Explicit Bounds to Some New Gronwall-Bellman-Type Delay Integral Inequalities in Two Independent Variables on Time Scales

Abstract

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.

1. Introduction

During the past decades, with the development of the theory of differential and integral equations, a lot of integral and difference inequalities have been discovered, 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. In these established inequalities, Gronwall-Bellman-type inequalities are of particular importance as these inequalities provide explicit bounds for unknown functions, and much effort has been done for developing such inequalities (e.g., see [1–13] and the references therein). On the other hand, Hilger [14] initiated the theory of time scales as a theory capable containing 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 (e.g., see [15–17] and the references therein). In these investigations, integral inequalities on time scales have been paid much attention by many authors, and a lot of integral inequalities on time scales have been established (e.g., see [18–26]), which have been designed to unify continuous and discrete analysis and play an important role in the research of qualitative and quantitative properties of solutions of certain dynamic equations on time scales. But to our knowledge, Gronwall-Bellman-type delay integral inequalities on time scales have been paid little attention in the literature so far. Recent results in this direction include the works of Li [27] and Ma and Pečarić [28] to our best knowledge. Furthermore, nobody has studied Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.

Our aim in this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which unify some known continuous and discrete analysis. New explicit bounds for unknown functions are obtained due to the presented inequalities. We will also present some applications for our results.

First we will give some preliminaries on time scales and some universal symbols for further use.

Throughout the 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).

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) by 𝜎(𝑡)=inf{𝑠∈𝕋,𝑠>𝑡}and𝜌(𝑡)=sup{𝑠∈𝕋,𝑠<𝑡}.

Definition 1.1. The graininess 𝜇∈(𝕋,ℝ₊) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡.

Remark 1.2. Obviously, 𝜇(𝑡)=0 if 𝕋=ℝ while 𝜇(𝑡)=1 if 𝕋=ℤ.

Definition 1.3. 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 1.4. 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 1.5. A function 𝑓∈(𝕋,ℝ) is called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at 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,∀𝑡∈𝕋}.

Definition 1.6. For some 𝑡∈𝕋^𝜅 and a function 𝑓∈(𝕋,ℝ), the delta derivative of 𝑓 is denoted by 𝑓^Δ(𝑡) and satisfies ||𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)||≤𝜀||𝜎(𝑡)−𝑠||for∀𝜀>0,(1.1) where 𝑠∈𝔘, and 𝔘 is a neighborhood of 𝑡 which can depend on 𝜀.
Similarly, for some 𝑦∈𝕋^𝜅 and a function 𝑓∈(𝕋×𝕋,ℝ), the partial delta derivative of 𝑓 with respect to 𝑦 is denoted by (𝑓(𝑥,𝑦))^Δ𝑦 or 𝑓^Δ𝑦(𝑥,𝑦) and satisfies ||𝑓(𝑥,𝜎(𝑦))−𝑓(𝑥,𝑠)−𝑓^Δ𝑦(𝑥,𝑦)(𝜎(𝑦)−𝑠)||≤𝜀||𝜎(𝑦)−𝑠||for∀𝜀>0,(1.2) where 𝑠∈𝔘, and 𝔘 is a neighborhood of 𝑦 which can depend on 𝜀.

Remark 1.7. 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 1.8. For 𝑎,𝑏∈𝕋 and a function 𝑓∈(𝕋,ℝ), the Cauchy integral of 𝑓 is defined by ^𝑏𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎),(1.3) where 𝐹^Δ(𝑡)=𝑓(𝑡),𝑡∈𝕋^𝜅.
Similarly, for 𝑎,𝑏∈𝕋 and a function 𝑓∈(𝕋×𝕋,ℝ), the Cauchy partial integral of 𝑓 with respect to 𝑦 is defined by ^𝑏𝑎𝑓(𝑥,𝑦)Δ𝑦=𝐹(𝑥,𝑏)−𝐹(𝑥,𝑎),(1.4) where 𝐹^Δ𝑦(𝑥,𝑦)=𝑓(𝑥,𝑦),𝑦∈𝕋^𝜅.

Definition 1.9. The cylinder transformation 𝜉_ℎ is defined by 𝜉_ℎ(𝑧)=𝑧L,og(1ℎ+ℎ𝑧),iiffℎℎ≠=00,for𝑧≠−ℎ1,(1.5) where Log is the principal logarithm function.

Definition 1.10. For 𝑝(𝑥,𝑦)∈ℜ with respect to 𝑦, the exponential function is defined by 𝑒_𝑝(𝑦,𝑠)=exp^𝑦𝑠𝜉_𝜇(𝜏)(𝑝(𝑥,𝜏))Δ𝜏,for𝑠,𝑦∈𝕋.(1.6)

Remark 1.11. If 𝕋=ℝ, then for 𝑦∈ℝ the following formula holds: 𝑒_𝑝(𝑦,𝑠)=exp^𝑦𝑠𝑝(𝑥,𝜏)𝑑𝜏,for𝑠∈𝕋.(1.7) If 𝕋=ℤ, then, for 𝑦∈ℤ, 𝑒_𝑝(𝑦,𝑠)=∏^{𝑦−1𝜏=𝑠}[1+𝑝(𝑥,𝜏)],for𝑠∈ℤ and 𝑠<𝑦.

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

Theorem 1.12. If 𝑝(𝑥,𝑦)∈ℜ with respect to 𝑦, then the following conclusions hold: (i)𝑒_𝑝(𝑦,𝑦)≡1 and 𝑒₀(𝑠,𝑦)≡1,(ii)𝑒_𝑝(𝑠,𝜎(𝑦))=(1+𝜇(𝑦)𝑝(𝑥,𝑦))𝑒_𝑝(𝑠,𝑦),(iii) if 𝑝∈ℜ⁺ with respect to 𝑦, then 𝑒_𝑝(𝑠,𝑦)>0 for all 𝑠,𝑦∈𝕋,(iv) if 𝑝∈ℜ⁺ with respect to 𝑦, then ⊖𝑝∈ℜ⁺,(v)𝑒_𝑝(𝑠,𝑦)=1/𝑒_𝑝(𝑦,𝑠)=𝑒_⊖𝑝(𝑦,𝑠), where (⊖𝑝)(𝑥,𝑦)=−(𝑝(𝑥,𝑦)/1+𝜇(𝑦)𝑝(𝑥,𝑦)).

Theorem 1.13. If 𝑝(𝑥,𝑦)∈ℜ with respect to 𝑦,𝑦₀∈𝕋 is a fixed number, then the exponential function 𝑒_𝑝(𝑦,𝑦₀) is the unique solution of the following initial value problem: 𝑧^Δ𝑦(𝑥,𝑦𝑧)=𝑥,𝑝𝑦(₀𝑥,𝑦)𝑧(𝑥,𝑦),=1.(1.8)

Theorems 1.12-1.13 are similar to [24, Theorems 5.1-5.2]. For more details about the calculus of time scales, we advise to refer to [29].

In the rest of this paper, for the convenience of notation, we always assume that 𝕋₀=[𝑥₀,∞)⋂𝕋,𝕋₀=[𝑦₀,∞)⋂𝕋, where 𝑥₀,𝑦₀∈𝕋^𝜅, and furthermore assume that 𝕋₀⊆𝕋^𝜅,𝕋₀⊆𝕋^𝜅.

2. Main Results

We will give some lemmas for further use.

Lemma 2.1. Suppose that 𝑋∈𝕋₀ is a fixed number and 𝑢(𝑋,𝑦),𝑏(𝑋,𝑦)∈𝐶_rd,𝑚(𝑋,𝑦)∈ℜ₊ with respect to 𝑦, 𝑚(𝑋,𝑦)≥0; then 𝑢(𝑋,𝑦)≤𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀𝑚(𝑋,𝑡)𝑢(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀,(2.1) implies 𝑢(𝑋,𝑦)≤𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀𝑒_𝑚(𝑦,𝜎(𝑡))𝑎(𝑋,𝑡)𝑚(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀,(2.2) where 𝑚(𝑋,𝑦)=𝑚(𝑋,𝑦)𝑏(𝑋,𝑦), and 𝑒_𝑚(𝑦,𝑦₀) is the unique solution of the following IVP 𝑧^Δ𝑦(𝑋,y)=𝑚(𝑋,𝑦)𝑧(𝑋,𝑦),𝑧𝑋,𝑦₀=1.(2.3)

The proof of Lemma 2.1 is similar to that of [24, Theorem 5.6], and we omit it here.

Lemma 2.2. Under the conditions of Lemma 2.1 and furthermore assuming that 𝑎(𝑥,𝑦) is nondecreasing in 𝑦 for every fixed 𝑥,𝑏(𝑥,𝑦)≡1, then one has 𝑢(𝑋,𝑦)≤𝑎(𝑋,𝑦)𝑒_𝑚𝑦,𝑦₀.(2.4)

Proof. Since 𝑎(𝑥,𝑦) is nondecreasing in 𝑦 for every fixed 𝑥, then from Lemma 2.1 we have 𝑢(𝑋,𝑦)≤≤𝑎𝑎((𝑋𝑋,,𝑦𝑦))+1+^𝑦𝑦₀𝑒_{𝑚𝑦𝑦0}(𝑦,𝜎(𝑡))𝑎(𝑋,𝑡)𝑚(𝑋,𝑡)Δ𝑡𝑒_𝑚(𝑦,𝜎(𝑡))𝑚(𝑋,𝑡)Δ𝑡.(2.5) On the other hand, from [29, Theorems 2.39 and 2.36 (i)] we have 1+∫^𝑦𝑦₀𝑒_𝑚(𝑦,𝜎(𝑡))𝑚(𝑋,𝑡)Δ𝑡=𝑒_𝑚(𝑦,𝑦₀).
Then collecting the above information we can obtain the desired inequality.

Lemma 2.3 (see [30]). Assume that 𝑎≥0,𝑝≥𝑞≥0, and 𝑝≠0, then for any 𝐾>0𝑎^𝑞/𝑝≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎+𝑝−𝑞𝑝𝐾^𝑞/𝑝.(2.6)

Theorem 2.4. Suppose that 𝑢,𝑓,𝑔,ℎ,𝑎,𝑏∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊) and 𝑎,𝑏 are nondecreasing. 𝑝,𝑞,𝑟,𝑚 are constants, and 𝑝≥𝑞≥0,𝑝≥𝑟≥0,𝑝≥𝑚≥0,𝑝≠0.𝜏₁∈(𝕋₀,𝕋),𝜏₁(𝑥)≤𝑥,−∞<𝛼=inf{𝜏₁(𝑥),𝑥∈𝕋₀}≤𝑥₀. 𝜏₂∈(𝕋₀,𝕋),𝜏₂(𝑦)≤𝑦,−∞<𝛽=inf{𝜏₂(𝑦),𝑦∈𝕋₀}≤𝑦₀. 𝜙∈𝐶_rd(([𝛼,𝑥₀]×[𝛽,𝑦₀])⋂𝕋²,ℝ₊). If for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁+𝑏(𝑥,𝑦)_𝑦𝑦0(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑟(𝑠,𝑡)Δ𝑠Δ𝑡^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.7) with the initial condition 𝜙𝜏𝑢(𝑥,𝑦)=𝜙(𝑥,𝑦),if𝑥∈𝛼,𝑥₀𝕋,or𝑦∈𝛽,𝑦₀𝕋,₁(𝑥),𝜏₂(𝑦)≤𝛼^1/𝑝(𝑥,𝑦),if𝜏₁(𝑥)≤𝑥₀or𝜏₂(𝑦)≤𝑦₀,∀(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.8) then 𝑢(𝑥,𝑦)≤𝐵₁(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡^1/𝑝,(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.9) where 𝐵₁(𝑥,𝑦)=𝑎(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑝−𝑝𝑞𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟+𝑏(𝑥,𝑦)_𝑦𝑦0𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡,∀𝐾>0,(2.10)𝐵₂(𝑥,𝑦)=^𝑥𝑥₀𝑓(𝑠,𝑦)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝑔(𝑠,𝑦)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+^𝑦𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}Δ𝜉Δ𝜂Δ𝑠,∀𝐾>0,(2.11)𝐵₂(𝑥,𝑦)=𝑏(𝑥,𝑦)𝐵₂(𝑥,𝑦).(2.12)

Proof. Let the right side of (2.7) be 𝑣(𝑥,𝑦). Then 𝑢(𝑥,𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝕋₀×𝕋₀.(2.13) If 𝜏₁(𝑥)≥𝑥₀ and 𝜏₂(𝑦)≥𝑦₀, then 𝜏₁(𝑥)∈𝕋₀,𝜏₂(𝑦)∈𝕋₀, and since 𝑎,𝑏 are nondecreasing we have 𝑢𝜏₁(𝑥),𝜏₂(𝑦)≤𝑣^1/𝑝𝜏₁(𝑥),𝜏₂(𝑦)≤𝑣^1/𝑝(𝑥,𝑦).(2.14) If 𝜏₁(𝑥)≤𝑥₀ or 𝜏₂(𝑦)≤𝑦₀, then from (2.8) we have 𝑢𝜏₁(𝑥),𝜏₂(𝑦)=𝜙𝜏₁(𝑥),𝜏₂(𝑦)≤𝑎^1/𝑝(𝑥,𝑦)≤𝑣^1/𝑝(𝑥,𝑦).(2.15) From (2.14) and (2.15) we have 𝑢𝜏₁(𝑥),𝜏₂(𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝕋₀×𝕋₀.(2.16) Fix 𝑋∈𝕋₀, and let 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈𝕋₀; then 𝑣(𝑋,𝑦)=𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀^𝑋𝑥₀𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁+𝑏(𝑋,𝑦)_𝑦𝑦0(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑟(𝑠,𝑡)Δ𝑠Δ𝑡^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀≤𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)_𝑦𝑦0ℎ(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝑓(𝑠,𝑡)𝑣^𝑞/𝑝+(𝑠,𝑡)+𝑔(𝑠,𝑡)𝑣^𝑟/𝑝(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ(𝜉,𝜂)𝑣^𝑚/𝑝(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.17) From Lemma 2.3, we have 𝑣^𝑞/𝑝(𝑥,𝑦)≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑥,𝑦)+𝑝−𝑞𝑝𝐾^𝑞/𝑝𝑣_𝑟/𝑝,(𝑥,𝑦)≤𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑥,𝑦)+𝑝−𝑟𝑝𝐾^𝑟/𝑝𝑣_𝑚/𝑝,(𝑥,𝑦)≤𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝑣(𝑥,𝑦)+𝑝−𝑚𝑝𝐾^𝑚/𝑝,∀𝐾>0.(2.18) So combining (2.17) and (2.18), it follows that 𝑣(𝑋,𝑦)≤𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀^𝑋𝑥₀𝑓(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑞+𝑏(𝑋,𝑦)_𝑦𝑦0𝑝𝐾^𝑞/𝑝Δ𝑠Δ𝑡^𝑋𝑥₀𝑔(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑟+𝑏(𝑋,𝑦)_𝑦𝑦0𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑚≤𝑎(𝑋,𝑦)+𝑏(𝑋,𝑦)_𝑦𝑦0𝑝𝐾^𝑚/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝑓(𝑠,𝑡)𝑝−𝑝𝑞𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟𝑝𝐾^𝑟/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝+𝑏(𝑋,𝑦)_𝑦𝑦0Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝑓(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝑔(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}=𝐵₁Δ𝜉Δ𝜂Δ𝑠𝑣(𝑋,𝑡)Δ𝑡(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀𝐵₂(𝑋,𝑡)𝑣(𝑋,𝑡)Δ𝑡,(2.19) where 𝐵₁(𝑥,𝑦),𝐵₂(𝑥,𝑦) are defined in (2.10) and (2.11), respectively. Considering 𝐵₂(𝑋,𝑦)=𝑏(𝑋,𝑦)𝐵₂(𝑋,𝑦), by application of Lemma 2.1, we have 𝑣(𝑋,𝑦)≤𝐵₁(𝑋,𝑦)+𝑏(𝑋,𝑦)^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑋,𝑡)𝐵₁(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀.(2.20) Since 𝑋∈𝕋₀ is arbitrary, then in fact (2.20) holds for all 𝑥∈𝕋₀, that is, 𝑣(𝑥,𝑦)≤𝐵₁(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡,(𝑥,𝑦)∈𝕋₀×𝕋₀.(2.21) Combining (2.13) and (2.21), we obtain 𝑢(𝑥,𝑦)≤𝐵₁(𝑥,𝑦)+𝑏(𝑥,𝑦)^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡^1/𝑝,(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.22) which is the desired inequality.

If we apply Lemma 2.2 instead of Lemma 2.1 at the end of the proof of Theorem 2.4, we obtain the following theorem.

Theorem 2.5. Suppose that 𝑢,𝑓,𝑔,ℎ,𝑎,𝑝,𝑞,𝑟,𝑚,𝜏₁,𝜏₂,𝛼,𝛽,𝜙 are defined as in Theorem 2.4. If that for (𝑥,𝑦)∈𝕋₀×𝕋₀,𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁+(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑟(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡(2.23) with the initial condition (2.8), then 𝑢(𝑥,𝑦)≤𝐵₁(𝑥,𝑦)𝑒_𝐵2𝑦,𝑦₀^1/𝑝,(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.24) where 𝐵₁(𝑥,𝑦)=𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑝−𝑝𝑞𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟+𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝𝐵Δ𝜉Δ𝜂Δ𝑠Δ𝑡,∀𝐾>0,₂(𝑥,𝑦)=^𝑥𝑥₀𝑓(𝑠,𝑦)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝑔(𝑠,𝑦)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+^𝑦𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}∀𝐾>0Δ.𝜉Δ𝜂Δ𝑠,(2.25)

From Theorems 2.4 and 2.5 we can obtain two direct corollaries.

Corollary 2.6. Under the conditions of Theorem 2.4, if, for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑢𝜏₁+(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢(𝑠,𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.26) with the initial condition (2.8) (p=1), then 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝑎(𝑥,𝑡)Δ𝑡,(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.27) where 𝐵₂(𝑥,𝑦)=^𝑥𝑥₀𝑓(𝑠,𝑦)+𝑔(𝑠,𝑦)+^𝑦𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠.(2.28)

Corollary 2.7. Under the conditions of Theorem 2.5, if, for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑢𝜏₁+(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢(𝑠,𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.29) with the initial condition (2.8) (𝑝=1), then 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)𝑒_𝐵2𝑦,𝑦₀,(𝑥,𝑦)∈𝕋₀×𝕋₀,(2.30) where 𝐵₂(𝑥,𝑦)=^𝑥𝑥₀𝑓(𝑠,𝑦)+𝑔(𝑠,𝑦)+^𝑦𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠.(2.31)

Theorem 2.8. Suppose that 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊), 𝑓,𝑔,ℎ,𝜏₁,𝜏₂ are defined as in Theorem 2.4, and 𝜏₁(𝑥)≥𝑥₀, 𝜏₂(𝑦)≥𝑦₀. If, for (𝑥,𝑦)∈𝕋₀×𝕋₀,𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢≤(𝑥,𝑦)_𝑦𝑦0^𝑥𝑥₀𝑓(𝑠,𝑡)𝑢𝜏¹(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢(𝑠,𝑡)+^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.32) then 𝑢(𝑥,𝑦)≡0.

The proof of Theorem 2.8 is similar to Theorem 2.4, and we omit it here.

Based on Theorem 2.4, we will establish a class of Volterra-Fredholm-type integral inequality on time scales.

Theorem 2.9. Suppose that 𝑢,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊),𝑖=1,2.𝑎,𝑝,𝑞,𝑟,𝑚,𝜙,𝜏₁,𝜏₂,𝛼,𝛽 are the same as in Theorem 2.4, and 𝑀∈𝕋₀,𝑁∈𝕋₀ are two fixed numbers. If, for (𝑥,𝑦)∈([𝑥₀,𝑀]⋂𝕋)×([𝑦₀,𝑁]⋂𝕋),𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝(𝑥,𝑦)≤𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓₁+(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔₁(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀+ℎ₁(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝑓₂+(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔₂(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.33) with the initial condition (2.8), then one has 𝑢(𝑥≤,𝑦)𝜆+𝐵₆1−𝐵₅𝐵³(𝑥,𝑦)+𝐵₄(𝑥,𝑦)^1/𝑝,(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋,(2.34) provided that 𝐵₅<1, where 𝜆=^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑔₂(𝑠,𝑡)𝑝−𝑟𝑝𝐾^𝑟/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝𝐵Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.35)₁(𝑥,𝑦)=𝑎(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓¹(𝑠,𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑔₁(𝑠,𝑡)𝑝−𝑟+𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝𝐵Δ𝜉Δ𝜂Δ𝑠Δ𝑡,∀𝐾>0,(2.36)₂(𝑥,𝑦)=^𝑥𝑥₀𝑓¹(𝑠,𝑦)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝑔₁(𝑠,𝑦)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+^𝑦𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝐵Δ𝜉Δ𝜂Δ𝑠,∀𝐾>0,(2.37)₃𝐵(𝑥,𝑦)=1+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)Δ𝑡,(2.38)₄(𝑥,𝐵𝑦)=𝐵₁(𝑥,𝑦)+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡,(2.39)₅=^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵₃(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝐵₃(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡_y0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝐵𝐵₃(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.40)₆=^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵₄(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝐵₄(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.41)

Proof. Let the right side of (2.33) be 𝑣(𝑥,𝑦) and 𝜇=^𝑁𝑦₀^𝑀𝑥₀𝑓₂+(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔₂(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.42) Then 𝑢(𝑥,𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋.(2.43) Similar to the process of (2.14)–(2.16) one has 𝑢𝜏₁(𝑥),𝜏₂(𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋.(2.44) Fix 𝑋∈[𝑥₀,𝑀]⋂𝕋, and let 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈[𝑦₀,𝑁]⋂𝕋. Then 𝑣(𝑋,𝑦)=𝑎(𝑋,𝑦)+𝜇+^𝑦𝑦₀^𝑋𝑥₀𝑓₁+(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔₁(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀≤𝑎(𝑋,𝑦)+𝜇+ℎ_1𝑦𝑦0(𝜉,𝜂)𝑢^𝑚𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝑓₁+(𝑠,𝑡)𝑣^𝑞/𝑝(𝑠,𝑡)+𝑔₁(𝑠,𝑡)𝑣^𝑟/𝑝(𝑠,𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑣^𝑚/𝑝(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.45) Considering the structure of (2.45) is similar to (2.17), then following in a same manner as the process of (2.17)–(2.20) we can deduce 𝑣(𝑋,𝑦)≤=𝜇𝜇+1𝐵+₁(𝑋_𝑦𝑦0,𝑦)+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑋,𝑡)𝜇+𝐵₁(𝑋,𝑡)Δ𝑡+𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑋,𝑡)Δ𝑡+𝐵₁(𝑋,𝑦)_𝑦𝑦0𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑋,𝑡)𝐵₁(𝑋,𝑡)Δ𝑡,𝑦∈𝑦₀,𝑁𝕋,(2.46) where 𝐵₁(𝑥,𝑦),𝐵₂(𝑥,𝑦) are defined in (2.36) and (2.37), respectively.
Since 𝑋 is selected from [𝑥₀,𝑀]⋂𝕋 arbitrarily, then in fact (2.46) holds for all 𝑥∈𝕋₀, that is, 𝑣(𝑥,𝑦)≤=𝜇𝜇𝐵1₃+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)Δ𝑡+𝐵₁(𝑥,𝑦)+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡(𝑥,𝑦)+𝐵₄(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋,(2.47) where 𝐵₃(𝑥,𝑦),𝐵₄(𝑥,𝑦) are defined in (2.38) and (2.39), respectively.
On the other hand, from (2.18), (2.42), and (2.44) we obtain 𝜇≤^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑣^𝑞/𝑝(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑣^𝑟/𝑝(𝑠,𝑡)+^𝑡𝑦₀^𝑠𝑥₀≤ℎ₂(𝜉,𝜂)𝑣^𝑚/𝑝(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑞+𝑔₂𝑝𝐾^𝑞/𝑝(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑟+𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑚=𝜆+_𝑁𝑦0𝑝𝐾^𝑚/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝑣(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝑣(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.48) where 𝜆 is defined in (2.35). Then using (2.47) in (2.48) yields 𝜇≤𝜆+^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝜇𝐵³+𝑔₂(𝑠,𝑡)+𝐵₄(𝑠,𝑡)(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝜇𝐵³+(𝑠,𝑡)+𝐵₄(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝜇𝐵³=𝜆+𝜇_𝑁𝑦0(𝜉,𝜂)+𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵₃(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝐵₃(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}+𝐵₃(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵₄(𝑠,𝑡)+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝐵₄(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}=𝜆+𝜇𝐵₅𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡+𝐵₆,(2.49) which implies𝜇≤𝜆+𝐵⁶1−𝐵₅.(2.50) Combining (2.43), (2.47), and (2.50) we can obtain the desired inequality (2.34).

In the proof of Theorem 2.9, if we let the right side of (2.33) be 𝑎(𝑥,𝑦)+𝑣(𝑥,𝑦) in the first statement, then following in a same process as in Theorem 2.9 we obtain another bound on the function 𝑢(𝑥,𝑦), which is shown in the following theorem.

Theorem 2.10. Under the conditions of Theorem 2.9, if, for (𝑥,𝑦)∈([𝑥₀,𝑀]⋂𝕋)×([𝑦₀,𝑁]⋂𝕋), 𝑢(𝑥,𝑦) satisfies (2.33) with the initial condition (2.8), then the following inequality holds: 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝜇+𝐽¹(𝑥,𝑦)1−𝜆̃𝑒^𝐽2𝑦,𝑦₀^1/𝑝,(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋,(2.51) provided that 𝜆̃<1, where 𝜆̃=^𝑁𝑦₀^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑒_𝐽2𝑡,𝑦₀+𝑔₂(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+𝑒_𝐽2𝑡,𝑦₀Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝜇=_𝑁𝑦0𝑒_𝐽2𝜂,𝑦₀Δ𝜉Δ𝜂Δ𝑠Δ𝑡,^𝑀𝑥₀𝑓²(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞+𝑔₂𝑝𝐾^𝑞/𝑝(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑟+_𝑡𝑦0𝑝𝐾^𝑟/𝑝^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}+𝑝−𝑚(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))𝐽𝑝𝐾^𝑚/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡,₁(𝑥,𝑦)=^𝑦𝑦₀^𝑥𝑥₀𝑓¹(𝑠,𝑡)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞+𝑔₁𝑝𝐾^𝑞/𝑝(𝑠,𝑡)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑟+𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑚𝐽𝑝𝐾^𝑚/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡,₂(𝑥,𝑦)=^𝑥𝑥₀𝑓¹(𝑠,𝑦)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝑔₁(𝑠,𝑦)𝑟𝑝𝐾^{(𝑟−𝑝)/𝑝}+^𝑦𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑚𝑝𝐾^{(𝑚−𝑝)/𝑝}Δ𝜉Δ𝜂Δ𝑠.(2.52)

Finally, we will establish a more general inequality than that in Theorems 2.9-2.10. Consider the following inequality:𝑢^𝑝(𝑥,𝑦)≤+𝑎(𝑥,𝑦)_𝑦𝑦0^𝑥𝑥₀𝐿𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡)+^𝑡𝑦₀^𝑠𝑥₀+ℎ₁(𝜉,𝜂)𝑢^𝑞𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐿𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡)+^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑢^𝑞𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.53) with the initial condition (2.8), where 𝑢,𝑎,𝑝,𝑞,𝜙,𝛼,𝛽,𝜏_𝑖,ℎ_𝑖,𝑖=1,2 are the same as in Theorem 2.4, 𝑀∈𝕋₀,𝑁∈𝕋₀ are two fixed numbers. 𝐿∈(𝕋₀×𝕋₀×ℝ₊,ℝ₊), and 0≤𝐿(𝑠,𝑡,𝑥)−𝐿(𝑠,𝑡,𝑦)≤𝐴(𝑠,𝑡,𝑦)(𝑥−𝑦) for 𝑥≥𝑦≥0, where 𝐴∈(𝕋₀×𝕋₀×ℝ₊,ℝ₊).

Theorem 2.11. If, for (𝑥,𝑦)∈([𝑥₀,𝑀]⋂𝕋)×([𝑦₀,𝑁]⋂𝕋),𝑢(𝑥,𝑦) satisfies (2.53), then the following inequality holds: 𝑢(𝑥,𝑦)≤𝜆̂+𝐵⁶1−𝐵₅𝐵³(𝑥,𝑦)+𝐵₄(𝑥,𝑦)^1/𝑝,(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋,(2.54) provided that 𝐵₅<1, where 𝜆̂=^𝑁𝑦₀^𝑀𝑥₀𝐿𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑝−𝑞𝑝𝐾^𝑞/𝑝𝐵Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.55)₁(𝑥,𝑦)=+𝑎(𝑥,𝑦)_𝑦𝑦0^𝑥𝑥₀𝐿𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑝−𝑞𝑝𝐾^𝑞/𝑝𝐵Δ𝜉Δ𝜂Δ𝑠Δ𝑡,∀𝐾>0,(2.56)₂(𝑥,𝑦)=^𝑥𝑥₀𝐴𝑠,𝑦,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+^𝑦𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵Δ𝜉Δ𝜂Δ𝑠,∀𝐾>0,(2.57)₃𝐵(𝑥,𝑦)=1+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)Δ𝑡,(2.58)₄(𝑥,𝐵𝑦)=𝐵₁(𝑥,𝑦)+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡,(2.59)₅=^𝑁𝑦₀^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+𝐵₃(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵𝐵₃(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.60)₆=^𝑁𝑦₀^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+𝐵₄(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.61)

Proof. Let the right side of (2.53) be 𝑣(𝑥,𝑦) and 𝜇=^𝑁𝑦₀^𝑀𝑥₀𝐿𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡)+^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑢^𝑞𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.62) Then 𝑢(𝑥,𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋.(2.63) Similar to the process of (2.14)–(2.16) we have 𝑢𝜏₁(𝑥),𝜏₂(𝑦)≤𝑣^1/𝑝(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋.(2.64) Fix 𝑋∈[𝑥₀,𝑀]⋂𝕋, and let 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈[𝑦₀,𝑁]⋂𝕋. Then 𝑣(𝑋,𝑦)=+𝑎(𝑋,𝑦)+𝜇_𝑦𝑦0^𝑋𝑥₀𝐿𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡)+^𝑡𝑦₀^𝑠𝑥₀≤𝑎(𝑋,𝑦)+𝜇+_𝑦𝑦0ℎ₁(𝜉,𝜂)𝑢^𝑞𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝐿𝑠,𝑡,𝑣^1/𝑝(𝑠,𝑡)+^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑣^𝑞/𝑝(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(2.65) From Lemma 2.3, we have 𝑣^𝑞/𝑝(𝑥,𝑦)≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑥,𝑦)+𝑝−𝑞𝑝𝐾^𝑞/𝑝𝑣_1/𝑝,(𝑥,𝑦)≤1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑥,𝑦)+𝑝−1𝑝𝐾^1/𝑝,∀𝐾>0.(2.66) Combining (2.65) and (2.66), it follows that𝑣(𝑋,𝑦)≤𝑎(𝑋,𝑦)+𝜇+^𝑦𝑦₀^𝑋𝑥₀𝐿𝑠,𝑡,1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑞=+𝑎(𝑋,𝑦)+𝜇_𝑦𝑦0𝑝𝐾^𝑞/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝐿𝑠,𝑡,1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1+𝐿𝑠,𝑡,𝑝−1𝑝𝐾^1/𝑝−𝐿𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑞≤𝑎+(𝑋,𝑦)+𝜇_𝑦𝑦0𝑝𝐾^𝑞/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝐿𝑠,𝑡,𝑝−1+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+Δ𝜉Δ𝜂𝑣(𝑋,𝑡)Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑝−𝑞𝑝𝐾^𝑞/𝑝≤𝑎(𝑋,𝑦)+𝜇+_𝑦𝑦0Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑋𝑥₀𝐴𝑠,𝑡,𝑝−1𝑝𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+Δ𝑠𝑣(𝑋,𝑡)Δ𝑡_𝑦𝑦0^𝑋𝑥₀𝐿𝑠,𝑡,𝑝−1+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+Δ𝜉Δ𝜂Δ𝑠𝑣(𝑋,𝑡)Δ𝑡_𝑦𝑦0^𝑋𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑝−𝑞𝑝𝐾^𝑞/𝑝=𝜇+𝐵₁Δ𝜉Δ𝜂Δ𝑠Δ𝑡(𝑋,𝑦)+^𝑦𝑦₀𝐵₂(𝑋,𝑡)𝑣(𝑋,𝑡)Δ𝑡,(2.67) where 𝐵₁(𝑥,𝑦),𝐵₂(𝑥,𝑦) are defined in (2.56) and (2.57), respectively.
We notice the structure of (2.67) is similar to (2.19), so following in a same manner as in (2.19)–(2.21) we obtain𝑣(𝑥≤,𝑦𝜇)1+_𝑦𝑦0=𝜇𝐵₃𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)Δ𝑡+𝐵₁(𝑥,𝑦)+^𝑦𝑦₀𝑒𝐵₂(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡(𝑥,𝑦)+𝐵₄(𝑥,𝑦),(𝑥,𝑦)∈𝑥₀,𝑀𝕋×𝑦₀,𝑁𝕋,(2.68) where 𝐵₃(𝑥,𝑦),𝐵₄(𝑥,𝑦) are defined in (2.58) and (2.59), respectively.
On the other hand, from (2.62), (2.64), and (2.66) we have 𝜇≤^𝑁𝑦₀^𝑀𝑥₀𝐿𝑠,𝑡,𝑣^1/𝑝+^𝑡𝑦₀^𝑠𝑥₀≤ℎ₂(𝜉,𝜂)𝑣^𝑞/𝑝(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐿𝑠,𝑡,1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1+_𝑡𝑦0𝑝𝐾^1/𝑝^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑞=𝑝𝐾^𝑞/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐿𝑠,𝑡,1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1+𝐿𝑠,𝑡,𝑝−1𝑝𝐾^1/𝑝−𝐿𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑞≤𝑝𝐾^𝑞/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝐿𝑠,𝑡,𝑝−1+𝑝𝐾^1/𝑝Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)+𝑝−𝑞=𝜆̂+_𝑁𝑦0𝑝𝐾^𝑞/𝑝Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(2.69) where 𝜆̂ is defined in (2.55). Then using (2.68) in (2.69) yields𝜇≤𝜆̂+^𝑁𝑦₀^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−1𝑝𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}𝜇𝐵³+(𝑠,𝑡)+𝐵₄(𝑠,𝑡)Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝜇𝐵³=𝜆̂+𝜇_𝑁𝑦0(𝜉,𝜂)+𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−1𝑝𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+𝐵₃(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}+𝐵₃(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐴𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝1𝑝𝐾^{(1−𝑝)/𝑝}+𝐵₄(𝑠,𝑡)_𝑡𝑦0^𝑠𝑥₀ℎ₂(𝜉,𝜂)𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}=𝜆̂+𝜇𝐵₅𝐵₄(𝜉,𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡+𝐵₆,(2.70) which implies 𝜇≤𝜆̂+𝐵⁶1−𝐵₅.(2.71) Combining (2.63), (2.68), and (2.71), we obtain the desired inequality (2.54).

Remark 2.12. The established above inequalities generalize many known results including both integral inequalities for continuous functions and discrete inequalities. For example, if we take 𝕋=ℝ,𝑝=𝑞=1,𝑔(𝑥,𝑦)=ℎ(𝑥,𝑦)≡0, then Theorem 2.4 reduces to [1, Theorem 2.2], which is one case of integral inequality for continuous function. If we take 𝕋=ℝ,ℎ(𝑥,𝑦)≡0,𝑎(𝑥,𝑦)≡𝐶, then Corollary 2.7 reduces to [2, Theorem 3 (c1)], which is another case of inequality for continuous function. If we take 𝕋=ℝ,𝑝=𝑞=1,𝑔₁(𝑥,𝑦)=𝑔₂(𝑥,𝑦)=𝑓₂(𝑥,𝑦)=ℎ₁(𝑥,𝑦)=ℎ₂(𝑥,𝑦)≡0, then Theorem 2.10 reduces to [1, Theorem 2.2] with slight difference. If we take 𝕋=ℤ,𝑔₁(𝑥,𝑦)=ℎ₁(𝑥,𝑦)=𝑓₂(𝑥,𝑦)=ℎ₂(𝑥,𝑦)≡0,𝜏₁(𝑥)=𝑥,𝜏₂(𝑦)=𝑦, then Theorem 2.10 reduces to [3, Theorem 2.1], which is a discrete inequality.

Remark 2.13. Since 𝕋 is an arbitrary time scale, then if we take 𝕋 for some peculiar cases, such as 𝕋=ℝ or 𝕋=ℤ, we can deduce a series of corollaries according to Theorem 2.4–2.11. Due to the limited space, we omit them here.

3. Some Applications

In this section, we will present some applications for the results we have established previously. New explicit bounds for solutions of certain dynamic equations are derived in the first two examples, while the quantitative property of solutions is focused on in the final example.

Example 3.1. Consider the following delay dynamic differential equation: (𝑢^𝑝(𝑥,𝑦))^ΔΔ𝑦𝑥=𝐹𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂,(𝑥,𝑦)∈𝕋₀×𝕋₀,(3.1) with the initial condition 𝑢(𝑥,𝑦)=𝑢𝜙^𝑝(𝑥𝑥,₀𝑦,𝑦),^Δ𝑦if=𝑥𝑏∈^Δ(𝑦𝛼),,𝑥₀𝑢^𝑝𝑥,𝑦₀=𝑎(𝑥),||𝜙𝜏𝕋,or𝑦∈𝛽,𝑦₀𝕋,₁(𝑥),𝜏₂(𝑦)||≤||𝑘(𝑥,𝑦)||^1/𝑝,if𝜏₁(𝑥)≤𝑥₀,or𝜏₂(𝑦)≤𝑦₀,∀(𝑥,𝑦)∈𝕋₀×𝕋₀,(3.2) where 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ), 𝑎∈𝐶_rd(𝕋₀,ℝ), 𝑏∈𝐶_rd(𝕋₀,ℝ), 𝑏 is delta differential, and 𝑏(𝑦₀)=0,𝑘∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊), 𝑝>0 is a constant, 𝜙∈𝐶_rd(([𝛼,𝑥₀]×[𝛽,𝑦₀])⋂𝕋²,ℝ),𝐹∈(𝕋₀×𝕋₀×ℝ²,ℝ),𝑊∈(𝕋₀×𝕋₀×ℝ,ℝ). 𝛼,𝛽,𝜏₁,𝜏₂ are the same as in Theorem 2.4.

Theorem 3.2. Suppose that 𝑢(𝑥,𝑦) is a solution of (3.1)-(3.2), |𝑎(𝑥)+𝑏(𝑦)|≤𝑘(𝑥,𝑦), and |𝐹(𝑠,𝑡,𝑥,𝑦)|≤𝑓(𝑠,𝑡)|𝑥|^𝑞+|𝑦|, |𝑊(𝜉,𝜂,𝑥)|≤ℎ(𝜉,𝜂)|𝑥|^𝑚, where 𝑓,ℎ,𝑞,𝑚 are defined as in Theorem 2.4; then ||𝑢(𝑥,𝑦)||≤𝐵₁(𝑥,𝑦)+^𝑦𝑦₀𝑒_𝐵2(𝑦,𝜎(𝑡))𝐵₂(𝑥,𝑡)𝐵₁(𝑥,𝑡)Δ𝑡^1/𝑝,(𝑥,𝑦)∈𝕋₀×𝕋₀,(3.3) where 𝐵₁(𝑥,𝑦)=𝑘(𝑥,𝑦)+^𝑦𝑦₀^𝑥𝑥₀𝑓(𝑠,𝑡)𝑝−𝑝𝑞𝐾^𝑞/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)𝑝−𝑚𝑝𝐾^𝑚/𝑝∀𝐾>Δ𝜉0Δ,𝜂Δ𝑠Δ𝑡,(3.4) and 𝐵₂(𝑥,𝑦) is defined as in Theorem 2.4 (with 𝑔(𝑥,𝑦)≡0).

Proof. The equivalent integral equation of (3.1) can be denoted by𝑢^𝑝(𝑥,𝑦)=+𝑎(𝑥)+𝑏(𝑦)_𝑦𝑦0^𝑥𝑥₀𝐹𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(3.5) Then ||𝑢^𝑝≤(𝑥𝑘,𝑦(𝑥)||,𝑦)+_𝑦𝑦0^𝑥𝑥₀||||𝐹𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀≤+𝑘(𝑥,𝑦)_𝑦𝑦0𝑊𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡^𝑥𝑥₀𝑓(𝑠,𝑡)||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||^𝑞+||||^𝑡𝑦₀^𝑠𝑥₀≤𝑘(𝑥,𝑦)+_𝑦𝑦0𝑊𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡^𝑥𝑥₀𝑓(𝑠,𝑡)||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||^𝑞+^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)||𝑢𝜏₁(𝜉),𝜏₂(𝜂)||^𝑚Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(3.6) and a suitable application of Theorem 2.4 to (3.6) yields the desired inequality (3.3).

Theorem 3.3. Under the conditions of Theorem 3.2, one has ||𝑢(𝑥,𝑦)||≤𝐵₁(𝑥,𝑦)𝑒_𝐵2𝑦,𝑦₀^1/𝑝,(𝑥,𝑦)∈𝕋₀×𝕋₀,(3.7) where 𝐵₁,𝐵₂ are defined as in Theorem 3.2.

Proof. The desired inequality can be obtained by an application of Theorem 2.5 to (3.6).

Example 3.4. Consider the following delay dynamic integral equation: 𝑢^𝑝(𝑥,𝑦)=𝐶+^𝑦𝑦₀^𝑥𝑥₀𝐹₁𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀+𝑊₁𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐹₂𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀(𝑥,𝑦)∈𝑊𝑥₂₀𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,,𝑀𝕋×𝑦₀,𝑁𝕋,(3.8) with the initial condition ||𝜙𝜏𝑢(𝑥,𝑦)=𝜙(𝑥,𝑦),if𝑥∈𝛼,𝑥₀𝕋,or𝑦∈𝛽,𝑦₀𝕋,₁(𝑥),𝜏₂(𝑦)||≤||𝐶||^1/𝑝,if𝜏₁(𝑥)≤𝑥₀,or𝜏₂(𝑦)≤𝑦₀,∀(𝑥,𝑦)∈𝕋₀×𝕋₀,(3.9) where 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ), 𝑝>0 is a constant, 𝐶=𝑢^𝑝(𝑥₀,𝑦₀), 𝑀∈𝕋₀,𝑁∈𝕋₀ are two fixed numbers. 𝜙∈𝐶_rd(([𝛼,𝑥₀]×[𝛽,𝑦₀])⋂𝕋²,ℝ),𝐹_𝑖∈(𝕋₀×𝕋₀×ℝ²,ℝ),𝑊_𝑖∈(𝕋₀×𝕋₀×ℝ,ℝ),𝑖=1,2. 𝛼,𝛽,𝜏₁,𝜏₂ are the same as in Theorem 2.4.

Theorem 3.5. Suppose 𝑢(𝑥,𝑦) is a solution of (3.8)-(3.9) and |𝐹_𝑖(𝑠,𝑡,𝑥,𝑦)|≤𝐿(𝑠,𝑡,|𝑥|)+|𝑦|,|𝑊_𝑖(𝜉,𝜂,𝑥)|≤ℎ_𝑖(𝜉,𝜂)|𝑥|^𝑞,𝑖=1,2, where 𝐿,ℎ_𝑖,𝑖=1,2,𝑞 are defined the same as in Theorem 2.11; then the following inequality holds: ||𝑢(𝑥,𝑦)||≤𝜆̂+𝐵⁶1−𝐵₅𝐵³(𝑥,𝑦)+𝐵₄(𝑥,𝑦)^1/𝑝,(𝑥,𝑦)∈x₀,𝑀𝕋×𝑦₀,𝑁𝕋,(3.10) provided that 𝐵₅<1, where 𝜆̂,𝐵₂(𝑥,𝑦),𝐵₃(𝑥,𝑦),𝐵₄(𝑥,𝑦),𝐵₅,𝐵₆ are defined the same as in Theorem 2.11, and 𝐵₁(𝑥,𝑦)=||𝐶||+^𝑦𝑦₀^𝑥𝑥₀𝐿𝑠,𝑡,𝑝−𝑝1𝐾^1/𝑝+^𝑡𝑦₀^𝑠𝑥₀ℎ₁(𝜉,𝜂)𝑝−𝑞𝑝𝐾^𝑞/𝑝∀𝐾>Δ0𝜉.Δ𝜂Δ𝑠Δ𝑡,(3.11)

Proof. From (3.8) we have ||𝑢^𝑝≤(𝑥+||,𝐶𝑦||)||_𝑦𝑦0^𝑥𝑥₀||||𝐹₁𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀+𝑊₁𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀||||𝐹₂𝑠,𝑡,𝑢𝜏¹(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀≤+||𝐶||𝑊₂𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀𝐿𝑠,𝑡,||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||+||||^𝑡𝑦₀^𝑠𝑥₀+𝑊₁𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐿𝑠,𝑡,||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||+||||^𝑡𝑦₀^𝑠𝑥₀≤+||𝐶||𝑊₂𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀𝐿𝑠,𝑡,||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||+^𝑡𝑦₀^𝑠𝑥₀+ℎ₁(𝜉,𝜂)||𝑢𝜏₁(𝜉),𝜏₂(𝜂)||^𝑞Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑁𝑦0^𝑀𝑥₀𝐿𝑠,𝑡,||𝑢𝜏¹(𝑠),𝜏₂(𝑡)||+^𝑡𝑦₀^𝑠𝑥₀ℎ₂(𝜉,𝜂)||𝑢𝜏₁(𝜉),𝜏₂(𝜂)||^𝑞Δ𝜉Δ𝜂Δ𝑠Δ𝑡.(3.12) So by use of Theorem 2.11 we obtain the desired inequality (3.10).

Example 3.6. Consider the following delay dynamic integral equation: 𝑢=(𝑥𝐶,𝑦+)_𝑦𝑦0^𝑥𝑥₀𝐹𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡,(3.13) where 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ), 𝐶=𝑢^𝑝(𝑥₀,𝑦₀),𝐹∈(𝕋₀×𝕋₀×ℝ²,ℝ),𝑊∈(𝕋₀×𝕋₀×ℝ,ℝ). 𝜏₁,𝜏₂ are the same as in Theorem 2.4.

Theorem 3.7. Assume that |𝐹(𝑠,𝑡,𝑢₁,𝑣₁)−𝐹(𝑠,𝑡,𝑢₂,𝑣₂)|≤𝑓(𝑠,𝑡)|𝑢₁−𝑢₂|+|𝑣₁−𝑣₂|,|𝑊(𝑠,𝑡,𝑢₁)−𝑊(𝑠,𝑡,𝑢₂)|≤ℎ(𝑠,𝑡)|𝑢₁−𝑢₂|, where 𝑓,ℎ are defined as in Theorem 2.4, and; furthermore, assume that 𝜏₁(𝑥)≥𝑥₀,𝜏₂(𝑦)≥𝑦₀, then (3.13) has at most one solution.

Proof. Suppose that 𝑢₁(𝑥,𝑦),𝑢₂(𝑥,𝑦) are two solutions of (3.13); then we have ||≤𝑢₁(||||𝑥,𝑦)−𝑢₂(𝑥,𝑦)_𝑦𝑦0||^𝑥𝑥₀𝐹𝑠,𝑡,𝑢¹𝜏₁(𝑠),𝜏₂(𝑡)^𝑡𝑦₀^𝑠𝑥₀−𝐹𝑠,𝑡,𝑢₂𝑊𝜉,𝜂,𝑢₁𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂≤𝜏₁(𝑠),𝜏₂(𝑡)^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢₂𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑦𝑦0||||^𝑥𝑥₀||||𝐹𝑠,𝑡,𝑢₁−𝐹𝑠,𝑡,𝑢₂𝜏₁(𝑠),𝜏₂(𝑡)^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢₁𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂≤𝜏₁(𝑠),𝜏₂(𝑡)^𝑡𝑦₀^𝑠𝑥₀𝑊𝜉,𝜂,𝑢₂𝜏₁(𝜉),𝜏₂(𝜂)Δ𝜉Δ𝜂||||Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀+𝑓(𝑠,𝑡)||𝑢₁𝜏₁(𝑠),𝜏₂(𝑡)−𝑢₂𝜏₁(𝑠),𝜏₂(𝑡)||Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀≤||𝑊𝜉,𝜂,𝑢₁𝜏₁(𝜉),𝜏₂(𝜂)−𝑊𝜉,𝜂,𝑢₂𝜏₁(𝜉),𝜏₂(𝜂)||Δ𝜉Δ𝜂Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀+𝑓(𝑠,𝑡)||𝑢₁𝜏₁(𝑠),𝜏₂(𝑡)−𝑢₂𝜏₁(𝑠),𝜏₂(𝑡)||Δ𝑠Δ𝑡_𝑦𝑦0^𝑥𝑥₀^𝑡𝑦₀^𝑠𝑥₀ℎ(𝜉,𝜂)||𝑢₁𝜏₁(𝜉),𝜏₂(𝜂)−𝑢₂𝜏₁(𝜉),𝜏₂(𝜂)||Δ𝜉Δ𝜂Δ𝑠Δ𝑡,.(3.14) Then a suitable application of Theorem 2.8 yields |𝑢₁(𝑥,𝑦)−𝑢₂(𝑥,𝑦)|≤0, that is, 𝑢₁(𝑥,𝑦)≡𝑢₂(𝑥,𝑦), and the proof is complete.

4. Conclusions

In this paper, we established some new Gronwall-Bellman type integral inequalities on time scales. As one can see, the presented results provide a handy tool in the quantitative as well as qualitative analysis of solutions of certain delay dynamic equations on time scales. The established inequalities unify some known continuous and discrete inequalities.

Acknowledgments

This work is supported by Natural Science Foundation of Shandong Province (ZR2009AM011) China and Specialized Research Fund for the Doctoral Program of Higher Education (20103705110003) China. The authors thank the referees very much for their careful comments and valuable suggestions on this paper.