Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 584951 | 24 pages | https://doi.org/10.1155/2011/584951

Some New Volterra-Fredholm-Type Discrete Inequalities and Their Applications in the Theory of Difference Equations

Academic Editor: Martin D. Schechter
Received21 Mar 2011
Accepted29 Jun 2011
Published29 Aug 2011

Abstract

Some new Volterra-Fredholm-type discrete inequalities in two independent variables are established, which provide a handy tool in the study of qualitative and quantitative properties of solutions of certain difference equations. The established results extend some known results in the literature.

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. Many such generalized inequalities (e.g., see [330] and the references therein) have been established in the literature including the known Ou-Liang's inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang's inequality in two variables to Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of solutions of certain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma [9] to our knowledge. We notice, in the analysis of some certain Volterra-Fredholm-type difference equations with more complicated forms, that the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.

Our aim in this paper is to establish some new generalized Volterra-Fredholm-type discrete inequalities, which extend Ma's work in [9], and provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain more complicated Volterra-Fredholm-type difference equations.

Throughout this paper, denotes the set of real numbers +=[0,), and denotes the set of integers, while 0 denotes the set of nonnegative integers. Let Ω=([𝑚0,𝑀]×[𝑛0,𝑁])2, where 𝑚0,𝑛0 and 𝑀,𝑁{} are two constants. 𝑙1,𝑙2 are two constants, and 𝐾𝑖>0,𝑖=1,2,3,4, are all constants. If 𝑈 is a lattice, then we denote the set of all -valued functions on 𝑈 by (𝑈) and denote the set of all +-valued functions on 𝑈 by +(𝑈). Finally, for a function 𝑓+(𝑈), we have 𝑚1𝑠=𝑚0𝑓=0 provided 𝑚0>𝑚1.

2. Main Results

Lemma 2.1 (see [15]). Assume that 𝑎0,𝑝𝑞0, and 𝑝0 then for any 𝐾>0𝑎𝑞/𝑝𝑞𝑝𝐾(𝑞𝑝)/𝑝𝑎+𝑝𝑞𝑝𝐾𝑞/𝑝.(2.1)

Lemma 2.2. Suppose that 𝑢(𝑚,𝑛)+(Ω),𝑏(𝑠,𝑡,𝑚,𝑛)+(Ω2), 𝛼0 is a constant. If 𝑏 is nondecreasing in the third variable, then, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛)𝛼+𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)(2.2) implies that 𝑢(𝑚,𝑛)𝛼exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏(𝑠,𝑡,𝑚,𝑛).(2.3)

Lemma 2.3. Suppose that 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛),𝑏(𝑚,𝑛)+(Ω). If 𝑎(𝑚,𝑛) is nondecreasing in the first variable, then, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑚1𝑠=𝑚0𝑏(𝑠,𝑛)𝑢(𝑠,𝑛)(2.4) implies that 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)𝑚1𝑠=𝑚0[].1+𝑏(𝑠,𝑛)(2.5)

Remark 2.4. Lemma 2.3 is a direct variation of [19, Lemma 2.5(𝛽1)], and we note 𝑎(𝑚,𝑛)0 here.

Theorem 2.5. Suppose that 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛)+(Ω), 𝑏𝑖(𝑠,𝑡,𝑚,𝑛),𝑐𝑖(𝑠,𝑡,𝑚,𝑛)+(Ω2), 𝑖=1,2,,𝑙1, 𝑑𝑖(𝑠,𝑡,𝑚,𝑛),𝑒𝑖(𝑠,𝑡,𝑚,𝑛)+(Ω2), 𝑖=1,2,,𝑙2with 𝑏𝑖,𝑐𝑖,𝑑𝑖,𝑒𝑖 nondecreasing in the last two variables. 𝑝,𝑞𝑖,𝑟𝑖 are nonnegative constants with 𝑝𝑞𝑖,𝑝𝑟𝑖,𝑖=1,2,,𝑙1,𝑝0, while 𝑖,𝑗𝑖 are nonnegative constants with 𝑝𝑖,𝑝𝑗𝑖,𝑖=1,2,,𝑙2. If, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛) satisfies 𝑢𝑝(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(,𝜉,𝜂)(2.6) then 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)1/𝑝,(2.7) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑞𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖p𝐾(𝑟𝑖2𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝,(2.8)𝜇(𝑚,𝑛)=𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝐶(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝑗𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝐶(𝜉,𝜂),(2.9)𝐶(𝑚,𝑛)=exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0,𝐵(𝑠,𝑡,𝑚,𝑛)(2.10)𝐵(𝑠,𝑡,𝑚,𝑛)=𝑙1𝑖=1𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝.(2.11)

Proof. Denote 𝑣(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(.𝜉,𝜂)(2.12) Then, we have []𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)1/𝑝,(2.13) and, furthermore, from Lemma 2.1 we have 𝑢𝑞𝑖[](𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑞𝑖/𝑝𝑞𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝[]+𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝,𝑖=1,2,,𝑙1,𝑢𝑟𝑖[](𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑟𝑖/𝑝𝑟𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝[]+𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝,𝑖=1,2,,𝑙1,𝑢𝑖[](𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑖/𝑝𝑖𝑝𝐾(𝑖3𝑝)/𝑝[]+𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑝𝑖𝑝𝐾𝑖3/𝑝,𝑖=1,2,,𝑙2,𝑢𝑗𝑖[](𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑗𝑖/𝑝𝑗𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝[]+𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝,𝑖=1,2,,𝑙2.(2.14) So 𝑣(𝑚,𝑛)𝑙1𝑖=1m1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑞𝑠,𝑡,𝑚,𝑛)i𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝=𝐻(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑣(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝,𝑣(𝜉,𝜂)(2.15) where 𝐻(𝑚,𝑛)=𝐽(𝑚,𝑛)+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0{𝑑𝑖(𝑠,𝑡,𝑚,𝑛)(𝑖/𝑝)𝐾(𝑖3𝑝)/𝑝𝑣(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)(𝑗𝑖/𝑝)𝐾(𝑗𝑖4𝑝)/𝑝𝑣(𝜉,𝜂)} and 𝐽(𝑚,𝑛) is defined in (2.8). Then, using that 𝐻(𝑚,𝑛) is nondecreasing in every variable, we obtain 𝑣(𝑚,𝑛)𝐻(𝑀,𝑁)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝𝑣+(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝𝑣(𝜉,𝜂)𝐻(𝑀,𝑁)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝𝑣(𝑠,𝑡)=𝐻(𝑀,𝑁)+𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛)𝑣(𝑠,𝑡),(2.16) where 𝐵(𝑠,𝑡,𝑚,𝑛) is defined in (2.11).
Since 𝑏𝑖(𝑠,𝑡,𝑚,𝑛),𝑐𝑖(𝑠,𝑡,𝑚,𝑛) are nondecreasing in the last two variables, then 𝐵(𝑠,𝑡,𝑚,𝑛) is also nondecreasing in the last two variables, and by a suitable application of Lemma 2.2 we obtain 𝑣(𝑚,𝑛)𝐻(𝑀,𝑁)exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛)=𝐻(𝑀,𝑁)𝐶(𝑚,𝑛),(2.17) where 𝐶(𝑚,𝑛) is defined in (2.10). Furthermore, considering the definition of 𝐻(𝑚,𝑛) and (2.17) we have 𝐻(𝑀,𝑁)=𝐽(𝑀,𝑁)+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑀,𝑁)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝑣(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝑗𝜉,𝜂,𝑀,𝑁)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑣(𝜉,𝜂)𝐽(𝑀,𝑁)+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑀,𝑁)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝐻(𝑀,𝑁)𝐶(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝜉,𝜂,𝑚1,𝑛1𝑗𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝐻(𝑀,𝑁)𝐶(𝜉,𝜂)=𝐽(𝑀,𝑁)+𝐻(𝑀,𝑁)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑀,𝑁)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝐶(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑀,𝑁)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝐶(𝜉,𝜂)=𝐽(𝑀,𝑁)+𝐻(𝑀,𝑁)𝜇(𝑀,𝑁),(2.18) where 𝜇(𝑚,𝑛) is defined in (2.9). Then, 𝐻(𝑀,𝑁)𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁).(2.19) Combining (2.17) and (2.19) we deduce 𝑣(𝑚,𝑛)𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛).(2.20) Then, combining (2.13) and (2.20), we obtain the desired result.

Corollary 2.6. Suppose that 𝑔1𝑖(𝑚,𝑛),𝑔2𝑖(𝑚,𝑛),𝑏1𝑖(𝑚,𝑛),𝑐1𝑖(𝑚,𝑛)+(Ω),𝑖=1,2,,𝑙1 with 𝑔1𝑖,𝑔2𝑖 nondecreasing in every variable. 𝑑1𝑖(𝑚,𝑛),𝑒1𝑖(𝑚,𝑛)+(Ω),𝑖=1,2,,𝑙2. 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛),𝑝,𝑞𝑖,𝑟𝑖,𝑖,𝑗𝑖 are defined as in Theorem 2.5. If, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛) satisfies 𝑢𝑝(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑙1𝑖=1𝑔1𝑖(𝑚,𝑛)𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏1𝑖(𝑠,𝑡)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐1𝑖(𝜉,𝜂)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑔2𝑖(𝑚,𝑛)𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑1𝑖(𝑠,𝑡)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒1𝑖(𝜉,𝜂)𝑢𝑗𝑖(,𝜉,𝜂)(2.21) then 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑀,𝑁)1/𝑝,(2.22) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=𝑙1𝑖=1𝑔1𝑖(𝑚,𝑛)𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏1𝑖(𝑞𝑠,𝑡)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐1𝑖(𝑟𝜉,𝜂)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑔2𝑖(𝑚,𝑛)𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑1𝑖(𝑠,𝑡)𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒1𝑖𝑗(𝜉,𝜂)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝,𝜇(𝑚,𝑛)=𝑙2𝑖=1𝑔2𝑖(𝑚,𝑛)𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑1𝑖(𝑠,𝑡)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝐶(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒1𝑖𝑗(𝜉,𝜂)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝,𝐶(𝜉,𝜂)𝐶(𝑚,𝑛)=exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0,𝐵𝐵(𝑠,𝑡,𝑚,𝑛)(𝑠,𝑡,𝑚,𝑛)=𝑙1𝑖=1𝑔1𝑖𝑏(𝑚,𝑛)1𝑖𝑞(𝑠,𝑡)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐1𝑖𝑟(𝜉,𝜂)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝.(2.23)

The proof of Corollary 2.6 can be completed by setting 𝑏𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔1𝑖(𝑚,𝑛)𝑏1𝑖(𝑠,𝑡), 𝑐𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔1𝑖(𝑚,𝑛)𝑐1𝑖(𝑠,𝑡),𝑑𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔2𝑖(𝑚,𝑛)𝑑1𝑖(𝑠,𝑡),𝑒𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔2𝑖(𝑚,𝑛)𝑒1𝑖(𝑠,𝑡) in Theorem 2.5.

Corollary 2.7. Suppose that 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛),𝑏𝑖(𝑠,𝑡,𝑚,𝑛),𝑐𝑖(𝑠,𝑡,𝑚,𝑛),𝑑𝑖(𝑠,𝑡,𝑚,𝑛),𝑒𝑖(𝑠,𝑡,𝑚,𝑛) are defined as in Theorem 2.5. If, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛) satisfies 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖+(𝜉,𝜂,𝑚,𝑛)𝑢(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(,𝜉,𝜂,𝑚,𝑛)𝑢(𝜉,𝜂)(2.24) then 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛),(2.25) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(+𝜉,𝜂,𝑚,𝑛)𝑎(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(,𝜉,𝜂,𝑚,𝑛)𝑎(𝜉,𝜂)𝜇(𝑚,𝑛)=𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝐶(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(,𝜉,𝜂,𝑚,𝑛)𝐶(𝜉,𝜂)𝐶(𝑚,𝑛)=exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0,𝐵(𝑠,𝑡,𝑚,𝑛)𝐵(𝑠,𝑡,𝑚,𝑛)=𝑙1𝑖=1𝑏𝑖(𝑠,𝑡,𝑚,𝑛)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖.(𝜉,𝜂,𝑚,𝑛)(2.26)

Theorem 2.8. Suppose that 𝑤(𝑚,𝑛)+(Ω), 𝑢,𝑎,𝑏𝑖,𝑐𝑖,𝑑𝑖,𝑒𝑖,𝑝,𝑞𝑖,𝑟𝑖,𝑖,𝑗𝑖 are defined as in Theorem 2.5. Furthermore, assume that 𝑎(𝑚,𝑛) is nondecreasing in the first variable. If, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛) satisfies 𝑢𝑝(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑚1𝑠=𝑚0𝑤(𝑠,𝑛)𝑢𝑝+(𝑚,𝑛)𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖,(𝜉,𝜂)(2.27) then 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)𝑤(𝑚,𝑛)1/𝑝,(2.28) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑞𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝,(2.29)𝑏𝑖(𝑠,𝑡,𝑚,𝑛)=𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑞𝑖/𝑝,𝑐𝑖(𝑠,𝑡,𝑚,𝑛)=𝑐𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑟𝑖/𝑝,𝑖=1,2,,𝑙1,(2.30)𝑑𝑖(𝑠,𝑡,𝑚,𝑛)=𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑖/𝑝,𝑒𝑖(𝑠,𝑡,𝑚,𝑛)=𝑒𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑗𝑖/𝑝,𝑖=1,2,,𝑙2,(2.31)𝑤(𝑚,𝑛)=𝑚1𝑠=𝑚0[],1+𝑤(𝑠,𝑛)𝜇(𝑚,𝑛)=𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝐶(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝐶(𝜉,𝜂)(2.32)𝐶(𝑚,𝑛)=exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0,𝐵(𝑠,𝑡,𝑚,𝑛)(2.33)𝐵(𝑠,𝑡,𝑚,𝑛)=𝑙1𝑖=1𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝.(2.34)

Proof. Denote 𝑧(𝑚,𝑛)=𝑎(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(.𝜉,𝜂)(2.35) Then, we have 𝑢𝑝(𝑚,𝑛)𝑧(𝑚,𝑛)+𝑚1𝑠=𝑚0𝑤(𝑠,𝑛)𝑢𝑝(𝑚,𝑛).(2.36) Obviously 𝑧(𝑚,𝑛) is nondecreasing in the first variable. So by Lemma 2.3 we obtain 𝑢𝑝(𝑚,𝑛)𝑧(𝑚,𝑛)𝑚1𝑠=𝑚0[]1+𝑤(𝑠,𝑛)=𝑧(𝑚,𝑛)𝑤(𝑚,𝑛),(2.37) where 𝑤(𝑚,𝑛)=𝑚1𝑠=𝑚0[1+𝑤(𝑠,𝑛)]. Define 𝑣(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(.𝜉,𝜂)(2.38) Then, 𝑢(𝑚,𝑛)(𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛))𝑤(𝑚,𝑛)1/𝑝,(2.39) and, furthermore, by (2.39) and Lemma 2.1 we have 𝑣(𝑚,𝑛)𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖((𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))𝑤(𝑠,𝑡)𝑞𝑖/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐i((𝜉,𝜂,𝑚,𝑛)𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))𝑤(𝜉,𝜂)𝑟𝑖/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖((𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))𝑤(𝑠,𝑡)𝑖/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))𝑤(𝜉,𝜂)𝑗𝑖/𝑝𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑞𝑖/𝑝𝑞𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑤(𝜉,𝜂)𝑟𝑖/𝑝×𝑟𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)𝑖/𝑝𝑖𝑝𝐾(𝑞𝑖3𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑤(𝜉,𝜂)𝑗𝑖/𝑝×𝑗𝑖𝑝𝐾𝑗𝑖4𝑝/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖3𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝑗𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝=𝐻(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑣(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝,𝑣(𝜉,𝜂)(2.40) where 𝐻(𝑚,𝑛)=𝐽(𝑚,𝑛)+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0{𝑑𝑖(𝑠,𝑡,𝑚,𝑛)(𝑖/𝑝)𝐾(𝑖3𝑝)/𝑝𝑣(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑗𝑖/𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑣(𝜉,𝜂)}, and 𝐽(𝑚,𝑛),𝑏𝑖,𝑐𝑖,𝑑𝑖,𝑒𝑖 are defined in (2.29)–(2.31) respectively.
Similar to the process of (2.15)–(2.20) we deduce 𝑣(𝑚,𝑛)𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛),(2.41) where 𝜇(𝑚,𝑛),𝐶(𝑚,𝑛) are defined in (2.32) and (2.33).
Combining (2.39) and (2.41), we get the desired result.

Remark 2.9. If we set 𝑏𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔1𝑖(𝑚,𝑛)𝑏1𝑖(𝑠,𝑡),𝑐𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔1𝑖(𝑚,𝑛)𝑐1𝑖(𝑠,𝑡),𝑑𝑖(𝑠,𝑡,𝑚,𝑛)= 𝑔2𝑖(𝑚,𝑛)𝑑1𝑖(𝑠,𝑡),𝑒𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔2𝑖(𝑚,𝑛)𝑒1𝑖(𝑠,𝑡) or set 𝑝=𝑞𝑖=𝑟𝑖=𝑖=𝑗𝑖=1 in Theorem 2.8, then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we omit the details for them.

Theorem 2.10. Suppose that 𝑢,𝑎,𝑏𝑖,𝑐𝑖,𝑑𝑖,𝑒𝑖,𝑝,𝑞𝑖,𝑟𝑖,𝑖,𝑗𝑖 are defined as in Theorem 2.5. 𝐿𝑖,𝑇𝑖Ω×++,𝑖=1,2,,𝑙2, satisfies 0𝐿𝑖(𝑚,𝑛,𝑢)𝐿𝑖(𝑚,𝑛,𝑣)𝑇𝑖(𝑚,𝑛,𝑣)(𝑢𝑣) for 𝑢𝑣0. If, for (𝑚,𝑛)Ω, 𝑢(𝑚,𝑛) satisfies 𝑢𝑝(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝐿𝑖𝑠,𝑡,𝑢𝑖(+𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(,𝜉,𝜂)(2.42) then 𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)1/𝑝,(2.43) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑞𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝐿𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝𝑎(𝜉,𝜂)+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝,(2.44)𝜇(𝑚,𝑛)=𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑖3𝑝)/𝑝+𝐶(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝑗𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝,𝐶(𝜉,𝜂)(2.45)𝑑𝑖(𝑠,𝑡,𝑚,𝑛)=𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝑇𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝,𝑖=1,2,,𝑙2,(2.46)𝐶(𝑚,𝑛)=exp𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0,𝐵(𝑠,𝑡,𝑚,𝑛)(2.47)𝐵(𝑠,𝑡,𝑚,𝑛)=𝑙1𝑖=1𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝.(2.48)

Proof. Denote 𝑣(𝑚,𝑛)=𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑠,𝑡,𝑚,𝑛)𝑢𝑞𝑖(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑟𝑖+(𝜉,𝜂)𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝐿𝑖𝑠,𝑡,𝑢𝑖(+𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)𝑢𝑗𝑖(.𝜉,𝜂)(2.49) Then []𝑢(𝑚,𝑛)𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)1/𝑝,(2.50) and, furthermore, from Lemma 2.1 we have 𝑣(𝑚,𝑛)𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖(𝑞𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖(𝑠,𝑡,𝑚,𝑛)𝐿𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖𝐿(𝑠,𝑡,𝑚,𝑛)𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑖𝑝𝐾𝑖3/𝑝𝐿𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝐿𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝𝑞𝑖𝑝𝐾𝑞𝑖1/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖𝑟(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑟𝑖𝑝𝐾𝑟𝑖2/𝑝+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0𝑑𝑖𝑇(𝑠,𝑡,𝑚,𝑛)𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝𝑖𝑝𝐾(𝑖3𝑝)/𝑝×𝑣(𝑠,𝑡)+𝐿𝑖𝑠,𝑡,𝑖𝑝𝐾(𝑖3𝑝)/𝑝𝑎(𝑠,𝑡)+𝑝𝑖𝑝𝐾𝑖3/𝑝+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖𝑗(𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑗𝑖4𝑝)/𝑝(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝𝑗𝑖𝑝𝐾𝑗𝑖4/𝑝=𝐻(𝑚,𝑛)+𝑙1𝑖=1𝑚1𝑠=𝑚0𝑛1𝑡=𝑛0𝑏𝑖𝑞(𝑠,𝑡,𝑚,𝑛)𝑖𝑝𝐾(𝑞𝑖1𝑝)/𝑝+𝑣(𝑠,𝑡)𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑐𝑖(𝑟𝜉,𝜂,𝑚,𝑛)𝑖𝑝𝐾(𝑟𝑖2𝑝)/𝑝,𝑣(𝜉,𝜂)(2.51) where 𝐻(𝑚,𝑛)= 𝐽(𝑚,𝑛)+𝑙2𝑖=1𝑀1𝑠=𝑚0𝑁1𝑡=𝑛0{𝑑𝑖(𝑠,𝑡,𝑚,𝑛)(𝑖/𝑝)𝐾(𝑖3𝑝)/𝑝𝑣(𝑠,𝑡)+𝑠𝜉=𝑚0𝑡𝜂=𝑛0𝑒𝑖(𝜉,𝜂,𝑚,𝑛)(𝑗𝑖/𝑝)𝐾(𝑗𝑖4𝑝)/𝑝𝑣(𝜉,𝜂)} and 𝑑𝐽(𝑚,𝑛),𝑖(𝑠,𝑡,𝑚,𝑛) are defined in (2.44) and (2.46) respectively.
Similar to the process of (2.15)–(2.20) we deduce 𝑣(𝑚,𝑛)𝐽(𝑀,𝑁)1𝜇(𝑀,𝑁)𝐶(𝑀,𝑁),(2.52) where 𝜇(𝑚,𝑛),𝐶(𝑚,𝑛) are defined in (2.45) and (2.47) respectively.
Combining (2.50) and (2.52), we get the desired result.

Theorem 2.11. Suppose that