𝑡=𝑟(𝑡)𝑦^Δ(𝑡)^𝛾𝑦(𝑡)^𝛽_𝛼−^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡=−^𝛽_𝛼𝑞(𝑡)𝑦^𝜎(𝑡)^𝛾+1Δ𝑡. (3.11) Using the assumptions that 𝑦(𝛼)=𝑦^Δ(𝛽)=0 and 𝑄(𝑡)=∫^𝛽_𝑡𝑞(𝑠)Δ𝑠, we have ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡=^𝛽_𝛼𝑞(𝑡)𝑦^𝜎(𝑡)^𝛾+1Δ𝑡=−^𝛽_𝛼𝑄^Δ(𝑡)𝑦^𝜎(𝑡)^𝛾+1Δ𝑡. (3.12) Integrating by parts the right-hand side (see (2.9)), we see that ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡=−𝑄(𝑡)𝑦(𝑡)^𝛾+1𝛽_𝛼+^𝛽_𝛼𝑄(𝑡)𝑦^𝛾+1(𝑡)^ΔΔ𝑡. (3.13) Again using the facts that 𝑦(𝛼)=0=𝑄(𝛽), we obtain ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1𝑑𝑡=^𝛽_𝛼𝑄(𝑡)𝑦^𝛾+1(𝑡)^Δ𝑑𝑡. (3.14) Applying the chain rule formula (2.11) and the inequality (2.2), we see that 𝑦^𝛾+1(𝑡)^Δ≤𝛾+1¹₀ℎ𝑦^𝜎(𝑡)+(1−ℎ)𝑦(𝑡)^𝛾𝑑ℎ𝑦^Δ(𝑡≤𝑦)𝛾+1^Δ(𝑡)¹₀ℎ𝑦^𝜎(𝑡)^𝛾𝑑ℎ+𝛾+1𝑦^Δ(𝑡)¹₀(1−ℎ)𝑦(𝑡)^𝛾=𝑦𝑑ℎ_Δ(𝑡)𝑦^𝜎(𝑡)^𝛾+𝑦^Δ(𝑡)𝑦(𝑡)^𝛾≤2^1−𝛾𝑦^𝜎(𝑡)+𝑦(𝑡)^𝛾𝑦^Δ(𝑡). (3.15) This and (3.14) imply that ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡≤2^1−𝛾𝛽_𝛼|𝑄(𝑡)|𝑦(𝑡)+𝑦^𝜎(𝑡)^𝛾𝑦^Δ(𝑡)Δ𝑡. (3.16) Applying the inequality (2.13) with 𝑠(𝑡)=|𝑄(𝑡)|,𝑝=𝛾 and 𝑞=1, we have ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡≤2^1−𝛾𝐾₁𝛼,𝛽,𝛾,1^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡, (3.17) where 𝐾₁𝛼,𝛽,𝛾,1=𝑀𝛽+2^𝛾1𝛾+1^1/(𝛾+1)×^𝛽_𝛼|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^−1/𝛾(𝑥)^𝑥_𝛼𝑟^−1/𝛾(𝑡)Δ𝑡^𝛾Δ𝑥^{𝛾/(𝛾+1)}. (3.18) Then, we have from (3.17) after cancelling the term ∫^𝛽_𝛼𝑟(𝑡)|𝑦^Δ(𝑡)|^𝛾+1Δ𝑡, that 2^1−𝛾𝑀𝛽+2𝛾+1^1/(𝛾+1)×^𝛽_𝛼|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^1/𝛾(𝑥)^𝑥_𝛼Δ𝑡𝑟^1/𝛾(𝑡)^𝛾Δ𝑥^{𝛾/(𝛾+1)}≥1, (3.19) which is the desired inequality (3.9). The proof of (3.10) is similar to (3.9) by using the integration by parts and (2.30) of Theorem 2.2 and (2.31) instead of (2.14). The proof is complete.

As a special case of Theorem 3.1, when 𝑟(𝑡)=1, we have the following result.

Corollary 3.2. Suppose that 𝑦 is a nontrivial solution of 𝑦^Δ(𝑡)^𝛾Δ+𝑞(𝑡)𝑦^𝜎(𝑡)^𝛾=0,𝑡∈𝛼,𝛽_𝕋. (3.20) If 𝑦(𝛼)=𝑦^Δ(𝛽)=0, then 2𝛾+1^1/(𝛾+1)×^𝛽_𝛼|𝑄(𝑡)|^{(1+𝛾)/𝛾}(𝑡−𝛼)^𝛾Δ𝑡^{𝛾/(𝛾+1)}+2^1−𝛾sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)|𝑄(𝑡)|≥1, (3.21) where 𝑄(𝑡)=∫^𝛽_𝑡𝑞(𝑠)Δ𝑠. If 𝑦^Δ(𝛼)=𝑦(𝛽)=0, then 2𝛾+1^{1/(𝛾+1)𝛽𝛼}|𝑄(𝑡)|^{(1+𝛾)/𝛾}𝛽−𝑡^𝛾Δ𝑡^{𝛾/(𝛾+1)}+2^1−𝛾sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)|𝑄(𝑡)|≥1, (3.22) where 𝑄(𝑡)=∫^𝑡_𝛼𝑞(𝑠)Δ𝑠.

As a special case of Theorem 3.1, when 𝛾=1, we have the following result.

Corollary 3.3. Suppose that 𝑦 is a nontrivial solution of 𝑟(𝑡)𝑦^Δ(𝑡)^Δ+𝑞(𝑡)𝑦^𝜎(𝑡)=0,𝑡∈𝛼,𝛽_𝕋. (3.23) If 𝑦(𝛼)=𝑦^Δ(𝛽)=0, then √2^𝛽_𝛼|𝑄(𝑥)|²𝑟(𝑥)^𝑥_𝛼Δ𝑡𝑟(𝑡)Δ𝑥^1/2+𝑀𝛽≥1, (3.24) where𝑄(𝑡)=∫^𝛽_𝑡𝑞(𝑠)Δ𝑠. If 𝑦^Δ(𝛼)=𝑦(𝛽)=0, then √2^𝛽_𝛼|𝑄(𝑥)|²𝑟(𝑥)^𝛽_𝑥Δ𝑡𝑟(𝑡)Δ𝑥^1/2+𝑀(𝛼)≥1, (3.25) where𝑄(𝑡)=∫^𝑡_𝛼𝑞(𝑠)Δ𝑠.

As a special case when 𝕋=ℝ, we have 𝑀(𝛼)=𝑀(𝛽)=0 and then the results in Corollary 3.3 reduce to the following results obtained by Brown and Hinton [29] for the second-order differential equation𝑦^′′+𝑞(𝑡)𝑦(𝑡)=0,𝛼≤𝑡≤𝛽. (3.26)

Corollary 3.4 (see [29]). If 𝑦 is a solution of (3.26) such that 𝑦(𝛼)=𝑦^′(𝛽)=0, then 2^𝛽_𝛼𝑄²(𝑠)(𝑠−𝛼)𝑑𝑠>1, (3.27) where 𝑄(𝑡)=∫^𝛽_𝑡𝑞(𝑠)𝑑𝑠. If instead 𝑦^′(𝛼)=𝑦(𝛽)=0, then 2^𝛽_𝛼𝑄²(𝑠)𝛽−𝑠𝑑𝑠>1, (3.28) where 𝑄(𝑡)=∫^𝑡_𝛼𝑞(𝑠)𝑑𝑠.

Remark 3.5. Note that if 𝕋=ℕ, then 𝜇(𝑡)=1 and (3.1) (when 𝑟(𝑡)=1) becomes Δ²𝑦(𝑛)+𝑞(𝑛)𝑦(𝑛+1)=0, (3.29) and as a special case of Corollary 3.3, we have the following result.

Corollary 3.6. If 𝑦 is a solution of (3.29) such that 𝑦(𝛼)=Δ𝑦(𝛽)=0, then √2^𝛽−1_𝑛=𝛼(𝑄(𝑛))²(𝑛−𝛼)^1/2+sup_{𝛼≤𝑛≤𝛽}|𝑄(𝑛)|>1, (3.30) where 𝑄(𝑛)=∑^𝛽−1_𝑠=𝑛𝑞(𝑠). If instead Δ𝑦(𝛼)=𝑦(𝛽)=0, then √2^𝛽−1_𝑛=𝛼(𝑄(𝑛))²𝛽−𝑛^1/2+sup_{𝛼≤𝑛≤𝛽}|𝑄(𝑛)|>1, (3.31) where 𝑄(𝑛)=∑^𝑛−1_𝑠=𝛼𝑞(𝑠).

Remark 3.7. By using the maximum of |𝑄| on [𝛼,𝛽]_𝕋 and ^𝑋_𝑎(𝑥−𝑎)^{(𝑝+𝑞−1)}Δ𝑥≤^𝑋_𝑎1()𝑝+𝑞𝑥−𝑎^𝑝+𝑞Δ()Δ𝑥=𝑋−𝑎^𝑝+𝑞,𝑝+𝑞 (3.32) in (3.21) and (3.22) with 𝑝=𝛾 and 𝑞=1, we have the following results.

Corollary 3.8. Suppose that 𝑦 is a nontrivial solution of (3.20), where 𝛾≤1 is a quotient of odd positive integers. If 𝑦(𝛼)=𝑦^Δ(𝛽)=0, then 2𝛽−𝛼^𝛾𝛾+1max_{𝛼≤𝑡≤𝛽}^𝛽_𝑡𝑞(𝑠)Δ𝑠+2^1−𝛾sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)^𝛽_𝑡𝑞(𝑠)Δ𝑠≥1, (3.33) and if 𝑦^Δ(𝛼)=𝑦(𝛽)=0, then 2𝛽−𝛼^𝛾𝛾+1max_{𝛼≤𝑡≤𝛽}^𝑡_𝛼𝑞(𝑠)Δ𝑠+2^1−𝛾sup_{𝛼≤𝑡≤𝛽}𝜇^𝛾(𝑡)^𝑡_𝛼𝑞(𝑠)Δ𝑠≥1. (3.34)

As a special when 𝕋=ℝ, we have 𝑀(𝛼)=𝑀(𝛽)=0 and then the results in Corollary 3.8 reduce to the following results for the second-order half-linear differential equation:𝑦^′(𝑡)^𝛾′+𝑞(𝑡)𝑦(𝑡)^𝛾=0,𝛼≤𝑡≤𝛽, (3.35) where 𝛾≤1 is a quotient of odd positive integers.

Corollary 3.9. Assume that 𝛾≤1 is a quotient of odd positive integers. Suppose that 𝑦 is a nontrivial solution of (3.35). If 𝑦(𝛼)=𝑦^′(𝛽)=0, then 2𝛾+1𝛽−𝛼^𝛾sup_{𝛼≤𝑡≤𝛽}^𝛽_𝑡𝑞(𝑠)𝑑𝑠≥1. (3.36) If instead 𝑦^′(𝛼)=𝑦(𝛽)=0, then 2𝛾+1𝛽−𝛼^𝛾sup_{𝛼≤𝑡≤𝛽}^𝑡_𝛼𝑞(𝑠)𝑑𝑠≥1. (3.37)

As a special case of Corollary 3.9 when 𝛾=1, we have the following results that has been established by Harris and Kong [30].

Corollary 3.10 (see [30]). If 𝑦 is a solution of the equation 𝑦^′′+𝑞(𝑡)𝑦(𝑡)=0,𝑎≤𝑡≤𝑏, (3.38) with no zeros in (𝑎,𝑏) and such that 𝑦(𝑎)=𝑦^′(𝑏)=0, then (𝑏−𝑎)sup_{𝑎≤𝑡≤𝑏}^𝑏_𝑡𝑞(𝑠)𝑑𝑠>1. (3.39) If instead 𝑦^′(𝑎)=𝑦(𝑏)=0, then (𝑏−𝑎)sup_{𝑎≤𝑡≤𝑏}^𝑡_𝑎𝑞(𝑠)𝑑𝑠>1. (3.40)

As a special when 𝕋=ℤ, we see that 𝑀(𝛼) and 𝑀(𝛽) are defined as in (3.8) and then the results in Corollary 3.8 reduce to the following results for the second-order half-linear difference equationΔΔ𝑦(𝑛)^𝛾+𝑞(𝑛)𝑦(𝑛+1)^𝛾=0,𝛼≤𝑛≤𝛽, (3.41) where 𝛾≤1 is a quotient of odd positive integers.

Corollary 3.11. Suppose that 𝑦 is a nontrivial solution of (3.41), where 𝛾≤1 is a quotient of odd positive integers. If 𝑦(𝛼)=Δ𝑦(𝛽)=0, then 2𝛽−𝛼^𝛾𝛾+1max_{𝛼≤𝑛≤𝛽}^𝛽−1_𝑠=𝑛𝑞(𝑠)+2^1−𝛾sup_{𝛼≤𝑛≤𝛽}^𝛽−1_𝑠=𝑛𝑞(𝑠)≥1, (3.42) and if Δ𝑦(𝛼)=𝑦(𝛽)=0, then 2𝛽−𝛼^𝛾𝛾+1max_{𝛼≤𝑛≤𝛽}^𝑛−1_𝑠=𝛼𝑞(𝑠)+2^1−𝛾sup_{𝛼≤𝑛≤𝛽}^𝑛−1_𝑠=𝛼𝑞(𝑠)≥1. (3.43)

Remark 3.12. The above results yield sufficient conditions for disfocality of (3.1), that is, sufficient conditions so that there does not exist a nontrivial solution 𝑦 satisfying either 𝑦(𝛼)=𝑦^Δ(𝛽)=0 or 𝑦^Δ(𝛼)=𝑦(𝛽)=0.

In the following, we employ Theorem 2.5, to determine the lower bound for the distance between consecutive zeros of solutions of (3.1). Note that the applications of the above results allow the use of arbitrary antiderivative 𝑄 in the above arguments. In the following, we assume that 𝑄^Δ(𝑡)=𝑞(𝑡) and there exists ℎ∈(𝛼,𝛽) which is the unique solution of the equation𝐾₁𝛼,𝛽=𝐾₁𝛼,𝛽,ℎ=𝐾₁𝛼,ℎ,𝛽<∞, (3.44) where𝐾₁=2𝛼,𝛽,ℎ^𝛾𝛾+1^1/(𝛾+1)×^𝛽_𝛼|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^1/𝛾(𝑥)^ℎ_𝛼Δt𝑟^1/𝛾(𝑡)^𝛾Δ𝑥^{𝛾/(𝛾+1)},𝐾₁=2𝛼,ℎ,𝛽^𝛾𝛾+1^{1/(𝛾+1)𝛽𝛼}|𝑄(𝑥)|^{(𝛾+1)/𝛾}𝑟^1/𝛾(𝑥)^𝛽_ℎΔ𝑡𝑟^1/𝛾(𝑡)^𝛾Δ𝑥^{𝛾/(𝛾+1)}. (3.45)

Theorem 3.13. Assume that 𝑄^Δ(𝑡)=𝑞(𝑡) and 𝑦 is a nontrivial solution of (3.1). If 𝑦(𝛼)=𝑦(𝛽)=0, then 𝐾₁𝛼,𝛽≥1, (3.46) where 𝐾₁(𝛼,𝛽) is defined as in (3.44).

Proof. Multiplying (3.1) by 𝑦^𝜎(𝑡), proceed as in Theorem 3.1 and use 𝑦(𝛼)=𝑦(𝛽)=0, to get ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡=^𝛽_𝛼𝑞(𝑡)𝑦(𝑡)^𝛾+1Δ𝑡=^𝛽_𝛼𝑄^Δ(𝑡)𝑦^𝜎(𝑡)^𝛾+1Δ𝑡. (3.47) Integrating by parts the right hand side (see (2.9)), we see that ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡=𝑄(𝑡)𝑦(𝑡)^𝛾+1𝛽_𝛼+^𝛽_𝛼(−𝑄(𝑡))𝑦^𝛾+1(𝑡)^ΔΔ𝑡. (3.48) Again using the facts that 𝑦(𝛼)=0=𝑦(𝛽), we obtain ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡≤^𝛽_𝛼|𝑄(𝑡)|𝑦(𝑡)+𝑦^𝜎(𝑡)^𝛾𝑦^Δ(𝑡)Δ𝑡. (3.49) Applying the inequality (2.37) with 𝑠(𝑡)=|𝑄(𝑡)|,𝑝=𝛾 and 𝑞=1, we have ^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1𝑑𝑡≤2^1−𝛾𝐾₁𝛼,𝛽^𝛽_𝛼𝑟(𝑡)𝑦^Δ(𝑡)^𝛾+1Δ𝑡. (3.50) From this inequality, after cancelling ∫^𝛽_𝛼|𝑦^Δ(𝑡)|^𝛾+1Δ𝑡, we get the desired inequality (3.46). This completes the proof.

Problem 1. It would be interesting to extend the above results to cover the delay equation with oscillatory coefficients 𝑟(𝑡)𝑥^Δ(𝑡)^𝛾−1𝑥^Δ(𝑡)^Δ+𝑞(𝑡)|𝑥(𝜏(𝑡))|^𝛾−1𝑥(𝜏(𝑡))=0, (3.51) where the delay function 𝜏(𝑡) satisfies 𝜏(𝑡)<𝑡 and lim_𝑡→∞𝜏(𝑡)=∞.

Acknowledgment

This project was supported by the Research Centre, College of Science, King Saud University.