Some New 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 can be used as a handy tool in the research of boundedness of solutions of delay dynamic equations on time scales. Some of the established results are 2D extensions of several known results in the literature, while some results unify existing continuous and discrete analysis.

1. Introduction

In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations, which provide explicit bounds, play a fundamental role in the research of this domain. During the past decades, much effort has been done for developing such inequalities (e.g., see [3–15] and the references therein). On the other hand, Hilger [16] 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 (e.g., see [17–19] and the references therein). In these investigations, integral inequalities on time scales have been paid much attention by many authors, which play a fundamental role in the research of quantitative as well as qualitative properties of solutions of certain dynamic equations on time scales. A lot of integral inequalities on time scales have been established (e.g., see [20–26]), which have been designed to unify continuous and discrete analysis. But to our best knowledge, the 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 work of Li [27] and that of Ma and Pečarić [28]. Furthermore, nobody has studied the Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.

The aim of this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which provide new bounds for the unknown functions concerned. Some of our results are 2D extensions of many known inequalities in the literature, while some results unify existing continuous and discrete analysis. For illustrating the validity of the established results, we will present some applications of them.

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

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).

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 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 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. (1.1)
Similarly, for some 𝑦∈𝕋^𝜅 and a function 𝑓∈(𝕋×𝕋,ℝ), the partial delta of 𝑓 with respect to 𝑦 is denoted by (𝑓(𝑥,𝑦))^Δ_𝑦 or 𝑓^Δ_𝑦(𝑥,𝑦) and satisfies 𝑓𝑥,𝜎𝑦−𝑓(𝑥,𝑠)−𝑓^Δ_𝑦𝑥,𝑦𝜎𝑦−𝑠≤𝜀𝜎𝑦−𝑠∀𝜀>0, (1.2) where 𝑠∈𝔘 and 𝔘 is a neighborhood of 𝑦.

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.

For more details about the calculus of time scales, see [29]. In the rest of this paper, for the convenience of notation, we always assume that 𝕋₀=[𝑥₀,∞)⋂𝕋,𝕋₀=[𝑦₀,∞)⋂𝕋, where 𝑥₀,𝑦₀∈𝕋^𝜅 and furthermore assume 𝕋₀⊆𝕋^𝜅,𝕋₀⊆𝕋^𝜅.

2. Main Results

We will give some lemmas for further use.

Lemma 2.1. Suppose 𝑋∈𝕋₀is a fixed number, and 𝑢(𝑋,𝑦),𝑎(𝑋,𝑦),𝑏(𝑋,𝑦)∈𝐶_rd,𝑚(𝑋,𝑦)∈ℜ₊ with respect to 𝑦, 𝑚(𝑋,𝑦)≥0, then 𝑢𝑋,𝑦≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑚(𝑋,𝑡)𝑢(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀, (2.1) implies 𝑢𝑋,𝑦≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒_𝑚𝑦,𝜎(𝑡)𝑎(𝑋,𝑡)𝑚(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀, (2.2) where 𝑚(𝑋,𝑦)=𝑚(𝑋,𝑦)𝑏(𝑋,𝑦): and 𝑒_𝑚(𝑦,𝑦₀) is the unique solution of the following equation 𝑧^Δ_𝑦𝑋,𝑦=𝑚𝑋,𝑦𝑧𝑋,𝑦,𝑧𝑋,𝑦₀=1. (2.3)

The proof of Lemma 2.1 is similar to [26, Theorem 5.6].

Lemma 2.2. Under the conditions of Lemma 2.1, and furthermore assuming 𝑎(𝑥,𝑦) is nondecreasing in 𝑦 for every fixed 𝑥, 𝑏(𝑥,𝑦)≡1, then one has 𝑢𝑋,𝑦≤𝑎𝑋,𝑦𝑒_𝑚𝑦,𝑦₀. (2.4)

Proof. Since 𝑎(𝑥,𝑦) is nondecreasing in 𝑦 for every fixed 𝑥, then from Lemma 2.1 we have 𝑢𝑋,𝑦≤𝑎𝑋,𝑦+^𝑦_𝑦0𝑒_𝑚𝑦,𝜎(𝑡)𝑎(𝑋,𝑡)𝑚(𝑋,𝑡)Δ𝑡≤𝑎𝑋,𝑦1+^𝑦_𝑦0𝑒_𝑚𝑦,𝜎(𝑡)𝑚(𝑋,𝑡)Δ𝑡. (2.5) On the other hand, from [29, Theorems 2.39 and 2.36 (i)] we have 1+∫^𝑦_𝑦0𝑒_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 [11]). Assume that 𝑎≥0,𝑝≥𝑞≥0, and 𝑝≠0; then, for any𝐾>0𝑎^𝑞/𝑝≤𝑞𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑎+𝑝−𝑞𝑝𝐾^𝑞/𝑝. (2.6)

Lemma 2.4. Let ℎ∶𝕋×ℝ→ℝ be continuous and nondecreasing in the second variable, and assume 𝑋 is a fixed number in 𝕋. Suppose 𝑣(𝑋,𝑦) and 𝑤(𝑋,𝑦) satisfy the dynamics inequalities: 𝑣^Δ_𝑦≤ℎ𝑦,𝑣,𝑤^Δ_𝑦≥ℎ𝑦,𝑤. (2.7) Then 𝑣(𝑋,𝑦₀)≤𝑤(𝑋,𝑦₀) for some 𝑦₀∈𝕋 implies 𝑣(𝑋,𝑦)≤𝑤(𝑋,𝑦) for all 𝑦∈𝕋.

The proof of Lemma 2.4 is similar to [26, Theorem 5.7].

Theorem 2.5. Suppose 𝑢,𝑎,𝑏,𝑓∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊), and 𝑎(𝑥,𝑦),𝑏(𝑥,𝑦) are nondecreasing. 𝑝 is a constant, and 𝑝≥1. 𝜏₁∈(𝕋₀,𝕋),𝜏₁(𝑥)≤𝑥,−∞<𝛼=inf{𝜏₁(𝑥),𝑥∈𝕋₀}≤𝑥₀. 𝜏₂∈(𝕋₀,𝕋),𝜏₂(𝑦)≤𝑦,−∞<𝛽=inf{𝜏₂(𝑦),𝑦∈𝕋₀}≤𝑦₀. 𝜙∈𝐶_𝑟𝑑(([𝛼,𝑥₀]×[𝛽,𝑦₀])⋂𝕋²,ℝ₊). If for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝𝑥,𝑦≤𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡, (2.8) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤𝑎^1/𝑝𝑥,𝑦,∀𝑥,𝑦∈𝕋₀×𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (2.9) then 𝑢𝑥,𝑦≤𝐻₁𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0𝑒_𝐻2𝑦,𝜎(𝑡)𝐻₂(𝑥,𝑡)𝐻₁(𝑥,𝑡)Δ𝑡^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀, (2.10) where 𝐻₁𝑥,𝑦=𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑝−1𝑝𝐾^1/𝑝𝐻Δ𝑠Δ𝑡,∀𝐾>0,₂𝑥,𝑦=^𝑥_𝑥0𝑓1𝑠,𝑦𝑝𝐾^{(1−𝑝)/𝑝}Δ𝑠,𝐻₂𝑥,𝑦=𝑏𝑥,𝑦𝐻₂𝑥,𝑦. (2.11)

Proof. Fix 𝑋∈𝕋₀, and 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈𝕋₀. Let 𝑣𝑥,𝑦=𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡. (2.12) Then 𝑢𝑥,𝑦≤𝑣^1/𝑝𝑥,𝑦≤𝑣^1/p𝑋,𝑦,∀𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.13) If 𝜏₁(𝑥)≥𝑥₀ and 𝜏₂(𝑦)≥𝑦₀, then 𝜏₁(𝑥)∈[𝑥₀,𝑋]⋂𝕋,𝜏₂(𝑦)∈𝕋₀, and 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣^1/𝑝𝜏₁(𝑥),𝜏₂𝑦≤𝑣^1/𝑝𝑥,𝑦. (2.14) If 𝜏₁(𝑥)≤𝑥₀ or 𝜏₂(𝑦)≤𝑦₀, then from (2.9) we have 𝑢𝜏₁(𝑥),𝜏₂𝑦=𝜙𝜏₁(𝑥),𝜏₂𝑦≤𝑎^1/𝑝𝑥,𝑦≤𝑣^1/𝑝𝑥,𝑦. (2.15) From (2.14) and (2.15) we always have 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣^1/𝑝𝑥,𝑦,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.16) Moreover 𝑣𝑋,𝑦=𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑣^1/𝑝(𝑠,𝑡)Δ𝑠Δ𝑡. (2.17) From Lemma 2.3, we have 𝑣^1/𝑝()≤1𝑠,𝑡𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1𝑝𝐾^1/𝑝,∀𝐾>0. (2.18) So 𝑣𝑋,𝑦≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(1𝑠,𝑡)𝑝𝐾^{(1−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−1𝑝𝐾^1/𝑝Δ𝑠Δ𝑡≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑝−1𝑝𝐾^1/𝑝Δ𝑠Δ𝑡+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓()1𝑠,𝑡𝑝𝐾^{(1−𝑝)/𝑝}Δ𝑠𝑣(𝑋,𝑡)Δ𝑡=𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝐻₂(𝑋,𝑡)𝑣(𝑋,𝑡)Δ𝑡. (2.19) Then applying Lemma 2.1 to (2.19), we obtain 𝑣𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒_𝐻2𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡. (2.20) So 𝑢𝑥,𝑦≤𝑣^1/𝑝𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,y^𝑦_𝑦0𝑒_𝐻2𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡^1/𝑝,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.21) Setting 𝑥=𝑋 in (2.21), it follows that 𝑢𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒_𝐻2𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡^1/𝑝. (2.22) Replacing 𝑋 with 𝑥 in (2.22), we obtain the desired inequality.

Remark 2.6. Theorem 2.5 is the 2D extension of [27, Theorem 1]. For its special case 𝕋=ℝ, the established bound for 𝑢(𝑥,𝑦) in (2.10) is a new bound compared with the result in [12, Theorem 2.2].

Remark 2.7. Assume 𝑏(𝑥,𝑦)≡1 in Theorem 2.5. If we apply Lemma 2.2 instead of Lemma 2.1 to (2.19) in the proof of Theorem 2.5, then we obtain another bound for 𝑢(𝑥,𝑦) as follows: 𝑢𝑥,𝑦≤𝐻₁𝑥,𝑦𝑒_𝐻2𝑦,𝑦₀^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀. (2.23)

Now we will establish a more general inequality than that in Theorem 2.5.

Theorem 2.8. Suppose 𝑢,𝑎,𝑏,𝑓,𝜙,𝜏₁,𝜏₂,𝛼,𝛽 are the same as in Theorem 2.5, and 𝑔∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊). 𝑝,𝑞,𝑟 are constants, and 𝑝≥𝑞≥0, 𝑝≥𝑟≥0, 𝜌≠0. If for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝𝑥,𝑦≤𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡, (2.24)

with the initial condition (2.9), then 𝑢𝑥,𝑦≤𝐻₁𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0𝑒𝐻₂𝑦,𝜎(𝑡)𝐻₂(𝑥,𝑡)𝐻₁(𝑥,𝑡)Δ𝑡^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀, (2.25) where𝐻₁𝑥,𝑦=𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻Δ𝑠Δ𝑡,₂𝑥,𝑦=^𝑥_𝑥0𝑓𝑞𝑠,𝑦𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑟+g𝑠,𝑦𝑝𝐾^{(𝑟−𝑝)/𝑝}Δ𝑠,𝐻₂𝑥,𝑦=𝑏𝑥,𝑦𝐻₂𝑥,𝑦. (2.26)

Proof. Fix 𝑋∈𝕋₀, and 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈𝕋₀. Let 𝑣𝑥,𝑦=𝑎𝑥,𝑦+𝑏𝑥,𝑦^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡. (2.27) Then 𝑢𝑥,𝑦≤𝑣^1/𝑝𝑥,𝑦≤𝑣^1/𝑝𝑋,𝑦,∀𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.28) Similar to (2.14)–(2.16), we obtain 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣^1/𝑝𝑥,𝑦,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.29) So 𝑣𝑋,𝑦=𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑢^𝑞𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(s,𝑡)𝑢^𝑟𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑣^𝑞/𝑝(𝑠,𝑡)+𝑔(𝑠,𝑡)𝑣^𝑟/𝑝(𝑠,𝑡)Δ𝑠Δ𝑡. (2.30) From Lemma 2.3, we have 𝑣^𝑞/𝑝()≤𝑞𝑠,𝑡𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑞𝑝𝐾^𝑞/𝑝𝑣,∀𝐾>0,_𝑟/𝑝()≤𝑟𝑠,𝑡𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑟𝑝𝐾^𝑟/𝑝,∀𝐾>0. (2.31) Combining (2.30) and (2.31) we get that 𝑣𝑋,𝑦≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓()𝑞𝑠,𝑡𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑞𝑝𝐾^𝑞/𝑝()𝑟+𝑔𝑠,𝑡𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣(𝑠,𝑡)+𝑝−𝑟𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡≤𝑎𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟𝑝𝐾^𝑟/𝑝Δ𝑠Δ𝑡+𝑏𝑋,𝑦^𝑦_𝑦0^𝑋_𝑥0𝑓()𝑞𝑠,𝑡𝑝𝐾^{(𝑞−𝑝)/𝑝}()𝑟+𝑔𝑠,𝑡𝑝𝐾^{(𝑟−𝑝)/𝑝}𝑣()=𝐻Δ𝑠𝑋,𝑡Δ𝑡₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝐻₂(𝑋,𝑡)𝑣(𝑋,𝑡)Δ𝑡. (2.32) Applying Lemma 2.1 to (2.32) yields 𝑣𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒𝐻₂𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡. (2.33) Then 𝑢𝑥,𝑦≤𝑣^1/𝑝𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒𝐻₂𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡^1/𝑝,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.34) Setting 𝑥=𝑋 in (2.34) yields 𝑢𝑋,𝑦≤𝐻₁𝑋,𝑦+𝑏𝑋,𝑦^𝑦_𝑦0𝑒𝐻₂𝑦,𝜎(𝑡)𝐻₂(𝑋,𝑡)𝐻₁(𝑋,𝑡)Δ𝑡^1/𝑝. (2.35) Considering 𝑋∈𝕋₀ is arbitrary and replacing 𝑋 with 𝑥 in (2.35), we obtain the desired inequality.

Remark 2.9. Assume 𝑏(𝑥,𝑦)≡1 in Theorem 2.8. If we apply Lemma 2.2 instead of Lemma 2.1 to (2.32) in the proof of Theorem 2.8, then we obtain another bound for 𝑢(𝑥,𝑦) as follows: 𝑢𝑥,𝑦≤𝐻₁𝑒𝐻𝑥,𝑦₂𝑦,𝑦₀^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀. (2.36)

Remark 2.10. Theorem 2.8 is the 2D extension of [27, Theorem 3].

Theorem 2.11. Suppose 𝑢,𝑓,𝛼,𝛽,𝜙,𝜏₁,𝜏₂ are the same as in Theorem 2.5, and 𝐶>0 is a constant. If for (𝑥,𝑦)∈(𝕋₀×𝕋₀), 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢²𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)+𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)Δ𝑠Δ𝑡 (2.37) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤𝐶^1/2,∀𝑥,𝑦∈𝕋₀×𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (2.38) then 𝑢𝑥,𝑦≤√𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡,𝑥,𝑦∈𝕋₀×𝕋₀. (2.39)

Proof. Let the right side of (2.37) be 𝑣²(𝑥,𝑦). Then 𝑢𝑥,𝑦≤𝑣𝑥,𝑦,∀𝑥,𝑦∈𝕋₀×𝕋₀. (2.40) For (𝑥,𝑦)∈(𝕋₀×𝕋₀), if 𝜏₁(𝑥)≥𝑥₀and𝜏₂(𝑦)≥𝑦₀, then 𝜏₁(𝑥)∈𝕋₀and𝜏₂(𝑦)∈𝕋₀, and from (2.40) we have 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣𝜏₁(𝑥),𝜏₂𝑦≤𝑣𝑥,𝑦. (2.41) If 𝜏₁(𝑥)≤𝑥₀ or 𝜏₂(𝑦)≤𝑦₀, from (2.38) we have 𝑢𝜏₁(𝑥),𝜏₂𝑦=𝜙𝜏₁(𝑥),𝜏₂𝑦≤𝑎^1/2𝜏₁(𝑥),𝜏₂𝑦≤𝑎^1/2𝑥,𝑦≤𝑣𝑥,𝑦. (2.42) So from (2.41) and (2.42), we always have 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣𝑥,𝑦,∀𝑥,𝑦∈𝕋₀×𝕋₀. (2.43) Similarly, when 𝜏₁(𝑥)≥𝑥₀ and 𝜎(𝜏₂(𝑦))≥𝑦₀, then 𝜏₁(𝑥)∈𝕋₀and𝜎(𝜏₂(𝑦))∈𝕋₀, and from (2.40) we have 𝑢𝜏₁(𝑥),𝜎𝜏₂𝑦≤𝑣𝜏₁(𝑥),𝜎𝜏₂𝑦≤𝑣𝑥,𝜎𝑦. (2.44) When 𝜏₁(𝑥)≤𝑥₀ or 𝜎(𝜏₂(𝑦))≤𝑦₀, considering 𝜎(𝜏₂(𝑦))≥𝜏₂(𝑦)≥𝛽, from (2.38) it follows that 𝑢𝜏₁(𝑥),𝜎𝜏₂𝑦=𝜙𝜏₁(𝑥),𝜎𝜏₂𝑦≤𝐶^1/2≤𝑣𝑥,𝑦≤𝑣𝑥,𝜎𝑦. (2.45) Combining (2.44) and (2.45), we always have 𝑢𝜏₁(𝑥),𝜎𝜏₂𝑦≤𝑣𝑥,𝜎𝑦,∀𝑥,𝑦∈𝕋₀×𝕋₀. (2.46) By (2.43) and (2.46), we obtain 𝑣²𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)[𝑣(𝑠,𝑡)+𝑣(𝑠,𝜎(𝑡))]Δ𝑠Δ𝑡,𝑥∈𝕋₀,𝑦∈𝕋₀. (2.47) Let the right side of (2.47) be 𝑧²(𝑥,𝑦). Then 𝑣𝑥,𝑦≤𝑧𝑥,𝑦,∀𝑥,𝑦∈𝕋₀×𝕋₀𝑧₂,𝑥,𝑦^Δ_𝑦=^𝑥_𝑥0𝑓𝑣𝑦≤𝑠,𝑦𝑠,𝑦+𝑣𝑠,𝜎Δ𝑠_𝑥𝑥0𝑓𝑣𝑦≤𝑠,𝑦Δ𝑠𝑥,𝑦+𝑣𝑥,𝜎_𝑥𝑥0𝑓𝑠,𝑦Δ𝑠𝑧𝑥,𝑦+𝑧𝑥,𝜎𝑦. (2.48)(2.49) Considering 𝑧(𝑥,𝑦)+𝑧(𝑥,𝜎(𝑦))≥𝑧(𝑥₀,𝑦₀)=𝐶>0, and (𝑧²(𝑥,𝑦))^Δ_𝑦=[𝑧(𝑥,𝑦)+𝑧(𝑥,𝜎(𝑦))](𝑧(𝑥,𝑦))^Δ_𝑦, from (2.49) it follows that 𝑧𝑥,𝑦^Δ_𝑦≤^𝑥_𝑥0𝑓𝑠,𝑦Δ𝑠. (2.50) An integration of (2.50) with respect to 𝑦 from 𝑦₀ to 𝑦 yields 𝑧(𝑥,𝑦)−𝑧(𝑥,𝑦₀)≤∫^𝑦_𝑦0∫^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡.
Considering 𝑧(𝑥,𝑦₀)=√𝐶, it follows that 𝑧𝑥,𝑦≤√𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡. (2.51) Then combining (2.40), (2.48), and (2.51), we obtain 𝑢𝑥,𝑦≤𝑣𝑥,𝑦≤𝑧𝑥,𝑦≤√𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡, (2.52) and the proof is complete.

Remark 2.12. If we take 𝕋=ℝ, then Theorem 2.11 becomes the extension of the known Ou-Iang's inequality [13] to the 2D case.

The following theorem provides a more general result than Theorem 2.11.

Theorem 2.13. Suppose 𝑝 is a positive integer, and 𝑝≥2. Under the conditions of Theorem 2.11, if 𝑢(𝑥,𝑦) satisfies 𝑢^𝑝𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)^𝑝−1_𝑙=0𝑢^𝑙𝜏₁(𝑠),𝜏₂(𝑡)𝑢^{𝑝−1−𝑙}𝜏₁(𝑠),𝜎𝜏₂(𝑡)∈𝕋Δ𝑠Δ𝑡,𝑥,𝑦₀×𝕋₀, (2.53) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤𝐶^1/𝑝,∀𝑥,𝑦∈𝕋₀×𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (2.54) then 𝑢𝑥,𝑦≤𝐶^1/𝑝+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡,𝑥∈𝕋₀,𝑦∈𝕋₀. (2.55)

The proof of Theorem 2.13 is similar to Theorem 2.11. As long as we notice a delta d ifferentiable function 𝑧(𝑥,𝑦), the following formula [26, Equation (6.2)] holds: 𝑧^𝑝𝑥,𝑦^Δ_𝑦=𝑧𝑥,𝑦^Δ_𝑦^𝑝−1_𝑙=0𝑧^𝑙𝑥,𝑦𝑧^{𝑝−1−𝑙}𝑥,𝜎𝑦. (2.56) Then following a similar manner as in Theorem 2.11, we can deduce the desired result.

Theorem 2.14. Suppose 𝑢,𝑓,𝜏₁,𝜏₂,𝜙,𝛼,𝛽 are the same as in Theorem 2.5, 𝜔∈𝐶(ℝ₊,ℝ₊) is nondecreasing, and 𝑝,𝐶 are constants with 𝑝≥1,𝐶>0. Furthermore, define a bijective function 𝐺 such that [𝐺(𝑧(𝑥,𝑦))]^Δ_𝑦=(𝑧(𝑥,𝑦))^Δ_𝑦/𝜔(𝑧^1/𝑝(𝑥,𝑦)). If for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝜔𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡, (2.57) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤𝐶^1/𝑝,∀𝑥,𝑦∈𝕋₀×𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (2.58) then 𝑢𝑥,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₁(𝑥,𝑡)Δ𝑡^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀, (2.59) where 𝜂₁(𝑥,𝑦)=∫^𝑥_𝑥0𝑓(𝑠,𝑦)Δ𝑠.

Proof. Fix 𝑋∈𝕋₀, and 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈𝕋₀. Let 𝑣𝑥,𝑦=𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.60) Then 𝑢𝑥,𝑦≤𝑣^1/𝑝𝑥,𝑦≤𝑣^1/𝑝𝑋,𝑦,∀𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.61) Similar to (2.14)–(2.16), we obtain 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣^1/𝑝𝑥,𝑦,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.62) Moreover, 𝑣^Δ_𝑦𝑋,𝑦=^𝑋_𝑥0𝑓𝑠,𝑦𝜔𝑢𝜏₁(𝑠),𝜏₂𝑦≤Δ𝑠_𝑋𝑥0𝑓𝑠,𝑦𝜔𝑣^1/𝑝≤𝑠,𝑦Δ𝑠_𝑋𝑥0𝑓𝑠,𝑦Δ𝑠𝜔𝑣^1/𝑝𝑋,𝑦=𝜂₁𝑋,𝑦𝜔𝑣^1/𝑝𝑋,𝑦. (2.63) Let 𝑣(𝑋,𝑦) be the solution of the following problem: 𝑣^Δ_𝑦𝑋,𝑦=𝜂₁𝑋,𝑦𝜔𝑣^1/𝑝𝑋,𝑦,𝑣𝑋,𝑦₀=𝐶. (2.64) Considering 𝑣(𝑋,𝑦₀)=𝐶 and 𝜔 is nondecreasing and continuous, then from (2.63), (2.64), and Lemma 2.4, we have 𝑣𝑋,𝑦≤𝑣𝑋,𝑦,𝑦∈𝕋₀. (2.65) On the other hand, from the definition of 𝐺 we have (𝐺(𝑣(𝑋,𝑦)))^Δ_𝑦=𝑣^Δ_𝑦(𝑋,𝑦)/𝜔(𝑣(𝑋,𝑦))=𝜂₁(𝑋,𝑦). Then an integration with respect to 𝑦 from 𝑦₀ to 𝑦 yields 𝐺𝑣𝑋,𝑦−𝐺𝑣𝑋,𝑦₀=^𝑦_𝑦0𝜂₁(𝑋,𝑡)Δ𝑡, (2.66) that is, 𝑣𝑋,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₁(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀. (2.67) Combining (2.61), (2.65), and (2.67), we have 𝑢𝑥,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₁(𝑋,𝑡)Δ𝑡^1/𝑝,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.68) Setting 𝑥=𝑋 in (2.68), we get the desired result.

Remark 2.15. If we take 𝕋=ℝ, then Theorem 2.14 reduces to [14, Theorem 2.1], while Theorem 2.14 reduces to [15, Theorem 2.1] if we take 𝕋=ℤ.

Theorem 2.16. Suppose 𝑢,𝑓,𝜏₁,𝜏₂,𝜙,𝛼,𝛽 are the same as in Theorem 2.5, and furthermore, 𝑢 is delta differential on 𝕋₀ with respect to 𝑦, 𝑔∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊). 𝜔∈𝐶(ℝ₊,ℝ₊) is nondecreasing, and 𝜔 is submultiplicative, that is, 𝜔(𝛼𝛽)≤𝜔(𝛼)𝜔(𝛽), for all 𝛼,𝛽∈ℝ₊. 𝐶>0 is a constant. 𝐺 is a bijective function such that [𝐺(𝑧(𝑥,𝑦))]^Δ_𝑦=(𝑧(𝑥,𝑦))^Δ_𝑦/𝜔(𝑧(𝑥,𝑦)). If for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝜔𝑢𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡, (2.69) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤𝐶,∀𝑥,𝑦∈𝕋₀×𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (2.70) then 𝑢𝑥,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₂(𝑥,𝑡)Δ𝑡𝑒_𝐵1𝑦,𝑦₀,𝑥,𝑦∈𝕋₀×𝕋₀, (2.71) where 𝐵₁(𝑥,𝑦)=∫^𝑥_𝑥0𝑔(𝑠,𝑦)Δ𝑠,𝜂₂(𝑥,𝑦)=𝜔(𝑒_𝐵1(𝑦,𝑦₀))∫^𝑥_𝑥0𝑓(s,𝑦)Δ𝑠 and 𝑒_𝐵1(𝑦,𝑦₀) is the unique solution of the following equation: 𝑧^Δ_𝑦𝑥,𝑦=𝐵₁𝑥,𝑦𝑧𝑥,𝑦,𝑧𝑥,𝑦₀=1. (2.72)

Proof. Fix 𝑋∈𝕋₀, and 𝑥∈[𝑥₀,𝑋]⋂𝕋,𝑦∈𝕋₀. Let 𝑣𝑥,𝑦=𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝜔𝑢𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡. (2.73) Then 𝑢𝑥,𝑦≤𝑣𝑥,𝑦≤𝑣𝑋,𝑦,∀𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.74) Similar to (2.14)–(2.16), we can obtain 𝑢𝜏₁(𝑥),𝜏₂𝑦≤𝑣𝑥,𝑦,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.75) Furthermore we have 𝑣𝑋,𝑦=𝐶+^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝜔𝑢𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡≤𝐶+^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝜔(𝑣(𝑠,𝑡))+𝑔(𝑠,𝑡)𝑣(𝑠,𝑡)Δ𝑠Δ𝑡≤𝐶+^𝑦_𝑦0^𝑋_𝑥0𝑓(𝑠,𝑡)𝜔(𝑣(𝑠,𝑡))Δ𝑠Δ𝑡+^𝑦_𝑦0^𝑋_𝑥0𝑔(𝑠,𝑡)Δ𝑠𝑣(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀. (2.76) Let 𝐵₂(𝑋,𝑦)=𝐶+∫^𝑦_𝑦0∫^𝑋_𝑥0𝑓(𝑠,𝑡)𝜔(𝑣(𝑠,𝑡))Δ𝑠Δ𝑡. Then from (2.76) it follows that 𝑣𝑋,𝑦≤𝐵₂𝑋,𝑦+^𝑦_𝑦0𝐵₁(𝑋,𝑡)𝑣(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀. (2.77) Considering 𝐵₂(𝑋,𝑦) is nondecreasing in 𝑦, by applying Lemma 2.2 to (2.77), we obtain 𝑣𝑋,𝑦≤𝐵₂𝑋,𝑦𝑒_𝐵1𝑦,𝑦₀,𝑦∈𝕋₀. (2.78) On the other hand, 𝐵₂𝑋,𝑦^Δ_𝑦=^𝑋_𝑥0𝑓𝑠,𝑦𝜔𝑣𝑠,𝑦Δ𝑠≤^𝑋_𝑥0𝑓𝜔𝑣≤𝑠,𝑦Δ𝑠𝑋,𝑦_𝑋𝑥0𝑓𝑠,𝑦Δ𝑠𝜔𝐵₂𝑋,𝑦𝑒_𝐵1𝑦,𝑦₀≤^𝑋_𝑥0𝑓𝑠,𝑦Δ𝑠𝜔𝐵₂𝑋,𝑦𝜔𝑒_𝐵1𝑦,𝑦₀=𝜔𝐵₂𝑋,𝑦𝜂₂𝑋,𝑦. (2.79) Let 𝑣(𝑋,𝑦) be the solution of the following equation: 𝑣^Δ_𝑦𝑋,𝑦=𝜂₂𝑋,𝑦𝜔𝑣𝑋,𝑦,𝑣𝑋,𝑦₀=𝐶. (2.80) Considering 𝐵₂(𝑋,𝑦₀)=𝐶 and 𝜔 is nondecreasing and continuous, then from (2.79), (2.80), and Lemma 2.4, we have 𝐵₂𝑋,𝑦≤𝑣𝑋,𝑦,𝑦∈𝕋₀. (2.81) From the definition of 𝐺 and (2.80), we have (𝐺(𝑣(𝑋,𝑦)))^Δ_𝑦=𝑣^Δ_𝑦(𝑋,𝑦)/𝜔(𝑣(𝑋,𝑦))=𝜂₂(𝑋,𝑦). Then similar to (2.66) and (2.67), we obtain 𝐵₂𝑋,𝑦≤𝑣𝑋,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₂(𝑋,𝑡)Δ𝑡,𝑦∈𝕋₀. (2.82) Combining (2.74), (2.78), and (2.82), we have 𝑢𝑥,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₂(𝑋,𝑡)Δ𝑡𝑒_𝐵1𝑦,𝑦₀,𝑥∈𝑥₀,𝑋𝕋,𝑦∈𝕋₀. (2.83) Setting 𝑥=𝑋 in (2.83), we obtain 𝑢𝑋,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₂(𝑋,𝑡)Δ𝑡𝑒_𝐵1𝑦,𝑦₀,𝑦∈𝕋₀. (2.84) Replacing 𝑋 with 𝑥 in (2.84) yields the desired inequality (2.71).

Theorem 2.17. Under the conditions of Theorem 2.16, if 𝑝,𝐶 are constants with 𝑝>0, 𝐶>0, and for (𝑥,𝑦)∈𝕋₀×𝕋₀, 𝑢(𝑥,𝑦) satisfies the following inequality: 𝑢^𝑝𝑥,𝑦≤𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝜔𝑢𝜏₁(𝑠),𝜏₂(𝑡)+𝑔(𝑠,𝑡)𝑢^𝑝𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡, (2.85) with the initial condition (2.58), then 𝑢𝑥,𝑦≤𝐺⁻¹𝐺(𝐶)+^𝑦_𝑦0𝜂₃(𝑥,𝑡)Δ𝑡𝑒_𝐽1𝑦,𝑦₀^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀, (2.86) where 𝐺 is defined as in Theorem 2.14, 𝐽₁(𝑥,𝑦)=∫^𝑥_𝑥0𝑔(𝑠,𝑦)Δ𝑠, 𝜂₃(𝑥,𝑦)=𝜔((𝑒_𝐽1(𝑦,𝑦₀))^1/𝑝)∫^𝑥_𝑥0𝑓(𝑠,𝑦)Δ𝑠, and 𝑒_𝐽1(𝑦,𝑦₀) is the unique solution of the following equation: 𝑧^Δ_𝑦𝑥,𝑦=𝐽₁𝑥,𝑦𝑧𝑥,𝑦,𝑧𝑥,𝑦₀=1. (2.87)

The proof of Theorem 2.17 is similar to that of Theorem 2.16, and we omit it here.

3. Some Simple Applications

In this section, we will present some examples to illustrate the validity of our results in deriving explicit bounds of solutions of certain delay dynamic equations on time scales.

Example 3.1. Consider the following delay dynamic integral equation: 𝑢^𝑝𝑥,𝑦=𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑀𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡)Δ𝑠Δ𝑡,𝑥,𝑦∈𝕋₀×𝕋₀, (3.1) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋,₁(𝑥),𝜏₂𝑦≤|𝐶|^1/𝑝,∀𝑥,𝑦∈𝕋₀,𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (3.2) where 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ), 𝜙,𝛼,𝛽,𝜏₁,𝜏₂ are the same as in Theorem 2.8, and 𝑀∈𝐶_rd(𝕋₀×𝕋₀×ℝ,ℝ). Furthermore, assume |𝑀(𝑠,𝑡,𝑢)|≤𝑓(𝑠,𝑡)|𝑢|^𝑞+𝑔(𝑠,𝑡)|𝑢|^𝑟, where 𝑓,𝑔∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊), and 𝑝,𝑞,𝑟 are the same as in Theorem 2.8.
From (3.1) we have 𝑢^𝑝𝑥,𝑦≤|𝐶|+^𝑦_𝑦0^𝑥_𝑥0𝑀𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡)≤|𝐶|+Δ𝑠Δ𝑡_{𝑦𝑦0𝑥𝑥0}𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)^𝑞+𝑔(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)^𝑟ΔsΔ𝑡. (3.3) Then according to Theorem 2.8, we can obtain the following estimate: 𝑢𝑥,𝑦≤𝐻₁𝑥,𝑦+^𝑦_𝑦0𝑒𝐻₂(𝑦,𝜎𝑡)𝐻₂(𝑥,𝑡)𝐻₁(𝑥,𝑡)Δ𝑡^1/𝑝,𝑥,𝑦∈𝕋₀×𝕋₀, (3.4) where 𝐻₁𝑥,𝑦=|𝐶|+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)𝑝−𝑞𝑝𝐾^𝑞/𝑝+𝑔(𝑠,𝑡)𝑝−𝑟𝑝𝐾^𝑟/𝑝𝐻Δ𝑠Δ𝑡,∀𝐾>0,₂𝑥,𝑦=^𝑥_𝑥0𝑓𝑞𝑠,𝑦𝑝𝐾^{(𝑞−𝑝)/𝑝}𝑟+𝑔𝑠,𝑦𝑝𝐾^{(𝑟−𝑝)/𝑝}Δ𝑠,∀𝐾>0. (3.5)

Example 3.2. Considering the following delay dynamic integral equation: 𝑢³𝑥,𝑦=𝐶+^𝑦_𝑦0^𝑥_𝑥0𝑁𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)Δ𝑠Δ𝑡,𝑥,𝑦∈𝕋₀×𝕋₀, (3.6) with the initial condition 𝑢𝑥,𝑦=𝜙𝑥,𝑦,if𝑥∈𝛼,𝑥₀𝕋or𝑦∈𝛽,𝑦₀𝜙𝜏𝕋;₁(𝑥),𝜏₂𝑦≤|𝐶|^1/3,∀𝑥∈𝕋₀,𝑦∈𝕋₀,if𝜏₁(𝑥)≤𝑥₀or𝜏₂𝑦≤𝑦₀, (3.7) where 𝑢∈𝐶_rd(𝕋₀×𝕋₀,ℝ), and 𝑁∈𝐶_rd(𝕋₀×𝕋₀×ℝ²,ℝ).

Assume |𝑁(𝑥,𝑦,𝑢,𝑣)|≤𝑓(𝑥,𝑦)(|𝑢|²+|𝑣|²), where 𝑓∈𝐶_rd(𝕋₀×𝕋₀,ℝ₊), then from (3.6) we have 𝑢³𝑥,𝑦≤|𝐶|+^𝑦_𝑦0^𝑥_𝑥0𝑁𝑠,𝑡,𝑢𝜏₁(𝑠),𝜏₂(𝑡),𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)≤|𝐶|+Δ𝑠Δ𝑡_{𝑦𝑦0𝑥𝑥0}𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)²+𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)²≤|𝐶|+Δ𝑠Δ𝑡_{𝑦𝑦0𝑥𝑥0}𝑓(𝑠,𝑡)𝑢𝜏₁(𝑠),𝜏₂(𝑡)²+𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)²+𝑢𝜏₁(𝑠),𝜏₂(𝑡)𝑢𝜏₁(𝑠),𝜎𝜏₂(𝑡)Δ𝑠Δ𝑡. (3.8) According to Theorem 2.13 (p = 3), we can reach the following estimate: 𝑢𝑥,𝑦≤|𝐶|^1/3+^𝑦_𝑦0^𝑥_𝑥0𝑓(𝑠,𝑡)Δ𝑠Δ𝑡,𝑥,𝑦∈𝕋₀×𝕋₀. (3.9)

4. Conclusions

In this paper, we established some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales. As one can see, the presented results provide a handy tool for deriving bounds for solutions of certain delay dynamic equations on time scales. Furthermore, the process of constructing Theorems 2.5, 2.8, 2.14, 2.16 and 2.17 can be applied to the situation with 𝑛 independent variables.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province (ZR2010AZ003) (China). The authors thank the referees very much for their careful comments and valuable suggestions on this paper.