Nonlinear Dynamical Integral Inequalities in Two Independent Variables and Their Applications

Abstract

In this paper, we investigate some nonlinear dynamical integral inequalities involving the forward jump operator in two independent variables. These inequalities provide explicit bounds on unknown functions, which can be used as handy tools to study the qualitative properties of solutions of certain partial dynamical systems on time scales pairs.

1. Introduction

Theory of dynamical equations on time scales, which goes back to Hilger's landmark paper [1], has received considerable attention in recent years. For example, see the monographs [2, 3] and the references cited therein. Since dynamical integral inequalities usually can be used as handy tools to study the qualitative theory of dynamical equations on time scales, many researchers devoted to the study of different types of integral inequalities on time scales. We refer the readers to [4–19].

To the best of our knowledge, the theory of partial dynamic equations on time scales has received less attention [20–24]. The main purpose of this paper is to investigate several nonlinear integral inequalities in two independent variables on time scale pairs, which can be used to estimate explicit bounds of solutions of certain partial dynamical equations on time scales. Unlike some existing results in the literature (e.g., [12]), the integral inequalities considered in this paper involve the forward jump operator 𝜎(𝑡) and 𝜎(𝑠) on a pair of time scales 𝕋 and 𝕋, which results in difficulties in the estimation on the explicit bounds of unknown functions 𝑢(𝑡,𝑠) for 𝑡∈𝕋 and 𝑠∈𝕋. As an application, we study the qualitative property of certain partial dynamical equations on time scales.

Throughout this paper, a knowledge and understanding of time scales and time scale notations is assumed. In what follows, 𝕋 and 𝕋 are two unbounded time scales, 𝑡₀∈𝕋 and 𝑠₀∈𝕋. 𝐶_rd(𝕋,𝕋) is the set of right-dense continuous functions on 𝕋×𝕋. For an excellent introduction to the calculus on time scales, we refer the reader to monographs [2, 3].

2. Problem Statements

Before establishing the main results of this paper, we first present two useful lemmas as follows.

Lemma 2.1. Let 𝑐≥0, 𝑥≥0, and 0<𝜆<1. Then, for any 𝑘>0, 𝑐𝑥^𝜆≤𝑘𝑥+𝜃(𝑐,𝑘,𝜆) (2.1) holds, where 𝜃(𝑐,𝑘,𝜆)=(1−𝜆)𝜆^{𝜆/(1−𝜆)}𝑐^1/(1−𝜆)𝑘^{𝜆/(𝜆−1)}.

Proof. Set 𝐹(𝑥)=𝑐𝑥^𝜆−𝑘𝑥. It is not difficult to see that 𝐹(𝑥) obtains its maximum at 𝑥=(𝜆𝑐/𝑘)^1/(1−𝜆) and 𝐹_max=(1−𝜆)𝜆^{𝜆/(1−𝜆)}𝑐^1/(1−𝜆)𝑘^{𝜆/(1−𝜆)}. (2.2) This completes the proof of Lemma 2.1.

Lemma 2.2. Let 𝑦,𝑝,𝑞,𝑟∈𝐶_𝑟𝑑(𝕋) with 𝑝(𝑡),𝑞(𝑡)≥0 for 𝑡∈𝕋. Then 𝑦^Δ(𝑡)(𝑡)𝑦(𝑡)+𝑞(𝑡)≤𝑝(𝑡)𝑞(𝑡)𝑦(𝜎(𝑡(𝑡)1+𝜇))+𝑟,𝑡∈𝕋, (2.3) implies 𝑦(𝑡)≤𝑦𝑡₀𝑒_𝑝⊕𝑞𝑡,𝑡₀+^𝑡_𝑡0𝑒_𝑝⊕𝑞(𝑡,𝜎(𝑠))1+𝜇(𝑠)𝑞(𝑠)𝑟(𝑠)Δ𝑠,𝑡∈𝕋, (2.4) where 𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞 and 𝜇(𝑡)=𝜎(𝑡)−𝑡.

Proof. Note that 𝑦(𝜎(𝑡))=𝑦(𝑡)+𝜇(𝑡)𝑦^Δ(𝑡), we have 𝑦^Δ(𝑡)(𝑡)𝑦(𝑡)+𝑞(𝑡)≤𝑝(𝑡)𝑞(𝑡)𝑦(𝑡)(𝑡)𝑦1+𝜇+𝜇^Δ(𝑡)+𝑟(𝑡) (2.5) that is, 𝑦^Δ(𝑡)≤𝑝⊕𝑞(𝑡)𝑦(𝑡)+1+𝜇(𝑡)𝑞(𝑡)𝑟(𝑡). (2.6) By Theorem  6.1 [2, page 255], we get that Lemma 2.2 holds.

Consider the following nonlinear integral inequalities in two independent variables on time scales 𝕋×𝕋: 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏)𝑢()()(),𝜂Δ𝜂Δ𝜏,𝑡,𝑠≤𝑎𝑡,𝑠+𝑏𝑡,𝑠^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏),𝜂+ℎ₂𝜏,𝜂𝑢^𝜆2𝜂𝑢()()()𝜏,𝜎Δ𝜂Δ𝜏,𝑡,𝑠≤𝑎𝑡,𝑠+𝑏𝑡,𝑠^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏),𝜂+ℎ₂𝜏,𝜂𝑢^𝜆2𝜏,𝜎𝜂+ℎ₃𝜏,𝜂𝑢^𝜆3𝜎(𝜏),𝜎𝜂Δ𝜂Δ𝜏, (2.7)(2.8)(2.9) where 𝑢(𝑡,𝑠),𝑎(𝑡,𝑠),𝑏(𝑡,𝑠),𝑔(𝑡,𝑠), and ℎ_𝑖(𝑡,𝑠) (𝑖=1,2,3) are nonnegative right-dense continuous functions on 𝕋×𝕋, 0<𝜆_𝑖<1 (𝑖=1,2,3) are constants.

The reason for studying inequalities of type (2.7)–(2.9) is that sometimes we may need to estimate the solutions of the following partial dynamical equation in the form 𝑢^Δ𝑡Δ𝑠(𝑡,𝑠)=𝑓(𝑡,𝑠,𝑢(𝑡,𝑠),𝑢(𝜎(𝑡),𝑠),𝑢(𝑡,𝜎(𝑠)),𝑢(𝜎(𝑡),𝜎(𝑠))) (2.10) with boundary conditions 𝑢(𝑡,𝑠₀)=𝛼(𝑡), 𝑢(𝑡₀,𝑠)=𝛽(𝑠), and 𝑢(𝑡₀,𝑠₀)=𝑢₀, where 𝑓∶𝕋×𝕋×ℝ³→ℝ is right-dense continuous, ℝ=(−∞,∞), and 𝑢₀ is a constant. Integrating (2.10) yields 𝑢(𝑡,𝑠)=𝛼(𝑡)+𝛽(𝑠)−𝑢₀+^𝑡_𝑡0^𝑠_𝑠0𝑓𝜏,𝜂,𝑢𝜏,𝜂,𝑢𝜎(𝜏),𝜂,𝑢𝜏,𝜎𝜂,𝑢𝜎(𝜏),𝜎𝜂Δ𝜂Δ𝜏. (2.11) Therefore, the study on the integral inequalities of type (2.7)–(2.9) can provide explicit bounds of solutions of system (2.10) in some cases.

3. Main Results

Now, let us present the main results of this paper.

Theorem 3.1. If there exists a positive function 𝑘₁(𝑡,𝑠)∈𝐶_rd(𝕋,𝕋), such that 𝜇(𝑡)𝑏(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)<1,(𝑡,𝑠)∈𝕋×𝕋, (3.1) then inequality (2.7) implies 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡_𝑡0𝑒_{(𝑝1⊕𝑞1)(⋅,𝑠)}(𝑡,𝜎(𝜏))1+𝜇(𝜏)𝑞₁(𝜏,𝑠)𝑟₁(𝜏,𝑠)Δ𝜏, (3.2) where 𝑝₁(𝑡,𝑠)=𝑏(𝑡,𝑠)^𝑠_𝑠0𝑔𝑞𝑡,𝜂Δ𝜂,₁()=𝑏(𝜎(𝑡))𝑘𝑡,𝑠,𝑠₁(𝑡,𝑠)1−𝜇(𝑡)𝑏(𝜎(𝑡),𝑠)𝑘₁(),𝑟𝑡,𝑠₁(𝑡,𝑠)=𝑎(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)+𝑎(𝑡,𝑠)^𝑠_𝑠0𝑔𝑡,𝜂Δ𝜂+𝜃^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂,𝑘₁(𝑡,𝑠),𝜆₁. (3.3)

Proof. Define a function 𝑣(𝑡,𝑠) by 𝑣(𝑡,𝑠)=^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏),𝜂Δ𝜂Δ𝜏. (3.4) Then, 𝑣(𝑡,𝑠)≥0 for (𝑡,𝑠)∈𝕋×𝕋, 𝑣(𝑡,𝑠) is nondecreasing with respect to 𝑡 and 𝑠, and 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑣(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.5) A delta derivative of 𝑣(𝑡,𝑠) with respect to 𝑡 yields 𝑣^Δ𝑡(𝑡,𝑠)=^𝑠_𝑠0𝑔𝑡,𝜂𝑢𝑡,𝜂+ℎ₁𝑡,𝜂𝑢^𝜆1𝜎(𝑡)≤𝑢(),𝜂Δ𝜂𝑡,𝑠+𝑢^𝜆1(𝜎(𝑡),𝑠)^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂. (3.6) By Lemma 2.1, we have 𝑢^𝜆1(𝜎(𝑡),𝑠)^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂≤𝑘₁(𝑡,𝑠)𝑢(𝜎(𝑡),𝑠)+𝜃^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂,𝑘₁(𝑡,𝑠),𝜆₁. (3.7) It follows from (3.5), (3.6), and (3.7) that 𝑣^Δ𝑡(𝑡,𝑠)≤[𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑣(𝑡,𝑠)]^𝑠_𝑠0𝑔+[𝑎(𝜎(𝑡))(𝜎(𝑡))𝑣(𝜎(𝑡)𝑘𝑡,𝜂Δ𝜂,𝑠+𝑏,𝑠,𝑠)]₁(𝑡,𝑠)+𝜃^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂,𝑘₁(𝑡,𝑠),𝜆₁. (3.8) Notice the definitions of 𝑝₁(𝑡,𝑠), 𝑞₁(𝑡,𝑠), and 𝑟₁(𝑡,𝑠), we have 𝑣^Δ𝑡(𝑡,𝑠)≤𝑝₁()𝑣()+𝑞𝑡,𝑠𝑡,𝑠¹(𝑡,𝑠)1+𝜇(𝑡)𝑞₁(𝑡,𝑠)+𝑟₁(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.9) Since 𝑣(𝑡₀,𝑠)=0, by Lemma 2.2 we get 𝑣(𝑡,𝑠)≤^𝑡_𝑡0𝑒_{(𝑝1⊕𝑞1)(⋅,𝑠)}(𝑡,𝜎(𝜏))1+𝜇(𝜏)𝑞₁(𝜏,𝑠)𝑟₁(𝜏,𝑠)Δ𝜏,(𝑡,𝑠)∈𝕋×𝕋. (3.10) Then, (3.5) and (3.10) imply (3.2).

Theorem 3.2. If there exist positive functions 𝑘₁(𝑡,𝑠),𝑘₂(𝑡,𝑠)∈𝐶_rd(𝕋,𝕋), such that 𝑘₂(𝑡,𝑠) is Δ-differentiable with respect to 𝑠, 𝑘^Δ𝑠₂(𝑡,𝑠)∈𝐶_rd(𝕋,𝕋), and 𝜇(𝑡)𝑏(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)<1,(𝑡,𝑠)∈𝕋×𝕋, (3.11) then inequality (2.8) implies 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡_𝑡0𝑒_{(𝑝2⊕𝑞2)(⋅,𝑠)}(𝑡,𝜎(𝜏))1+𝜇(𝜏)𝑞₂(𝜏,𝑠)𝑟₂(𝜏,𝑠)Δ𝜏, (3.12) where 𝑝₂(𝑡,𝑠)=𝑝₁(𝑡,𝑠)+𝑘₂(𝑡,𝑠)+^𝑠_𝑠0𝑘^Δ𝜂₂+𝑘𝑡,𝜂²𝑡,𝜎𝜂𝑏𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂Δ𝜂,𝑘^Δ𝑠₂(𝑡,𝑠)=max0,−𝑘^Δ𝑠₂(𝑡,𝑠),𝑞₂(𝑡,𝑠)=𝑞₁(),𝑟𝑡,𝑠₂(𝑡,𝑠)=𝑟₁(𝑡,𝑠)+^𝑠_𝑠0𝜃ℎ₂,𝑘𝑡,𝜂²𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂,𝜆₂+𝑘₂𝑡,𝜎𝜂𝑎𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂Δ𝜂. (3.13)

Proof. Set 𝑧(𝑡,𝑠)=^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏),𝜂+ℎ₂𝜏,𝜂𝑢^𝜆2𝜏,𝜎𝜂Δ𝜂Δ𝜏. (3.14) Then, 𝑧(𝑡,𝑠) is nonnegative and nondecreasing with respect to 𝑡 and 𝑠 on 𝕋×𝕋, and 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑧(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.15) By Lemma 2.1, we have 𝑧^Δ𝑡(𝑡,𝑠)≤𝑢(𝑡,𝑠)^𝑠_𝑠0𝑔𝑡,𝜂Δ𝜂+𝑢^𝜆1(𝜎(𝑡),𝑠)^𝑠_𝑠0ℎ₁+𝑡,𝜂Δ𝜂_𝑠𝑠0ℎ₂𝑡,𝜂𝑢^𝜆2𝜂()𝑡,𝜎Δ𝜂≤𝑢𝑡,𝑠^𝑠_𝑠0𝑔𝑡,𝜂Δ𝜂+𝑘₁(𝑡,𝑠)𝑢(𝜎(𝑡),𝑠)+𝜃^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂,𝑘₁(𝑡,𝑠),𝜆₁+^𝑠_𝑠0𝑘₂𝑡,𝜎𝜂𝜂𝑏𝜂𝑢𝜂+1+𝜇𝑡,𝜎𝑡,𝜎Δ𝜂_𝑠𝑠0𝜃ℎ₂,𝑘𝑡,𝜂²𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂,𝜆₂Δ𝜂. (3.16) Substituting (3.15) into (3.16), we get 𝑧^Δ𝑡(𝑡,𝑠)≤[𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑧(𝑡,𝑠)]^𝑠_𝑠0𝑔+[𝑎(𝜎(𝑡))(𝜎(𝑡))𝑧(𝜎(𝑡)𝑘𝑡,𝜂Δ𝜂,𝑠+𝑏,𝑠,𝑠)]₁(𝑡,𝑠)+𝜃^𝑠_𝑠0ℎ₁𝑡,𝜂Δ𝜂,𝑘₁(𝑡,𝑠),𝜆₁+^𝑠_𝑠0𝑘₂𝑡,𝜎𝜂𝜂𝑏𝜂𝑎𝜂𝜂𝑧𝜂+1+𝜇𝑡,𝜎𝑡,𝜎+𝑏𝑡,𝜎𝑡,𝜎Δ𝜂_𝑠𝑠0𝜃ℎ₂,𝑘𝑡,𝜂²𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂,𝜆₂Δ𝜂. (3.17) Note that 𝑧𝑡,𝜎𝜂=𝑧𝑡,𝜂+𝜇𝜂𝑧^Δ𝜂𝑡,𝜂. (3.18) Integrating by parts, we have ^𝑠_𝑠0𝑘₂𝑡,𝜎𝜂𝜂𝑏𝜂𝑎𝜂𝜂𝑧𝜂≤1+𝜇𝑡,𝜎𝑡,𝜎+𝑏𝑡,𝜎𝑡,𝜎Δ𝜂_𝑠𝑠0𝑘₂𝑡,𝜎𝜂𝑎𝑡,𝜎𝜂1+𝜇𝜂𝑏𝑡,𝜎𝜂Δ𝜂+^𝑠_𝑠0𝑘₂𝑡,𝜎𝜂𝑏𝑡,𝜎𝜂𝜂𝑏𝜂𝑧+1+𝜇𝑡,𝜎𝑡,𝜂Δ𝜂_𝑠𝑠0𝑘₂𝑡,𝜎𝜂𝑧^Δ𝜂≤𝑡,𝜂Δ𝜂_𝑠𝑠0𝑘₂𝑡,𝜎𝜂𝑎𝑡,𝜎𝜂𝜂𝑏𝜂()1+𝜇𝑡,𝜎Δ𝜂+𝑧𝑡,𝑠^𝑠_𝑠0𝑘₂𝑡,𝜎𝜂𝑏𝑡,𝜎𝜂𝜂𝑏𝜂(𝑘1+𝜇𝑡,𝜎Δ𝜂+𝑧𝑡,𝑠)₂(𝑡,𝑠)+^𝑠_𝑠0𝑘^Δ𝜂₂𝑡,𝜂Δ𝜂. (3.19) Therefore, it follows from (3.17) and (3.19) that 𝑧^Δ𝑡(𝑡,𝑠)≤𝑝₂()𝑧()+𝑞𝑡,𝑠𝑡,𝑠²(𝑡,𝑠)1+𝜇(𝑡)𝑞₂()𝑧(𝜎(𝑡))𝑡,𝑠,𝑠+𝑟₂(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.20) This together with Lemma 2.2 and (3.15) yields (3.12).

Theorem 3.3. If there exist positive functions 𝑘₁(𝑡,𝑠),𝑘₂(𝑡,𝑠),𝑘₃(𝑡,𝑠)∈𝐶_rd(𝕋,𝕋), such that 𝑘₂(𝑡,𝑠),k₃(𝑡,𝑠) are Δ-differentiable with respect to s,𝑘^Δ𝑠₂(𝑡,𝑠),𝑘^Δ𝑠₃(𝑡,𝑠)∈𝐶_rd(𝕋,𝕋), and 𝜇(𝑡)Λ(𝑡,𝑠)<1,(𝑡,𝑠)∈𝕋×𝕋, (3.21) then inequality (2.9) implies 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)^𝑡_𝑡0𝑒_{(𝑝3⊕𝑞3)(⋅,𝑠)}(𝑡,𝜎(𝜏))1+𝜇(𝜏)𝑞₃(𝜏,𝑠)𝑟₃(𝜏,𝑠)Δ𝜏, (3.22) where Λ(𝑡,𝑠)=𝑏(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)+𝑘₃(𝑡,𝑠)+^𝑠_𝑠0𝑘^Δ𝜂₃+𝑘𝑡,𝜂³𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂𝑝1+𝜇,𝜎Δ𝜂,₃(𝑡,𝑠)=𝑝₂(),𝑞𝑡,𝑠₃()=Λ()𝑡,𝑠𝑡,𝑠(𝑡)Λ(),𝑟1−𝜇𝑡,𝑠₃(𝑡,𝑠)=𝑟₂(𝑡,𝑠)+^𝑠_𝑠0𝑘₃𝑡,𝜎𝜂𝑎𝜎(𝑡),𝜎𝜂1+𝜇𝜂𝑏𝜎(𝑡),𝜎𝜂Δ𝜂+^𝑠_𝑠0𝜃₃ℎ₃,𝑘𝑡,𝜂³𝑡,𝜎𝜂1+𝜇𝜂𝑏𝜎(𝑡),𝜎𝜂,𝜆₃Δ𝜂, (3.23) and 𝑘^Δ𝑠₃(𝑡,𝑠)=max{0,−𝑘^Δ𝑠₃(𝑡,𝑠)}.

Proof. Let the nonnegative and nondecreasing function 𝑤(𝑡,𝑠) be defined by 𝑤(𝑡,𝑠)=^𝑡_𝑡0^𝑠_𝑠0𝑔𝜏,𝜂𝑢𝜏,𝜂+ℎ₁𝜏,𝜂𝑢^𝜆1𝜎(𝜏),𝜂+ℎ₂𝜏,𝜂𝑢^𝜆2𝜏,𝜎𝜂+ℎ₃𝜏,𝜂𝑢^𝜆3𝜎(𝜏),𝜎𝜂Δ𝜂Δ𝜏. (3.24) Then, 𝑢(𝑡,𝑠)≤𝑎(𝑡,𝑠)+𝑏(𝑡,𝑠)𝑤(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.25) Based on the same arguments as in Theorem 3.2, we have 𝑤^Δ𝑡(𝑡,𝑠)≤𝑝₂(𝑡,𝑠)𝑧(𝑡,𝑠)+𝑏(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)𝑤(𝜎(𝑡),𝑠)+𝑟₂()+𝑡,𝑠_𝑠𝑠0𝑘₃𝑡,𝜎𝜂𝑎𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂+1+𝜇,𝜎Δ𝜂_𝑠𝑠0𝜃₃ℎ₃,𝑘𝑡,𝜂³𝑡,𝜎𝜂1+𝜇𝜂𝑏𝜎(𝑡),𝜎𝜂,𝜆₃+Δ𝜂_𝑠𝑠0𝑘₃𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂𝑧𝜎(𝑡)𝜂1+𝜇,𝜎,𝜎Δ𝜂. (3.26) Notice that ^𝑠_𝑠0𝑘₃𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂𝑧𝜎(𝑡)𝜂=1+𝜇,𝜎,𝜎Δ𝜂_𝑠𝑠0𝑘₃𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂𝑧𝜎(𝑡)𝜂𝑧1+𝜇,𝜎,𝜂+𝜇^Δ𝜂𝜎(𝑡)(𝜎(𝑡)),𝜂Δ𝜂≤𝑧,𝑠^𝑠_𝑠0𝑘₃𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂1+𝜇𝜂𝑏𝜎(𝑡),𝜎𝜂Δ𝜂+^𝑠_𝑠0𝑘₃𝑡,𝜎𝜂𝑧^Δ𝜂𝜎(𝑡)(𝜎(𝑡)𝑘,𝜂Δ𝜂≤𝑧,𝑠)₃(𝑡,𝑠)+^𝑠_𝑠0𝑘^Δ𝜂₃+𝑘𝑡,𝜂³𝑡,𝜎𝜂𝑏𝜎(𝑡),𝜎𝜂𝜂𝑏𝜎(𝑡)𝜂.1+𝜇,𝜎Δ𝜂 (3.27) By (3.26) and (3.27), we have 𝑤^Δ𝑡(𝑡,𝑠)≤𝑝₃(𝑡,𝑠)𝑤(𝑡,𝑠)+Λ(𝑡,𝑠)𝑤(𝜎(𝑡),𝑠)+𝑟₃(𝑡,𝑠),(𝑡,𝑠)∈𝕋×𝕋. (3.28) Using the fact Λ(𝑡,𝑠)=𝑞₃(𝑡,𝑠)/(1+𝜇(𝑡)𝑞₃(𝑡,𝑠)), Lemma 2.2 and (3.25), we get that (3.22) holds.
It is worthy to mention that although some additional assumptions such as 𝜇(𝑡)𝑏(𝜎(𝑡),𝑠)𝑘₁(𝑡,𝑠)<1 and 𝜇(𝑡)Λ(𝑡,𝑠)<1 are imposed in Theorems 3.1–3.3, they are easy to be satisfied by choosing appropriate adjusting functions 𝑘₁(𝑡,𝑠) and 𝑘₃(𝑡,𝑠).

4. Applications

We now consider some applications of the main results in the partial dynamical system (2.10) under the boundary condition 𝑢𝑡,𝑠₀=𝛼(𝑡),𝑢𝑡₀,𝑠=𝛽(𝑠),𝑢𝑡₀,𝑠₀=𝑢₀. (4.1) Denote 𝑎(𝑡,𝑠)=|𝛼(𝑡)|+|𝛽(𝑠)|+|𝑢₀|. We have the following corollaries.

Corollary 4.1. Let 𝕋=𝑍={0,1,2,…}, 𝕋=𝑅₊=[0,∞), and 𝑓(𝑡,𝑠,𝑢(𝑡,𝑠),𝑢(𝑡+1,𝑠)≤|𝑢(𝑡,𝑠)|+|𝑢(𝑡+1,𝑠)|^𝜆1,(𝑡,𝑠)∈𝕋×𝕋. (4.2) Then, the solution of system (2.10) under the boundary condition (4.1) satisfies |𝑢(𝑡,𝑠)|≤𝑎(𝑡,𝑠)+2^𝑡−1_𝜏=0(2+2𝑠)^{𝑡−1−𝜏}𝑎(𝜏+1,𝑠)2()1+𝑎𝜏,𝑠𝑠+𝜃𝑠,2,𝜆₁, (4.3) for (𝑡,𝑠)∈𝕋×𝕋.

Proof. For (𝑡,𝑠)∈𝕋×𝕋, it follows from (2.11) and (4.2) that |𝑢(𝑡,𝑠)|≤𝑎(𝑡,𝑠)+^𝑡−1_𝜏=0^𝑠₀𝑢𝜏,𝜂+𝑢𝜏+1,𝜂^𝜆1. (4.4) Let 𝑘₁(𝑡,𝑠)=1/2 be a constant. A straightforward computation yields 𝑝₁(𝑡,𝑠)=𝑠,𝑞₁()𝑟𝑡,𝑠=1,₁()=𝑎()𝑡,𝑠𝑡+1,𝑠2()1+𝑎𝑡,𝑠𝑠+𝜃𝑠,2,𝜆₁. (4.5) Since (𝑝₁⊕𝑞₁)(𝑡,𝑠)=2+2𝑠, we get (4.3) by Theorem 3.1.

Corollary 4.2. Let 𝕋=𝕋=𝑍, and 𝑓(𝑡,𝑠,𝑢(𝑡,𝑠),𝑢(𝑡+1,𝑠),𝑢(𝑡,𝑠+1)≤|𝑢(𝑡,𝑠)|+|𝑢(𝑡+1,𝑠)|^𝜆1+|𝑢(𝑡,𝑠+1)|^𝜆2. (4.6) Then, the solution of system (2.10) under the boundary condition (4.1) satisfies |𝑢(𝑡,𝑠)|≤𝑎(𝑡,𝑠)+2^𝑡−1_𝜏=0(3+3𝑠)^{𝑡−1−𝜏}𝑟₁()1𝜏,𝑠+𝑠𝜃𝑠,2,𝜆₂+∑^𝑠−1_𝜂=0𝑎𝜏,𝜂+12, (4.7) where 𝑟₁(𝑡,𝑠) is defined as in Corollary 4.1 for (𝑡,𝑠)∈𝕋×𝕋.

Proof. For (𝑡,𝑠)∈𝕋×𝕋, it follows from (2.11) and (4.6) that |𝑢(𝑡,𝑠)|≤𝑎(𝑡,𝑠)+^𝑡−1_𝜏=0^𝑠−1_𝜂=0𝑢𝜏,𝜂+𝑢𝜏+1,𝜂^𝜆1+𝑢𝜏,𝜂+1^𝜆2 (4.8) holds for (𝑡,𝑠)∈𝕋×𝕋. Let 𝑘₁(𝑡,𝑠)=1/2 and 𝑘₂(𝑡,𝑠)=1. A straightforward computation yields 𝑝₂(𝑡,𝑠)=1+3𝑠2,𝑞₂()𝑟𝑡,𝑠=1,₂(𝑡,𝑠)=𝑟₁()1𝑡,𝑠+𝑠𝜃𝑠,2,𝜆₂+∑^𝑠−1_𝜂=0𝑎𝑡,𝜂+12. (4.9) Hence, 𝑝₂⊕𝑞₂=3+3𝑠. By Theorem 3.2, we have that (4.7) holds.
For the case when 𝑓 satisfies 𝑓(()()()(≤|𝑢(+|𝑢(𝑡,𝑠,𝑢𝑡,𝑠,𝑢𝑡+1,𝑠,𝑢𝑡,𝑠+1,𝑢𝑡+1,𝑠+1))𝑡,𝑠)|𝑡+1,𝑠)|^𝜆1+|𝑢(𝑡,𝑠+1)|^𝜆2+|𝑢(𝑡+1,𝑠+1)|^𝜆3 (4.10) on 𝑍×𝑍, the solution of system (2.10) under the boundary condition (4.1) can be similarly estimated by Theorem 3.3. We omit it here.

Acknowledgment

This paper was supported by the Natural Science Foundations of Shandong Province (ZR2010AL002, JQ201119) and the National Natural Science Foundation of China (61174217).