Some Delay Integral Inequalities on Time Scales and Their Applications in the Theory of Dynamic Equations

Abstract

We establish some delay integral inequalities on time scales, which on one hand provide a handy tool in the study of qualitative as well as quantitative properties of solutions of certain delay dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.

1. Introduction

During the past decades, with the development of the theory of differential and integral equations as well as difference equations, a lot of integral and difference inequalities have been discovered (e.g., see [1–13] and the references therein), which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. On the other hand, Hilger [14] initiated the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales including various inequalities on time scales (e.g., see [15–24], and the references therein). However, delay integral inequalities on time scales have been paid little attention so far. Recent results in this direction include the works of Li [25] and Ma and Pečarić [26] to our best knowledge.

In this paper, we will establish some new delay integral inequalities on time scales, which unify some known continuous and discrete results in the literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities. Some applications will be presented for the established results.

Throughout this paper, ℝ denotes the set of real numbers and ℝ₊=[0,∞), while ℤ denotes the set of integers. For two given sets 𝐺,𝐻, we denote the set of maps from 𝐺 to 𝐻 by (𝐺,𝐻).

2. Some Preliminaries on Time Scales

A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, 𝕋 denotes an arbitrary time scale. On 𝕋 we define the forward and backward jump operators 𝜎∈(𝕋,𝕋) and 𝜌∈(𝕋,𝕋) such that 𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡}, 𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}.

Definition 2.1. The graininess 𝜇∈(𝕋,ℝ₊) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡.

Remark 2.2. Obviously, 𝜇(𝑡)=0 if 𝕋=ℝ, while 𝜇(𝑡)=1 if 𝕋=ℤ.

Definition 2.3. A point 𝑡∈𝕋 is said to be left-dense if 𝜌(𝑡)=𝑡 and 𝑡≠inf𝕋, right-dense if 𝜎(𝑡)=𝑡 and 𝑡≠sup𝕋, left-scattered if 𝜌(𝑡)<𝑡, and right-scattered if 𝜎(𝑡)>𝑡.

Definition 2.4. The set 𝕋^𝜅 is defined to be 𝕋 if 𝕋 does not have a left-scattered maximum, otherwise it is 𝕋 without the left-scattered maximum.

Definition 2.5. A function 𝑓∈(𝕋,ℝ) is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while 𝑓 is called regressive if 1+𝜇(𝑡)𝑓(𝑡)≠0. 𝐶_rd denotes the set of rd-continuous functions, while ℜ denotes the set of all regressive and rd-continuous functions, and ℜ⁺={𝑓|𝑓∈ℜ,1+𝜇(𝑡)𝑓(𝑡)>0,forall𝑡∈𝕋}.

Definition 2.6. For some 𝑡∈𝕋^𝜅, and a function 𝑓∈(𝕋,ℝ), the delta derivative of 𝑓 at 𝑡 is denoted by 𝑓^Δ(𝑡) (provided it exists) with the property such that for every 𝜀>0 there exists a neighborhood 𝔘 of 𝑡 satisfying 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|∀𝑠∈𝔘. (2.1)

Remark 2.7. If 𝕋=ℝ, then 𝑓^Δ(𝑡) becomes the usual derivative 𝑓^′(𝑡), while 𝑓^Δ(𝑡)=𝑓(𝑡+1)−𝑓(𝑡) if 𝕋=ℤ, which represents the forward difference.

Definition 2.8. For 𝑎,𝑏∈𝕋 and a function 𝑓∈(𝕋,ℝ), the Cauchy integral of 𝑓 is defined by ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎), (2.2) where 𝐹^Δ(𝑡)=𝑓(𝑡),𝑡∈𝕋^𝜅.

The following two theorems include some important properties for delta derivative and the Cauchy integral on time scales.

Theorem 2.9 (see [27]). If 𝑓,𝑔∈(𝕋,ℝ), and 𝑡∈𝕋^𝜅, then(i)𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜇(𝑡)if𝜇(𝑡)≠0,lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)(𝑡)𝑡−𝑠if𝜇=0. (2.3)(ii)If 𝑓,𝑔 are delta differentials at t, then 𝑓𝑔 is also delta differential at t, and 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡). (2.4)

Theorem 2.10 (see [27]). If 𝑎,𝑏,𝑐∈𝕋,𝛼∈ℝ, and 𝑓,𝑔∈𝐶_rd, then(i)∫^𝑏_𝑎[𝑓(𝑡)+𝑔(𝑡)]Δ𝑡=∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑎𝑔(𝑡)Δ𝑡, (ii)∫^𝑏_𝑎(𝛼𝑓)(𝑡)Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡, (iii)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=−∫^𝑎_𝑏𝑓(𝑡)Δ𝑡, (iv)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∫^𝑐_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑐𝑓(𝑡)Δ𝑡, (v)∫^𝑎_𝑎𝑓(𝑡)Δ𝑡=0, (vi)if 𝑓(𝑡)≥0 for all 𝑎≤𝑡≤𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0.

Definition 2.11. The cylinder transformation 𝜉_ℎ∶ℂ_ℎ→ℤ_ℎ is defined by 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ1,ifℎ≠0for𝑧≠−ℎ,𝑧,ifℎ=0, (2.5) where Log is the principal logarithm function.

Definition 2.12. For 𝑝∈ℜ and 𝑠,𝑡∈𝕋, the exponential function is defined by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏. (2.6)

Remark 2.13. If 𝕋=ℝ, then for 𝑠,𝑡∈ℝ, 𝑒_𝑝(𝑡,𝑠)=exp(∫^𝑡_𝑠𝑝(𝜏)𝑑𝜏). If 𝕋=ℤ, then for 𝑠,𝑡∈ℤ and 𝑠<𝑡, 𝑒_𝑝(𝑡,𝑠)=∏^𝑡−1_𝜏=𝑠[1+𝑝(𝜏)].

The following two theorems include some known properties on the exponential function.

Theorem 2.14 (see [28]). If 𝑝∈ℜ, then the following conclusions hold:(i)𝑒_𝑝(𝑡,𝑡)≡1, and 𝑒₀(𝑡,𝑠)≡1,(ii)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠),(iii)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡),where ⊖𝑝=−𝑝/(1+𝜇𝑝).

Theorem 2.15 (see [28]). If 𝑝∈ℜ, and fix 𝑡₀∈𝕋, then the exponential function 𝑒_𝑝(𝑡,𝑡₀) is the unique solution of the following initial value problem: 𝑦^Δ(𝑡)(𝑡)𝑦(𝑡),𝑦𝑡=𝑝₀=1. (2.7)

For more details about the calculus of time scales, we advise to refer to [29].

3. Main Results

In the rest of this paper, for the sake of convenience, we denote 𝕋₀=[𝑡₀,∞)∩𝕋, where 𝑡₀∈𝕋, and always assume 𝕋₀⊂𝕋^𝜅.

Lemma 3.1 (see [30, Theorem  2.2]). Let 𝑡₀∈𝕋^𝜅 and 𝜔∶𝕋×𝕋^𝜅→ℝ be continuous at (𝑡,𝑡), where 𝑡≥𝑡₀, 𝑡∈𝕋^𝜅 with 𝑡>𝑡₀. Assume that 𝜔^Δ(𝑡,⋅) is rd-continuous on [𝑡₀,𝜎(𝑡)]. If for any 𝜀>0, there exists a neighborhood 𝑈 of 𝑡, independent of 𝜏∈[𝑡₀,𝜎(𝑡)], such that 𝜔(𝜎(𝑡),𝜏)−𝜔(𝑠,𝜏)−𝜔^Δ(𝑡,𝜏)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|∀𝑠∈𝑈, (3.1) where 𝜔^Δ denotes the derivative of 𝜔 with respect to the first variable, then 𝑔(𝑡)∶=^𝑡_𝑡0𝜔(𝑡,𝜏)Δ𝜏 (3.2) implies 𝑔^Δ(𝑡)=^𝑡_𝑡0𝜔^Δ(𝑡,𝜏)Δ𝜏+𝜔(𝜎(𝑡),𝑡). (3.3)

Theorem 3.2. Suppose 𝑢∈𝐶_rd(𝕋₀,ℝ₊),𝜔∈(ℝ₊,ℝ₊) with 𝜔(𝑢)>0 for 𝑢>0, and 𝜔 is nondecreasing. 𝜏_𝑖∈(𝕋₀,𝕋) with 𝜏_𝑖(𝑡)≤𝑡,𝑖=1,2, and −∞<𝛼=inf{min{𝜏_𝑖(𝑡),𝑖=1,2},𝑡∈𝕋₀}≤𝑡₀. 𝑓,𝑔∈𝐶_rd([𝛼,∞)∩𝕋,ℝ₊). 𝑝,𝑞,𝐶 are constants, and 𝑝>𝑞>0,𝐶≥0. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)≤𝐶^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)𝜔𝑢𝜏₁(𝑠)+𝑔𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠 (3.4) with the initial condition 𝑢(𝑡)=𝜙(𝑡),𝑡∈𝛼,𝑡₀𝜙𝜏∩𝕋,_𝑖(𝑡)≤𝐶^{1/(𝑝−𝑞)},∀𝑡∈𝕋₀,𝜏_𝑖(𝑡)≤𝑡₀,𝑖=1,2, (3.5) where 𝜙∈𝐶_rd([𝛼,𝑡₀]∩𝕋,ℝ₊), then, 𝑢(𝑡)≤𝐺⁻¹𝐺𝐶+^𝑡_𝑡0𝑔𝜏₂(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)Δ𝑠^{1/(𝑝−𝑞)},𝑡∈𝕋₀, (3.6) where 𝐺(𝑥)=∫^𝑥₁1/𝜔(𝑟^{1/(𝑝−𝑞)})𝑑𝑟,𝑥>0 with 𝐺(∞)=∞, and 𝐺⁻¹ is the inverse of 𝐺.

Proof. Assume 𝐶>0. Denote the right side of (3.4) by 𝑧(𝑡). Then 𝑢(𝑡)≤𝑧^1/𝑝(𝑡),𝑡∈𝕋₀. (3.7) If 𝜏_𝑖(𝑡)≥𝑡₀, for 𝑡∈𝕋₀, since 𝜏_𝑖(𝑡)≤𝑡, then 𝜏_𝑖(𝑡)∈𝕋₀, and from (3.7) we have 𝑢𝜏_𝑖(𝑡)≤𝑧^1/𝑝𝜏_𝑖(𝑡)≤𝑧^1/𝑝(𝑡),𝑖=1,2. (3.8) If 𝜏_𝑖(𝑡)≤𝑡₀, from (3.5) we obtain 𝑢𝜏_𝑖(𝑡)=𝜙𝜏_𝑖(𝑡)≤𝐶^{1/(𝑝−𝑞)}≤𝑧^1/𝑝(𝑡),𝑖=1,2. (3.9) So from (3.8) and (3.9) we always have 𝑢𝜏_𝑖(𝑡)≤𝑧^1/𝑝(𝑡),𝑖=1,2,𝑡∈𝕋₀. (3.10) Furthermore, 𝑧^Δ(𝑡)=𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑢^𝑞𝜏₁(𝑡)𝜔𝑢𝜏₁(𝑡)+𝑔𝜏₂(𝑡)𝑢^𝑞𝜏₂(𝑡)≤𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑧^𝑞/𝑝(𝑡)𝜔𝑧^1/𝑝(𝑡)+𝑔𝜏₂(𝑡)𝑧^𝑞/𝑝(𝑡), (3.11) that is, 𝑧^Δ(𝑡)𝑧^𝑞/𝑝(𝑡)≤𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝜔𝑧^1/𝑝(𝑡)+𝑔𝜏₂(𝑡), (3.12) According to [29, Theorem  1.90], considering 𝑧^Δ(𝑡)≥0, we have 𝑝𝑧𝑝−𝑞^{(𝑝−𝑞)/𝑝}(𝑡)^Δ=𝑧^Δ(𝑡)¹₀𝑧(𝑡)+ℎ𝜇(𝑡)𝑧^Δ(𝑡)^−𝑞/𝑝=𝑧𝑑ℎ_Δ(𝑡)𝑧^𝑞/𝑝(𝑡)¹₀(𝑡)𝑧1+ℎ𝜇^Δ(𝑡)𝑧(𝑡)^−𝑞/𝑝≤𝑧𝑑ℎ_Δ(𝑡)𝑧^𝑞/𝑝(𝑡). (3.13) Combining (3.12) and (3.13), we obtain 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)^Δ≤𝑓𝜏₁(𝑡)𝜔𝑧^1/𝑝(𝑡)+𝑔𝜏₂(𝑡). (3.14) Setting 𝑡=𝑠 in (3.14), an integration with respect to 𝑠 from 𝑡₀ to 𝑡 yields 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)−𝑧^{(𝑝−𝑞)/𝑝}𝑡₀≤^𝑡_𝑡0𝑓𝜏₁(𝑠)𝜔𝑧^1/𝑝(𝑠)+𝑔𝜏₂(𝑠)Δ𝑠. (3.15) Since 𝑧(𝑡₀)=𝐶^{𝑝/(𝑝−𝑞)}, then (3.15) implies 𝑧(𝑡)≤𝐶+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝜔𝑧^1/𝑝(𝑠)+𝑔𝜏₂(𝑠)Δ𝑠^{𝑝/(𝑝−𝑞)}. (3.16) Fix 𝑇∈𝕋₀, and let 𝑡∈[𝑡₀,𝑇]∩𝕋. Then, 𝑧(𝑡)≤𝐶+^𝑇_𝑡0𝑔𝜏₂(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝜔𝑧^1/𝑝(𝑠)Δ𝑠^{𝑝/(𝑝−𝑞)}. (3.17) Denote 𝑣(𝑡) by 𝐶+∫^𝑇_𝑡0𝑔(𝜏₂(𝑠))Δ𝑠+∫^𝑡_𝑡0𝑓(𝜏₁(𝑠))𝜔(𝑧^1/𝑝(𝑠))Δ𝑠. Then, 𝑧(𝑡)≤𝑣^{𝑝/(𝑝−𝑞)}(𝑡),𝑡∈𝑡₀,𝑇∩𝕋, (3.18)𝑣^Δ(𝑡)=𝑓𝜏₁(𝑡)𝜔𝑧^1/𝑝(𝑡)≤𝑓𝜏₁(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡), (3.19) that is, 𝑣^Δ(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡)≤𝑓𝜏₁(𝑡). (3.20) On the other hand, for 𝑡∈[𝑡₀,𝑇]∩𝕋, if 𝜎(𝑡)>𝑡, then [𝐺(𝑣(𝑡))]^Δ=𝐺(𝑣(𝜎(𝑡)))−𝐺(𝑣(𝑡))𝜎(𝑡)=1−𝑡𝜎(𝑡)−𝑡_𝑣(𝑡)^{𝑣(𝜎(𝑡))}1𝜔𝑟^{1/(𝑝−𝑞)}≤𝑣(𝜎(𝑡(𝑡)𝑑𝑟))−𝑣𝜎(𝑡)1−𝑡𝜔𝑣^{1/(𝑝−𝑞)}(𝑡=𝑣)^Δ(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡.) (3.21) If 𝜎(𝑡)=𝑡, then [𝐺(𝑣(𝑡))]^Δ=lim_𝑠→𝑡𝐺(𝑣(𝑡))−𝐺(𝑣(𝑠))𝑡−𝑠=lim_𝑠→𝑡1𝑡−𝑠_𝑣(𝑠)^𝑣(𝑡)1𝜔𝑟^{1/(𝑝−𝑞)}𝑑𝑟=lim_𝑠→𝑡𝑣(𝑡)−𝑣(𝑠)1𝑡−𝑠𝜔𝜉^{1/(𝑝−𝑞)}=𝑣^Δ(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡,) (3.22) where 𝜉 lies between 𝑣(𝑠) and 𝑣(𝑡). So from (3.21) and (3.22) we always have [𝐺(𝑣(𝑡))]^Δ≤𝑣^Δ(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡). (3.23) Combining (3.20) and (3.23), we deduce [𝐺(𝑣(𝑡))]^Δ≤𝑓𝜏₁(𝑡). (3.24) Setting 𝑡=𝑠 in (3.24), an integration with respect to 𝑠 from 𝑡₀ to 𝑡 yields 𝐺(𝑣(𝑡))−𝐺𝑣𝑡₀≤^𝑡_𝑡0𝑓𝜏₁(𝑠)Δ𝑠. (3.25) Considering 𝑣(𝑡₀)=𝐶+∫^𝑇_𝑡0𝑔(𝜏₂(𝑠))Δ𝑠, and 𝐺 is increasing, then we obtain 𝑣(𝑡)≤𝐺⁻¹𝐺𝐶+^𝑇_𝑡0𝑔𝜏₂(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)Δ𝑠,𝑡∈𝑡₀,𝑇∩𝕋. (3.26) Combining (3.7), (3.18), and (3.26), we have 𝑢(𝑡)≤𝐺⁻¹𝐺𝐶+^𝑇_𝑡0𝑔𝜏₂(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)Δ𝑠^{1/(𝑝−𝑞)},𝑡∈𝑡₀,𝑇∩𝕋. (3.27) Setting 𝑡=𝑇 in (3.27), considering that 𝑇∈𝕋₀ is selected arbitrarily, after substituting 𝑇 with 𝑡, we get the desired result.
If 𝐶=0, then we carry out the process above with 𝐶 replaced by 𝜀, where 𝜀>0, and after letting 𝜀→0, we also get the desired result. So the proof is complete.

Remark 3.3. If we take 𝕋=ℝ,𝑡₀=0, then Theorem 3.2 reduces to [31, Theorem  2.1]. If 𝕋=ℝ, 𝑡₀=0, 𝑝=2, 𝑞=1, then Theorem 3.2 reduces to [32, Theorem  1]. If 𝕋=ℝ, 𝑡₀=0, 𝑝=2,𝑞=1, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem 3.2 reduces to [33, Theorem  3(a6)]. If we take 𝕋=ℤ, 𝑡₀=0, 𝑝=2, 𝑞=1, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem 3.2 reduces to [33, Theorem  6(b6)].

Theorem 3.4. Suppose 𝑢,𝑎∈𝐶_rd(𝕋₀,ℝ₊), and 𝑎 is nondecreasing. 𝛼,𝜙,𝜏_𝑖,𝑖=1,2 are defined as in Theorem 3.2. 𝑓,ℎ,𝑓^Δ_𝑡,ℎ^Δ_𝑡∈𝐶_rd(𝕋₀×([𝛼,∞)∩𝕋),ℝ₊),𝑔,𝑑,𝑔^Δ_𝑡,𝑑^Δ_𝑡∈𝐶_rd(𝕋²₀,ℝ₊), where 𝑓^Δ_𝑡,ℎ^Δ_𝑡,𝑔^Δ_𝑡,𝑑^Δ_𝑡 denote the delta derivative of 𝑓,ℎ,𝑔,𝑑 with respect to the first variable. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the following inequality: 𝑢(𝑡)≤𝑎(𝑡)+^t_𝑡0𝑓𝑡,𝜏₁(𝑠)𝑢𝜏₁(𝑠)()𝑢(𝑠++𝑔𝑡,𝑠)Δ𝑠_𝑡𝑡0ℎ𝑡,𝜏₂(𝑠)𝑢𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑(𝑡,𝑠)𝑢(𝑠)Δ𝑠 (3.28) with the initial condition 𝑢(𝑡)=𝜙(𝑡),𝑡∈𝛼,𝑡₀𝜙𝜏∩𝕋,_𝑖(𝑡)≤𝑎(𝑡),∀𝑡∈𝕋₀,𝜏_𝑖(𝑡)≤𝑡₀,𝑖=1,2, (3.29) then 𝑢(𝑡)≤𝑎(𝑡)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀1+𝑎(𝑡)∫^𝑡_𝑡0𝑒_{⊖(−𝐹1)}𝜎(𝑠),𝑡₀𝐹₂(𝑠)Δ𝑠,𝑡∈𝕋₀ (3.30) provided that 1+𝑎(𝑡)∫^𝑡_𝑡0𝑒_{⊖(−𝐹1)}(𝜎(𝑠),𝑡₀)𝐹₂(𝑠)Δ𝑠>0 and 1−𝜇(𝑡)𝐹₁(𝑡)>0 for ∀𝑡∈𝕋₀, where 𝐹₁(𝑡)=^𝑡_𝑡0𝑓𝑡,𝜏₁(𝑠)+𝑔(𝑡,𝑠)Δ𝑠^Δ,𝐹₂(𝑡)=^𝑡_𝑡0ℎ𝑡,𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑(𝑡,𝑠)Δ𝑠^Δ. (3.31)

Proof. Assume 𝑎(0)>0. Fix 𝑇∈𝕋₀, and let 𝑡∈[𝑡₀,𝑇]∩𝕋. If the right side of (3.28) is 𝑎(𝑡)+𝑧(𝑡), then 𝑢(𝑡)≤𝑎(𝑡)+𝑧(𝑡), (3.32) and similar to the process of (3.8)–(3.10), we have 𝑢𝜏_𝑖(𝑡)≤𝑎(𝑡)+𝑧(𝑡),𝑖=1,2. (3.33) Furthermore, by Lemma 3.1 and Theorem 2.9(ii) 𝑧^Δ(𝑡)=^𝑡_𝑡0𝑓^Δ_𝑡𝑡,𝜏₁(𝑠)𝑢𝜏₁(𝑠)+𝑔^Δ_𝑡(𝑡,𝑠)𝑢(𝑠)Δ𝑠+𝑓𝜎(𝑡),𝜏₁(𝑡)𝑢𝜏₁(𝑡)(𝜎(𝑡))𝑢(𝑡)++𝑔,𝑡_𝑡𝑡0ℎ^Δ_𝑡𝑡,𝜏₂(𝑠)𝑢𝜏₂(𝑠)Δ𝑠+ℎ𝜎(𝑡),𝜏₂(𝑡)𝑢𝜏₂(𝑡)_𝑡^𝜎(𝑡)₀𝑑(𝜎(𝑡))𝑢(𝑠)+,𝑠Δ𝑠_𝑡𝑡0ℎ𝑡,𝜏₂(𝑠)𝑢𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑^Δ_𝑡()𝑢(𝑠)(𝜎(𝑡)≤[𝑎(𝑡)(𝑡𝑡,𝑠Δ𝑠+𝑑,𝑡)+𝑧)]^𝑡_𝑡0𝑓^Δ_𝑡𝑡,𝜏₁(𝑠)+𝑔^Δ_𝑡(𝑡,𝑠)Δ𝑠+𝑓𝜎(𝑡),𝜏₁(𝑡)(𝜎(𝑡)+[𝑎(𝑡)(𝑡+𝑔,𝑡)+𝑧)]^2𝑡_𝑡0ℎ^Δ_𝑡𝑡,𝜏₂(𝑠)Δ𝑠+ℎ𝜎(𝑡),𝜏₂(𝑡)_𝑡^𝜎(𝑡)₀𝑑(𝜎(𝑡))+,𝑠Δ𝑠_𝑡𝑡0ℎ𝑡,𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑^Δ_𝑡()(𝜎(𝑡)≤[𝑎(𝑇)(𝑡𝑡,𝑠Δ𝑠+𝑑,𝑡)+𝑧)]^𝑡_𝑡0𝑓𝑡,𝜏₁(𝑠)+𝑔(𝑡,𝑠)Δ𝑠^Δ+[𝑎(𝑇)+𝑧(𝑡)]^2𝑡_𝑡0ℎ𝑡,𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑(𝑡,𝑠)Δ𝑠^Δ, (3.34) that is, 𝑧^Δ(𝑡)[𝑎(𝑇)+𝑧(𝑡)]²≤1𝑎(𝑇)+𝑧(𝑡)^𝑡_𝑡0𝑓𝑡,𝜏₁(𝑠)+𝑔(𝑡,𝑠)Δ𝑠^Δ+^𝑡_𝑡0ℎ𝑡,𝜏₂(𝑠)Δ𝑠^𝑡_𝑡0𝑑(𝑡,𝑠)Δ𝑠^Δ≤1𝑎(𝑇)(𝑡)𝐹+𝑧¹(𝑡)+𝐹₂(𝑡), (3.35) where 𝐹₁,𝐹₂ are defined in (3.31).
Considering 𝑧^Δ(𝑡)≥0, from (3.35), we deduce𝑧^Δ(𝑡)[𝑎(𝑇)(𝑡𝑎(𝑇)(𝜎(𝑡≤𝑧+𝑧)][+𝑧))]^Δ(𝑡)[𝑎(𝑇)+𝑧(𝑡)]²≤1𝑎(𝑇)(𝑡)𝐹+𝑧¹(𝑡)+𝐹₂(𝑡). (3.36) Let 𝑣(𝑡)=1/(𝑎(𝑇)+𝑧(𝑡)). Then, by Theorem 2.9(ii), 𝑣^Δ(𝑡)=−𝑧^Δ(𝑡)/[𝑎(𝑇)+𝑧(𝑡)][𝑎(𝑇)+𝑧(𝜎(𝑡))], and (3.36) implies 𝑣^Δ(𝑡)+𝑣(𝑡)𝐹₁(𝑡)≥𝐹₂(𝑡). (3.37) On the other hand, since 1−𝜇(𝑡)𝐹₁(𝑡)>0, then 1+𝜇(𝑡)[⊖(−𝐹₁)(𝑡)]=1/(1−𝜇(𝑡)𝐹₁(𝑡))>0. So ⊖(−𝐹₁)∈ℜ₊, and 𝑒_{⊖(−𝐹1)}(𝑡,𝑡₀)>0,∀𝑡∈𝕋₀. By Theorem 2.14(i), we have 𝑒_{⊖(−𝐹1)}(𝑡₀,𝑡₀)=1. Furthermore, by a combination of Theorem 2.9(ii), Theorems 2.15, and 2.14, we obtain 𝑣(𝑡)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀^Δ=𝑒_{⊖(−𝐹1)}𝑡,𝑡₀^Δ𝑣(𝑡)+𝑒_{⊖(−𝐹1)}𝜎(𝑡),𝑡₀𝑣^Δ(𝑡)=⊖−𝐹₁(𝑡)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀𝑣(𝑡)+𝑒_{⊖(−𝐹1)}𝜎(𝑡),𝑡₀𝑣^Δ(𝑡)=𝑒_{⊖(−𝐹1)}𝜎(𝑡),𝑡₀⊖−𝐹₁(𝑡)1+𝜇(𝑡)⊖−𝐹₁(𝑡)𝑣(𝑡)+𝑣^Δ(𝑡)=𝑒_{⊖(−𝐹1)}𝜎(𝑡),𝑡₀𝑣^Δ(𝑡)+𝐹₁(𝑡)𝑣(𝑡). (3.38) Combining (3.37) and (3.38), we deduce 𝑣(𝑡)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀^Δ≥𝑒_{⊖(−𝐹1)}𝜎(𝑡),𝑡₀𝐹₂(𝑡). (3.39) Setting 𝑡=𝑠 in (3.39), an integration with respect to 𝑠 from 𝑡₀ to 𝑡 yields 𝑣(𝑡)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀−𝑣𝑡₀𝑒_{⊖(−𝐹1)}𝑡₀,𝑡₀≥^𝑡_𝑡0𝑒_{⊖(−𝐹1)}𝜎(𝑠),𝑡₀𝐹₂(𝑠)Δ𝑠. (3.40) Considering 𝑣(𝑡₀)=1/𝑎(𝑇), it is then followed by 𝑣(𝑡)≥1+𝑎(𝑇)∫^𝑡_𝑡0𝑒_{⊖(−𝐹1)}𝜎(𝑠),𝑡₀𝐹₂(𝑠)Δ𝑠𝑎(𝑇)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀, (3.41) and furthermore 𝑎(𝑇)(𝑡)≤𝑎(𝑇)𝑒+𝑧^{⊖(−𝐹1)}𝑡,𝑡₀1+𝑎(𝑇)∫^𝑡_𝑡0𝑒_{⊖(−𝐹1)}𝜎(𝑠),𝑡₀𝐹₂(𝑠)Δ𝑠. (3.42) Combining (3.32) and (3.42), we obtain 𝑢(𝑡)≤𝑎(𝑇)𝑒_{⊖(−𝐹1)}𝑡,𝑡₀1+𝑎(𝑇)∫^𝑡_𝑡0𝑒_{⊖(−𝐹1)}𝜎(𝑠),𝑡₀𝐹₂(𝑠)𝑡Δ𝑠,𝑡∈₀,𝑇∩𝕋. (3.43) Setting 𝑡=𝑇 in (3.43), since 𝑇∈𝕋₀ is selected arbitrarily, after substituting 𝑇 with 𝑡, we get the desired result.
If 𝑎(0)=0, then we carry out the process above with 𝑎(𝑡) replaced by 𝑎(𝑡)+𝜀, and after letting 𝜀→0, we also get the desired result. So the proof is complete.

Remark 3.5. If we take 𝕋=ℝ,𝑡₀=0, then Theorem 3.4 reduces to [34, Corollary  2.5]. If 𝕋=ℝ, 𝑡₀=0, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, 𝑎(𝑡)≡𝐶, where 𝐶 is a nonnegative constant, and 𝑓(𝑡,𝑠),𝑔(𝑡,𝑠),ℎ(𝑡,𝑠) are replaced by 𝑓(𝑠), 𝑔(𝑠), ℎ(𝑠), then Theorem 3.4 reduces to [35, Theorem  1]. If we take 𝕋=ℤ,𝑡₀=0,𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡,(𝑡)≡𝐶, where 𝐶 is a nonnegative constant, and 𝑓(𝑡,𝑠), (𝑡,𝑠), ℎ(𝑡,𝑠) are replaced by 𝑓(𝑠), 𝑔(𝑠), ℎ(𝑠), then Theorem 3.4 reduces to [35, Theorem  5].
Next we will study the delay integral inequality on time scales with the following form 𝑢^𝑝(𝑡)≤𝐶^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑝𝜏₁(𝑠)+𝑝Δs𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑢^𝑝−𝑞𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠, (3.44) where 𝑢,𝑓,𝑔,𝑝,𝑞,𝐶,𝛼,𝜙,𝜏_𝑖,𝑖=1,2 are defined as in Theorem 3.2, and ℎ∈𝐶_rd([𝛼,∞)∩𝕋,ℝ₊).

Lemma 3.6. Suppose 𝑢,𝑓,𝑔,𝜏₁ are defined as in Theorem 3.2, and (𝑓+𝑔)(𝜏₁(⋅))∈ℜ₊, then for 𝑡∈𝕋₀, 𝑢(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑢(𝜉)Δ𝜉Δ𝑠 (3.45) implies 𝑢(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑠,𝑡₀Δ𝑠. (3.46)

Proof. Denote the right side of (3.46) by 𝑧(𝑡). Then, 𝑢(𝑡)≤𝑧(𝑡),𝑡∈𝕋₀, and 𝑧^Δ(𝑡)=𝑓𝜏₁(𝑡)𝑢(𝑡)+𝑓𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝑢(𝜉)𝜏Δ𝜉≤𝑓₁(𝑡)𝑧(𝑡)+𝑓𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝑧(𝜉)Δ𝜉. (3.47) Let 𝑚(𝑡)=𝑧(𝑡)+∫^𝑡_𝑡0𝑔(𝜏₁(𝜉))𝑧(𝜉)Δ𝜉. Then, 𝑧(𝑡)≤𝑚(𝑡), and 𝑧^Δ(𝑡)≤𝑓(𝜏₁(𝑡))𝑚(𝑡). Furthermore, 𝑚^Δ(𝑡)=𝑧^Δ(𝑡)+𝑔𝜏₁(𝑡)𝑧(𝑡)≤𝑓𝜏₁(𝑡)𝑚(𝑡)+𝑔𝜏₁(𝑡)𝑚(𝑡). (3.48) Since (𝑓+𝑔)(𝜏₁(⋅))∈ℜ₊, by [28, Theorem  5.4], we have 𝑚(𝑡)≤𝑒_{(𝑓+𝑔)(𝜏1(⋅))}(𝑡,𝑡₀). So, 𝑧^Δ(𝑡)≤𝑓𝜏₁(𝑡)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑡,𝑡₀. (3.49) Using 𝑧(𝑡₀)=1, an integration for (3.49) from 𝑡₀ to 𝑡 yields 𝑧(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑠,𝑡₀Δ𝑠, (3.50) which confirms the desired inequality. So the proof is complete.

Theorem 3.7. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the inequality (3.44) with the initial condition (3.5), then 𝑢(𝑡)≤𝐽(𝑡)1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑠,𝑡₀Δ𝑠^{1/(𝑝−𝑞)},𝑡∈𝕋₀ (3.51) provided (𝑓+𝑔)(𝜏₁(⋅))∈ℜ₊, where 𝐽(𝑡)=𝐶+^𝑡_𝑡0ℎ𝜏₂(𝑠)Δ𝑠. (3.52)

Proof. Let 𝑧(𝑡)=(𝐶+𝜀)^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑝𝜏₁(𝑠)𝑝Δ𝑠+𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)×^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑢^𝑝−𝑞𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠, (3.53) where 𝜀>0 is an arbitrary small constant. Then, 𝑢(𝑡)≤𝑧^1/𝑝(𝑡),𝑡∈𝕋₀, (3.54) and similar to (3.8)–(3.10) 𝑢𝜏_𝑖(𝑡)≤𝑧^1/𝑝(𝑡),𝑖=1,2,𝑡∈𝕋₀. (3.55) Furthermore, 𝑧^Δ(𝑡)=𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑢^𝑝𝜏₁(𝑡)+𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑢^𝑞𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝑢^𝑝−𝑞𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑡)𝑢^𝑞𝜏₂(𝑡)≤𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑧(𝑡)+𝑝𝑓𝜏𝑝−𝑞¹(𝑡)𝑧^𝑞/𝑝(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝑧^{(𝑝−𝑞)/𝑝}(𝜉)Δ𝜉+ℎ𝜏₂(𝑡)𝑧^𝑞/𝑝(𝑡). (3.56) Using (3.13), we obtain ((𝑝/(𝑝−𝑞))𝑧^{(𝑝−𝑞)/𝑝}(𝑡))^Δ≤(𝑧^Δ(𝑡)/𝑧^𝑞/𝑝(𝑡)). So (3.56) implies 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)^Δ≤𝑓𝜏₁(𝑡)𝑧^{(𝑝−𝑞)/𝑝}(𝑡)+𝑓𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝑧^{(𝑝−𝑞)/𝑝}(𝜉)Δ𝜉+ℎ𝜏₂(𝑡). (3.57) Considering 𝑧(𝑡₀)=(𝐶+𝜀)^{𝑝/(𝑝−𝑞)}, an integration for (3.57) from 𝑡₀ to 𝑡 yields 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)≤𝐶+𝜀+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑧^{(𝑝−𝑞)/𝑝}(𝑠)+Δ𝑠_𝑡𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑧^{(𝑝−𝑞)/𝑝}(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)Δ𝑠=𝐽_𝜀(𝑡)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑧^{(𝑝−𝑞)/𝑝}(𝑠)+Δ𝑠_𝑡𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑧^{(𝑝−𝑞)/𝑝}(𝜉)Δ𝜉Δ𝑠, (3.58) where 𝐽_𝜀(𝑡)=𝐽(𝑡)+𝜀, and 𝐽(𝑡) is defined in (3.52). Then, 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)𝐽_𝜀(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑧^{(𝑝−𝑞)/𝑝}(𝑠)𝐽_𝜀(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑧^{(𝑝−𝑞)/𝑝}(𝜉)𝐽_𝜀(𝜉)Δ𝜉Δ𝑠. (3.59) Denote 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)/𝐽_𝜀(𝑡)=𝑣(𝑡). Then, 𝑣(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑣(𝑠)Δ𝑠+^𝑡_𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑣(𝜉)Δ𝜉Δ𝑠. (3.60) A suitable application of Lemma 3.6 to (3.60) yields 𝑣(𝑡)≤1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑠,𝑡₀Δ𝑠. (3.61) Combining (3.54) and (3.61), we obtain 𝑢(𝑡)≤𝐽_𝜀(𝑡)1+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑒_{(𝑓+𝑔)(𝜏1(⋅))}𝑠,𝑡₀Δ𝑠^{1/(𝑝−𝑞)}. (3.62) After letting 𝜀→0, we get the desired result. So the proof is complete.

Remark 3.8. If we take 𝕋=ℝ, 𝑡₀=0, 𝑝=2, 𝑞=1, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem 3.7 reduces to [33, Theorem  1(a2)]. If we take 𝕋=ℤ, 𝑡₀=0, 𝑝=2, 𝑞=1, 𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem 3.7 reduces to [33, Theorem  4(b2)].
Now we present a more general inequality than in Theorem 3.7.

Theorem 3.9. Suppose 𝑢,𝑓,𝑔,𝜔,𝑝,𝑞,𝐶,𝛼,𝜙,𝜏_𝑖,𝑖=1,2 are defined as in Theorem 3.2, and ℎ∈𝐶_rd([𝛼,∞)∩𝕋,ℝ₊). If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)≤𝐶^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑝𝜏₁(𝑠)𝑝Δ𝑠+𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)×^𝑠_𝑡0𝑔𝜏₁(𝜉)𝜔𝑢𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠 (3.63) with the initial condition (3.5), then for 𝑡∈𝕋₀, 𝑢(𝑡)≤𝐽(𝑡)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝐺⁻¹𝐺(𝐽(𝑠))+^𝑠_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉Δ𝑠^{1/(𝑝−𝑞)}, (3.64) where 𝐽(𝑡) is defined as in Theorem 3.7, and 𝐺(𝑥)=^𝑥₁1𝑟+𝜔𝑟^{1/(𝑝−𝑞)}𝑑𝑟,𝑥>0 (3.65) with 𝐺(∞)=∞, and 𝐺⁻¹ is the inverse of 𝐺.

Proof. Let the right side of (3.63) is 𝑧(𝑡), 𝜀>0 is an arbitrary small constant, and 𝐽_𝜀(𝑡) is defined as in Theorem 3.7. Then, 𝑢(𝑡)≤𝑧^1/𝑝(𝑡),𝑡∈𝕋₀, and similar to the process of (3.53)–(3.58), we obtain 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)≤𝐽_𝜀(𝑡)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑧^{(𝑝−𝑞)/𝑝}(𝑠)+Δ𝑠_𝑡𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝜔𝑧^1/𝑝(𝜉)Δ𝜉Δ𝑠. (3.66) Fix 𝑇∈𝕋₀, and let 𝑡∈[𝑡₀,𝑇]∩𝕋. Considering 𝐽_𝜀(𝑡) is nondecreasing, we deduce that 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)≤𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑧^{(𝑝−𝑞)/𝑝}(𝑠)+Δ𝑠_𝑡𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)𝜔𝑧^1/𝑝(𝜉)Δ𝜉Δ𝑠. (3.67) Denote the right side of (3.66) by 𝑣(𝑡). Then, 𝑧^{(𝑝−𝑞)/𝑝}(𝑡)≤𝑣(𝑡),𝑡∈[𝑡₀,𝑇]∩𝕋, and 𝑣^Δ(𝑡)=𝑓𝜏₁(𝑡)𝑧^{(𝑝−𝑞)/𝑝}(𝑡)+𝑓𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝜔𝑧^1/𝑝(𝜉)𝜏Δ𝜉≤𝑓₁(𝑡)𝑣(𝑡)+𝑓𝜏₁(𝑡)^𝑡_𝑡0𝑔𝜏₁(𝜉)𝜔𝑣^{1/(𝑝−𝑞)}(𝜉)Δ𝜉. (3.68) Denote 𝑚(𝑡)=𝑣(𝑡)+∫^𝑡_𝑡0𝑔(𝜏₁(𝜉))𝜔(𝑣^{1/(𝑝−𝑞)}(𝜉))Δ𝜉. Then, 𝑣(𝑡)≤𝑚(𝑡), and 𝑣^Δ(𝑡)≤𝑓𝜏₁(𝑡)𝑚(𝑡). (3.69) Furthermore, 𝑚^Δ(𝑡)=𝑣^Δ(𝑡)+𝑔𝜏₁(𝑡)𝜔𝑣^{1/(𝑝−𝑞)}(𝑡)≤𝑓𝜏₁(𝑡)𝑚(𝑡)+𝑔𝜏₁(𝑡)𝜔𝑚^{1/(𝑝−𝑞)}(𝑡)≤𝑓𝜏₁(𝑡)+𝑔𝜏₁(𝑡)𝑚(𝑡)+𝜔𝑚^{1/(𝑝−𝑞)}(𝑡), (3.70) that is, 𝑚^Δ(𝑡)𝑚(𝑡)+𝜔𝑚^{1/(𝑝−𝑞)}(𝑡)≤𝑓𝜏₁(𝑡)+𝑔𝜏₁(𝑡). (3.71) On the other hand, similar to (3.21)–(3.23), we have 𝐺(𝑚(𝑡))^Δ≤𝑚^Δ(𝑡)𝑚(𝑡)+𝜔𝑚^{1/(𝑝−𝑞)}(𝑡). (3.72) So combining (3.71) and (3.72), we deduce 𝐺(𝑚(𝑡))^Δ≤𝑓𝜏₁(𝑡)+𝑔𝜏₁(𝑡). (3.73) Considering 𝑚(𝑡₀)=𝑧(𝑡₀)=𝐽_𝜀(𝑇), an integration for (3.73) from 𝑡₀ to 𝑡 yields 𝐺(𝑚(𝑡))≤𝐺𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉. (3.74) Since 𝐺 is increasing, then 𝑚(𝑡)≤𝐺⁻¹𝐺𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉. (3.75) Combining (3.69) and (3.75), we obtain 𝑣^Δ(𝑡)≤𝑓𝜏₁(𝑡)𝐺⁻¹𝐺𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉. (3.76) Setting 𝑡=𝑇 in (3.76), since 𝑇 is selected from 𝕋₀ arbitrarily, then in fact (3.76) holds for all 𝑡∈𝕋₀, that is, 𝑣^Δ(𝑡)≤𝑓𝜏₁(𝑡)𝐺⁻¹𝐺𝐽_𝜀(𝑡)+^𝑡_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉,∀𝑡∈𝕋₀. (3.77) Considering 𝑣(𝑡₀)=𝐽_𝜀(𝑇), an integration for (3.77) from 𝑡₀ to 𝑡 yields 𝑣(𝑡)≤𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝐺⁻¹𝐺𝐽_𝜀(𝑠)+^𝑠_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉Δ𝑠, (3.78) which implies 𝑢(𝑡)≤𝐽_𝜀(𝑇)+^𝑡_𝑡0𝑓𝜏₁(𝑠)𝐺⁻¹×𝐺𝐽_𝜀(𝑠)+^𝑠_𝑡0𝑓𝜏₁(𝜉)+𝑔𝜏₁(𝜉)Δ𝜉Δ𝑠^{1/(𝑝−𝑞)},𝑡∈𝑡₀,𝑇∩𝕋. (3.79) Setting 𝑡=𝑇 in the above inequality, considering 𝑇 is selected from 𝕋₀ arbitrarily, after replacing 𝑇 with 𝑡 and letting 𝜀→0, we get the desired result. So the proof is complete.

Remark 3.10. If we take 𝕋=ℝ,𝑡₀=0,𝑝=2,𝑞=1,𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem 3.9 reduces to [33, Theorem  3(a7)]. If we take 𝕋=ℤ,𝑡₀=0,𝑝=2,𝑞=1,𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡, then Theorem  3.9 reduces to [33, Theorem  6(b7)].
Following in a similar manner as the proof of Theorem 3.7 and 3.9, then we present two more theorems as follows, the special cases of which (𝑝=2,𝑞=1,𝑡₀=0,𝜏₁(𝑡)=𝜏₂(𝑡)=𝑡) unify the continuous [33, Theorems  1(a3),  3(a8)] and discrete inequalities [33, Theorems  4(b3),  6(b8)].

Theorem 3.11. Suppose 𝑢,𝑓,𝑔,ℎ,𝑝,𝑞,𝐶,𝛼,𝜙,𝜏_𝑖,𝑖=1,2 are defined as in Theorem 3.9. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)≤𝐶^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)×^𝑠_𝑡0𝑔𝜏₁(𝜉)𝑢^𝑝−𝑞(𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠 (3.80) with the initial condition (3.5), then 𝑢(𝑡)≤𝐽(𝑡)𝑒_𝐻𝑡,𝑡₀^{1/(𝑝−𝑞)},𝑡∈𝕋₀ (3.81) provided 𝐻∈ℜ₊, where 𝐻(𝑡)=𝑓(𝜏₁(𝑡))∫^𝑡_𝑡0𝑔(𝜏₁(𝜉))Δ𝜉, and 𝐽(𝑡) is defined as in Theorem 3.7.

Theorem 3.12. Suppose 𝑢,𝑓,𝑔,ℎ,𝜔,𝑝,𝑞,𝐶,𝛼,𝜙,𝜏_𝑖,𝑖=1,2 are defined as in Theorem 3.9. If for 𝑡∈𝕋₀, 𝑢(𝑡) satisfies the following inequality: 𝑢^𝑝(𝑡)≤𝐶^{𝑝/(𝑝−𝑞)}+𝑝𝑝−𝑞^𝑡_𝑡0𝑓𝜏₁(𝑠)𝑢^𝑞𝜏₁(𝑠)×^𝑠_𝑡0𝑔𝜏₁(𝜉)𝜔𝑢𝜏₁(𝜉)Δ𝜉+ℎ𝜏₂(𝑠)𝑢^𝑞𝜏₂(𝑠)Δ𝑠 (3.82) with the initial condition (3.5), then 𝑢(𝑡)≤𝐺⁻¹𝐺(𝐽(𝑡))+^𝑡_𝑡0𝑓𝜏₁(𝑠)^𝑠_𝑡0𝑔𝜏₁(𝜉)Δ𝜉Δ𝑠^{1/(𝑝−𝑞)},𝑡∈𝕋₀,𝑡∈𝕋₀, (3.83) where 𝐺 is defined as in Theorem 3.2, and 𝐽(𝑡) is defined as in Theorem 3.7.

4. Some Applications

In this section, we will present some applications for the results which we have established above and apply them to qualitative and quantitative analysis of solutions of certain delay dynamic equations on time scales.

Example 4.1. Consider the following delay dynamic integral equation on time scales: 𝑢^𝑝(𝑡)=𝐶+^𝑡_𝑡0𝐹(𝜏(𝑠),𝑢(𝜏(𝑠)))Δ𝑠,𝑡∈𝕋₀ (4.1) with the initial condition 𝑢(𝑡)=𝜙(𝑡),𝑡∈𝛼,𝑡₀𝜙(𝜏(𝑡≤|𝐶|∩𝕋,))^1/𝑝,∀𝑡∈𝕋₀,𝜏(𝑡)≤𝑡₀, (4.2) where 𝑢∈𝐶_rd(𝕋₀,ℝ), 𝜏∈(𝕋₀,𝕋) with 𝜏(𝑡)≤𝑡, −∞<𝛼=inf{𝜏(𝑡),𝑡∈𝕋₀}≤𝑡₀, 𝜙∈𝐶_rd([𝛼,𝑡₀]∩𝕋,ℝ), 𝐹∈𝐶_rd(([𝛼,∞)∩𝕋)×ℝ,ℝ), and 𝑝 is a constant with 𝑝>1.

Theorem 4.2. Suppose 𝑢(𝑡) is a solution of (4.1)-(4.2), and |𝐹(𝑡,𝑢)|≤𝑓(𝑡)|𝑢|^𝑝+𝑔(𝑡)|𝑢|^𝑝−1, where 𝑓,𝑔∈𝐶_𝑟𝑑([𝛼,∞)∩𝕋,ℝ₊), then 𝑢(𝑡)≤𝐶+^𝑡_𝑡0𝑔(𝜏(𝑠))Δ𝑠exp^𝑡_𝑡0𝑓(𝜏(𝑠))Δ𝑠,𝑡∈𝕋₀. (4.3)

Proof. In fact, from (4.1) we have |𝑢(𝑡)|^𝑝≤|𝐶|+^𝑡_𝑡0|𝐹(𝜏(𝑠)(𝜏(𝑠≤|𝐶|+,𝑢)))|Δ𝑠_𝑡𝑡0𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑝+𝑔(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑝−1=|𝐶|+Δ𝑠_𝑡𝑡0𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑝−1𝜔(|𝑢(𝜏(𝑠))|)+𝑔(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑝−1Δ𝑠, (4.4) where 𝜔(𝑟)=𝑟. Then a suitable application of Theorem 3.2 (with 𝜏₁=𝜏₂=𝜏, 𝑞=𝑝−1) to (4.4) yields 𝑢(𝑡)≤𝐺⁻¹𝐺𝐶+^𝑡_𝑡0𝑔(𝜏(𝑠))Δ𝑠+^𝑡_𝑡0𝑓(𝜏(𝑠))Δ𝑠,𝑡∈𝕋₀, (4.5) where 𝐺(𝑥)=∫^𝑥₁(1/𝑟)𝑑𝑟=𝑙𝑛𝑥,forall𝑥>0. Using the expression of 𝐺 in (4.5), we obtain the desired result, and the proof is complete.

Example 4.3. Consider the following delay dynamic differential equation on time scales 𝑢³(𝑡)^Δ=𝐹𝜏(𝑡),𝑢(𝜏(𝑡)),^𝑡_𝑡0𝑀(𝜏(𝜉),𝑢(𝜏(𝜉)))Δ𝜉,𝑡∈𝕋₀ (4.6) with the initial condition 𝑢𝑡₀=𝑢₀,𝑢(𝑡)=𝜙(𝑡),𝑡∈𝛼,𝑡₀𝜙(𝜏(𝑡≤𝑢∩𝕋,))₀,∀𝑡∈𝕋₀,𝜏(𝑡)≤𝑡₀, (4.7) where 𝑢,𝜏,𝛼,𝜙 are defined as in Example 4.1, 𝐹∈𝐶_rd(([𝛼,∞)∩𝕋)×ℝ²,ℝ), 𝑀∈𝐶_rd(([𝛼,∞)∩𝕋)×ℝ,ℝ).

Theorem 4.4. Suppose 𝑢(𝑡) is a solution of (4.6)-(4.7), and |𝐹(𝑡,𝑢,𝑣)|≤𝑓(𝑡)|𝑢|³+𝑓(𝑡)|𝑢||𝑣|,|𝑀(𝑡,𝑢)|≤𝑔(𝑡)|𝑢|², where 𝑓,𝑔∈𝐶_rd([𝛼,∞)∩𝕋,ℝ₊), then 𝑢(𝑡)≤𝑢₀1+^𝑡_𝑡0𝑓(𝜏(𝑠))𝑒_{(𝑓+𝑔)(𝜏(⋅))}𝑠,𝑡₀Δ𝑠,𝑡∈𝕋₀ (4.8) provided (𝑓+𝑔)(𝜏(⋅))∈ℜ₊.

Proof. The equivalent integral form of (4.6)-(4.7) is denoted by 𝑢³(𝑡)=𝑢³₀+^𝑡_𝑡0𝐹𝜏(𝑠),𝑢(𝜏(𝑠)),^𝑠_𝑡0𝑀(𝜏(𝜉),𝑢(𝜏(𝜉)))Δ𝜉Δ𝑠. (4.9) So, |𝑢(𝑡)|³≤𝑢₀³+^𝑡_𝑡0𝐹𝜏(𝑠),𝑢(𝜏(𝑠)),^𝑠_𝑡0𝑀(𝜏(𝜉)(𝜏(𝜉≤𝑢,𝑢)))Δ𝜉Δ𝑠₀₃+^𝑡_𝑡0𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|³+𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑠_𝑡0𝑀(𝜏(𝜉)(𝜏(𝜉≤𝑢,𝑢)))Δ𝜉Δ𝑠_203/2+^𝑡_𝑡0𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|³+𝑓(𝜏(𝑠))|𝑢(𝜏(𝑠))|^𝑠_𝑡0𝑔(𝜏(𝜉))𝑢(𝜏(𝜉))²Δ𝜉Δ𝑠. (4.10) Then a suitable application of Theorem 3.7 (with 𝑝=3,𝑞=1,𝜏₁=𝜏₂=𝜏,ℎ(𝑡)≡0) to (4.10) yields the desired inequality (4.8), and the proof is complete.

Theorem 4.5. If under the conditions of Theorem 4.4, for 𝑡∈𝕋₀, ∫^𝑡_𝑡0𝑓(𝜏₁(𝑠))𝑒_{(𝑓+𝑔)(𝜏1(.))}(𝑠,𝑡₀)Δ𝑠≤𝐿, where 𝐿>0 is a constant, then the trivial solution of (4.6)-(4.7) is uniformly stable.

Theorem 4.6. Assume |𝐹(𝑡,𝑢₁,𝑣₁)−𝐹(𝑡,𝑢₂,𝑣₂)|≤𝑓(𝑡)|𝑢³₁−𝑢³₂|+𝑓(𝑡)|𝑢³₁−𝑢³₂||𝑣₁−𝑣₂|,|𝑀(𝑡,𝑢₁)−𝑀(𝑡,𝑢₂)|≤𝑔(𝑡)||𝑢³₁−𝑢³₂|, where 𝑓,𝑔 are defined as in Theorem 4.4, then (4.6)-(4.7), have at most one solution.

Proof. Suppose 𝑢₁(𝑡),𝑢₂(𝑡) are two solutions of (4.6)-(4.7). By (4.9) we have 𝑢³₁(𝑡)−𝑢³₂(𝑡)≤^𝑡_𝑡0𝐹𝜏(𝑠),𝑢₁(𝜏(𝑠)),^𝑠_𝑡0𝑀𝜏(𝜉),𝑢₁(𝜏(𝜉𝜏(𝑠)))Δ𝜉−𝐹,𝑢₂(𝜏(𝑠)),^𝑠_𝑡0𝑀𝜏(𝜉),𝑢₂(𝜏(𝜉≤))Δ𝜉Δ𝑠_𝑡𝑡0𝑓(𝜏(𝑠))𝑢³₁(𝜏(𝑠))−𝑢³₂(𝜏(𝑠))+𝑓(𝜏(𝑠))𝑢³₁(𝜏(𝑠))−𝑢³₂(𝜏(𝑠×))_𝑠𝑡0𝑀𝜏(𝜉),𝑢₁(𝜏(𝜉))−𝑀𝜏(𝜉),𝑢₂(𝜏(𝜉≤))Δ𝜉Δ𝑠_𝑡𝑡0𝑓(𝜏(𝑠))𝑢³₁(𝜏(𝑠))−𝑢³₂(𝜏(𝑠))+𝑓(𝜏(𝑠))𝑢³₁(𝜏(𝑠))−𝑢³₂(𝜏(𝑠))×^𝑠_𝑡0𝑔(𝜏(𝜉))𝑢³₁(𝜏(𝜉))−𝑢³₂(𝜏(𝜉))Δ𝜉Δ𝑠. (4.11) Treat |𝑢³₁(𝑡)−𝑢³₂(𝑡)| as one variable, and applying Theorem 3.7 to (4.11) (with 𝑝=1,𝑞=1/2,𝜏₁=𝜏₂=𝜏,ℎ(𝑡)≡0) yields |𝑢³₁−𝑢³₂|≤0, which implies 𝑢₁(𝑡)≡𝑢₂(𝑡), and the proof is complete.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (11171178 and 11101248), Natural Science Foundation of Shandong Province (ZR2009AM011, ZR2010AQ026, and ZR2010AZ003) (China), and Specialized Research Fund for the Doctoral Program of Higher Education (20103705 110003)(China). The authors thank the referees very much for their careful comments and valuable suggestions on this paper.