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

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 [3–30] 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, ℝ denotes the set of real numbers ℝ₊=[0,∞), and ℤ denotes the set of integers, while ℕ₀ denotes the set of nonnegative integers. Let Ω∶=([𝑚₀,𝑀]×[𝑛₀,𝑁])⋂ℤ², where 𝑚₀,𝑛₀∈ℤ and 𝑀,𝑁∈ℤ⋃{∞} are two constants. 𝑙₁,𝑙₂∈ℤ are two constants, and 𝐾_𝑖>0,𝑖=1,2,3,4, are all constants. If 𝑈 is a lattice, then we denote the set of all ℝ-valued functions on 𝑈 by ℘(𝑈) and denote the set of all ℝ₊-valued functions on 𝑈 by ℘₊(𝑈). Finally, for a function 𝑓∈℘₊(𝑈), we have ∑^𝑚1_𝑠=𝑚0𝑓=0 provided 𝑚₀>𝑚₁.

2. Main Results

Lemma 2.1 (see [15]). 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.(2.1)

Lemma 2.2. Suppose that 𝑢(𝑚,𝑛)∈℘₊(Ω),𝑏(𝑠,𝑡,𝑚,𝑛)∈℘₊(Ω²), 𝛼≥0 is a constant. If 𝑏 is nondecreasing in the third variable, then, for (𝑚,𝑛)∈Ω, 𝑢(𝑚,𝑛)≤𝛼+^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝑏(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)(2.2) implies that 𝑢(𝑚,𝑛)≤𝛼exp^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=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)+^𝑚−1_𝑠_=𝑚0𝑏(𝑠,𝑛)𝑢(𝑠,𝑛)(2.4) implies that 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)^𝑚−1_𝑠_=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), 𝑏_𝑖(𝑠,𝑡,𝑚,𝑛),𝑐_𝑖(𝑠,𝑡,𝑚,𝑛)∈℘₊(Ω²), 𝑖=1,2,…,𝑙₁, 𝑑_𝑖(𝑠,𝑡,𝑚,𝑛),𝑒_𝑖(𝑠,𝑡,𝑚,𝑛)∈℘₊(Ω²), 𝑖=1,2,…,𝑙₂with 𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑒_𝑖 nondecreasing in the last two variables. 𝑝,𝑞_𝑖,𝑟_𝑖 are nonnegative constants with 𝑝≥𝑞_𝑖,𝑝≥𝑟_𝑖,𝑖=1,2,…,𝑙₁,𝑝≠0, while ℎ_𝑖,𝑗_𝑖 are nonnegative constants with 𝑝≥ℎ_𝑖,𝑝≥𝑗_𝑖,𝑖=1,2,…,𝑙₂. If, for (𝑚,𝑛)∈Ω, 𝑢(𝑚,𝑛) satisfies 𝑢^𝑝(𝑚,𝑛)≤𝑎(𝑚,𝑛)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦,(2.6) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)^1/𝑝,(2.7) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖p𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑗^𝑖𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭,(2.8)𝜇(𝑚,𝑛)=^𝑙2_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝐶(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭,(2.9)𝐶(𝑚,𝑛)=exp^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛),(2.10)𝐵(𝑠,𝑡,𝑚,𝑛)=^𝑙1_𝑖=1⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}⎤⎥⎥⎦.(2.11)

Proof. Denote 𝑣(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(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 𝑢^𝑞𝑖(𝑚,𝑛)≤[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]^{𝑞𝑖/𝑝}≤𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+𝑝−𝑞_𝑖[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]𝑝𝐾^{𝑞𝑖1/𝑝}𝑢_𝑟𝑖,𝑖=1,2,…,𝑙₁,(𝑚,𝑛)≤[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]^{𝑟𝑖/𝑝}≤𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}+𝑝−𝑟_𝑖[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]𝑝𝐾^{𝑟𝑖2/𝑝}𝑢_ℎ𝑖,𝑖=1,2,…,𝑙₁,(𝑚,𝑛)≤[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]^ℎ𝑖/𝑝≤ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝑝−ℎ_𝑖[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]𝑝𝐾^{ℎ𝑖3/𝑝}𝑢_𝑗𝑖,𝑖=1,2,…,𝑙₂,(𝑚,𝑛)≤[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]^{𝑗𝑖/𝑝}≤𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}+𝑝−𝑗_𝑖[𝑎(𝑚,𝑛)+𝑣(𝑚,𝑛)]𝑝𝐾^{𝑗𝑖4/𝑝},𝑖=1,2,…,𝑙₂.(2.14) So 𝑣(𝑚,𝑛)≤^𝑙1_𝑖=1^m−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞ⁱ𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖=𝐻(𝑚,𝑛)+_𝑙1𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑣(𝜉,𝜂)⎤⎥⎥⎦,(2.15) where 𝐻(𝑚,𝑛)=𝐽(𝑚,𝑛)+∑^𝑙2_𝑖=1∑^{𝑀−1𝑠=𝑚}₀∑^{𝑁−1𝑡=𝑛}₀{𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)(ℎ_𝑖/𝑝)𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑣(𝑠,𝑡)+∑^{𝑠𝜉=𝑚}₀∑^{𝑡𝜂=𝑛}₀𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)(𝑗_𝑖/𝑝)𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑣(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)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}≤𝐻(𝑀,𝑁)+_𝑙1𝑣(𝜉,𝜂)⎤⎥⎥⎦_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}=𝐻(𝑀,𝑁)+_𝑚−1⎤⎥⎥⎦𝑣(𝑠,𝑡)_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛)𝑣(𝑠,𝑡),(2.16) where 𝐵(𝑠,𝑡,𝑚,𝑛) is defined in (2.11).
Since 𝑏_𝑖(𝑠,𝑡,𝑚,𝑛),𝑐_𝑖(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)exp^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=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)+^𝑙2_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑀,𝑁)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑀,𝑁)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}≤𝐽(𝑀,𝑁)+_𝑙2𝑣(𝜉,𝜂)⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑀,𝑁)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝐻(𝑀,𝑁)𝐶(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖𝜉,𝜂,𝑚₁,𝑛₁𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}=𝐽(𝑀,𝑁)+𝐻(𝑀,𝑁)_𝑙2𝐻(𝑀,𝑁)𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑀,𝑁)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝐶(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑀,𝑁)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}=𝐽(𝑀,𝑁)+𝐻(𝑀,𝑁)𝜇(𝑀,𝑁),𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭(2.18) where 𝜇(𝑚,𝑛) is defined in (2.9). Then, 𝐻(𝑀,𝑁)≤𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁).(2.19) Combining (2.17) and (2.19) we deduce 𝑣(𝑚,𝑛)≤𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛).(2.20) Then, combining (2.13) and (2.20), we obtain the desired result.

Corollary 2.6. Suppose that 𝑔_1𝑖(𝑚,𝑛),𝑔_2𝑖(𝑚,𝑛),𝑏_1𝑖(𝑚,𝑛),𝑐_1𝑖(𝑚,𝑛)∈℘₊(Ω),𝑖=1,2,…,𝑙₁ with 𝑔_1𝑖,𝑔_2𝑖 nondecreasing in every variable. 𝑑_1𝑖(𝑚,𝑛),𝑒_1𝑖(𝑚,𝑛)∈℘₊(Ω),𝑖=1,2,…,𝑙₂. 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛),𝑝,𝑞_𝑖,𝑟_𝑖,ℎ_𝑖,𝑗_𝑖 are defined as in Theorem 2.5. If, for (𝑚,𝑛)∈Ω, 𝑢(𝑚,𝑛) satisfies 𝑢^𝑝(𝑚,𝑛)≤𝑎(𝑚,𝑛)+^𝑙1_𝑖=1𝑔_1𝑖(𝑚,𝑛)^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_1𝑖(𝑠,𝑡)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_1𝑖(𝜉,𝜂)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_𝑖=1𝑔_2𝑖(𝑚,𝑛)^𝑀−1_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_1𝑖(𝑠,𝑡)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_1𝑖(𝜉,𝜂)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦,(2.21) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑀,𝑁)^1/𝑝,(2.22) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_𝑖=1𝑔_1𝑖(𝑚,𝑛)^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_1𝑖(𝑠,𝑡)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_1𝑖(𝜉,𝜂)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_𝑖=1𝑔_2𝑖(𝑚,𝑛)^𝑀−1_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_1𝑖(𝑠,𝑡)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_1𝑖(𝜉,𝜂)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑗^𝑖𝜇(𝑚,𝑛)=_𝑙2𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭,_𝑖=1⎧⎪⎨⎪⎩𝑔_2𝑖(𝑚,𝑛)^𝑀−1_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_1𝑖(𝑠,𝑡)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝐶(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_1𝑖(𝜉,𝜂)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝐶(𝑚,𝑛)=exp_𝑚−1𝐶(𝜉,𝜂)⎤⎥⎥⎦⎫⎪⎬⎪⎭,_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛)=_𝑙1𝐵(𝑠,𝑡,𝑚,𝑛),_𝑖=1𝑔_1𝑖(𝑚,𝑛)⎡⎢⎢⎣𝑏_1𝑖(𝑠,𝑡)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_1𝑖(𝜉,𝜂)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}⎤⎥⎥⎦.(2.23)

The proof of Corollary 2.6 can be completed by setting 𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_1𝑖(𝑚,𝑛)𝑏_1𝑖(𝑠,𝑡), 𝑐_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_1𝑖(𝑚,𝑛)𝑐_1𝑖(𝑠,𝑡),𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_2𝑖(𝑚,𝑛)𝑑_1𝑖(𝑠,𝑡),𝑒_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_2𝑖(𝑚,𝑛)𝑒_1𝑖(𝑠,𝑡) in Theorem 2.5.

Corollary 2.7. Suppose that 𝑢(𝑚,𝑛),𝑎(𝑚,𝑛),𝑏_𝑖(𝑠,𝑡,𝑚,𝑛),𝑐_𝑖(𝑠,𝑡,𝑚,𝑛),𝑑_𝑖(𝑠,𝑡,𝑚,𝑛),𝑒_𝑖(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)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢(𝜉,𝜂)⎤⎥⎥⎦,(2.24) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛),(2.25) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑎(𝜉,𝜂)_𝑙2⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑎(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝜇(𝑚,𝑛)=_𝑙2𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑎(𝜉,𝜂)⎫⎪⎬⎪⎭,_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐶(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝐶(𝑚,𝑛)=exp_𝑚−1𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭,_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛)=_𝑙1𝐵(𝑠,𝑡,𝑚,𝑛),_𝑖=1⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)⎤⎥⎥⎦.(2.26)

Theorem 2.8. Suppose that 𝑤(𝑚,𝑛)∈℘₊(Ω), 𝑢,𝑎,𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑒_𝑖,𝑝,𝑞_𝑖,𝑟_𝑖,ℎ_𝑖,𝑗_𝑖 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 𝑢^𝑝(𝑚,𝑛)≤𝑎(𝑚,𝑛)+^𝑚−1_𝑠_=𝑚0+𝑤(𝑠,𝑛)𝑢^𝑝(𝑚,𝑛)_𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦,(2.27) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)𝑤(𝑚,𝑛)^1/𝑝,(2.28) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑗^𝑖𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭,(2.29)𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^{𝑞𝑖/𝑝},𝑐_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑐_𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^{𝑟𝑖/𝑝},𝑖=1,2,…,𝑙₁,(2.30)𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^ℎ𝑖/𝑝,𝑒_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑒_𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^{𝑗𝑖/𝑝},𝑖=1,2,…,𝑙₂,(2.31)𝑤(𝑚,𝑛)=^𝑚−1_𝑠_=𝑚0[1+𝑤(𝑠,𝑛)],𝜇(𝑚,𝑛)=^𝑙2_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝐶(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝐶(𝑚,𝑛)=exp_𝑚−1𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭(2.32)_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛),(2.33)𝐵(𝑠,𝑡,𝑚,𝑛)=^𝑙1_𝑖=1⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}⎤⎥⎥⎦.(2.34)

Proof. Denote 𝑧(𝑚,𝑛)=𝑎(𝑚,𝑛)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦.(2.35) Then, we have 𝑢^𝑝(𝑚,𝑛)≤𝑧(𝑚,𝑛)+^𝑚−1_𝑠_=𝑚0𝑤(𝑠,𝑛)𝑢^𝑝(𝑚,𝑛).(2.36) Obviously 𝑧(𝑚,𝑛) is nondecreasing in the first variable. So by Lemma 2.3 we obtain 𝑢^𝑝(𝑚,𝑛)≤𝑧(𝑚,𝑛)^𝑚−1_𝑠_=𝑚0[1+𝑤(𝑠,𝑛)]=𝑧(𝑚,𝑛)𝑤(𝑚,𝑛),(2.37) where 𝑤(𝑚,𝑛)=∏^{𝑚−1𝑠=𝑚}₀[1+𝑤(𝑠,𝑛)]. Define 𝑣(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(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)≤^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))𝑤(𝑠,𝑡)^{𝑞𝑖/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_i(𝜉,𝜂,𝑚,𝑛)(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))𝑤(𝜉,𝜂)^{𝑟𝑖/𝑝}+_𝑙2⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))𝑤(𝑠,𝑡)^ℎ𝑖/𝑝+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))𝑤(𝜉,𝜂)^{𝑗𝑖/𝑝}≤_𝑙1⎫⎪⎬⎪⎭_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝑏^𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^{𝑞𝑖/𝑝}𝑞𝑝^𝑖𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑤(𝜉,𝜂)^{𝑟𝑖/𝑝}×𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+𝑝𝐾^{𝑟𝑖2/𝑝}_𝑙2_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0𝑑^𝑖(𝑠,𝑡,𝑚,𝑛)𝑤(𝑠,𝑡)^ℎ𝑖/𝑝ℎ𝑝^𝑖𝐾^{(𝑞𝑖3−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑤(𝜉,𝜂)^{𝑗𝑖/𝑝}×𝑗^𝑖𝑝𝐾^{𝑗𝑖4−𝑝/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖=𝑝𝐾^{𝑗𝑖4/𝑝}_𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(𝑞𝑖3−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖=𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭𝐻(𝑚,𝑛)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑣(𝜉,𝜂)⎤⎥⎥⎦,(2.40) where 𝐻∑(𝑚,𝑛)=𝐽(𝑚,𝑛)+∑^𝑙2_𝑖=1∑^{𝑀−1𝑠=𝑚}₀∑^{𝑁−1𝑡=𝑛}₀{𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)(ℎ_𝑖/𝑝)𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑣(𝑠,𝑡)+_{𝑠𝜉=𝑚0}∑^{𝑡𝜂=𝑛}₀𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗_𝑖/𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑣(𝜉,𝜂)}, and 𝐽(𝑚,𝑛),𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑒_𝑖 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)1−𝜇(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 𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_1𝑖(𝑚,𝑛)𝑏_1𝑖(𝑠,𝑡),𝑐_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_1𝑖(𝑚,𝑛)𝑐_1𝑖(𝑠,𝑡),𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)= 𝑔_2𝑖(𝑚,𝑛)𝑑_1𝑖(𝑠,𝑡),𝑒_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑔_2𝑖(𝑚,𝑛)𝑒_1𝑖(𝑠,𝑡) or set 𝑝=𝑞_𝑖=𝑟_𝑖=ℎ_𝑖=𝑗_𝑖=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,𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑒_𝑖,𝑝,𝑞_𝑖,𝑟_𝑖,ℎ_𝑖,𝑗_𝑖 are defined as in Theorem 2.5. 𝐿_𝑖,𝑇_𝑖∶Ω×ℝ₊→ℝ₊,𝑖=1,2,…,𝑙₂, satisfies 0≤𝐿_𝑖(𝑚,𝑛,𝑢)−𝐿_𝑖(𝑚,𝑛,𝑣)≤𝑇_𝑖(𝑚,𝑛,𝑣)(𝑢−𝑣) for 𝑢≥𝑣≥0. If, for (𝑚,𝑛)∈Ω, 𝑢(𝑚,𝑛) satisfies 𝑢^𝑝(𝑚,𝑛)≤𝑎(𝑚,𝑛)+^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦,(2.42) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)^1/𝑝,(2.43) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑗^𝑖𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭,(2.44)𝜇(𝑚,𝑛)=^𝑙2_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)ℎ^𝑖𝑝𝐾^{(ℎ𝑖3−𝑝)/𝑝}+𝐶(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝐶(𝜉,𝜂)⎫⎪⎬⎪⎭,(2.45)𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)=𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑇_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖𝑝𝐾^{ℎ𝑖3/𝑝},𝑖=1,2,…,𝑙²,(2.46)𝐶(𝑚,𝑛)=exp^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝐵(𝑠,𝑡,𝑚,𝑛),(2.47)𝐵(𝑠,𝑡,𝑚,𝑛)=^𝑙1_𝑖=1⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}⎤⎥⎥⎦.(2.48)

Proof. Denote 𝑣(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(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)≤^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−ℎ^𝑖+_𝑠𝑝𝐾^{ℎ𝑖3/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖≤_𝑙1𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−ℎ^𝑖−𝐿_𝑖𝑝𝐾^{ℎ𝑖3/𝑝}𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+𝐿_𝑖𝑝𝐾^{ℎ𝑖3/𝑝}𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+𝑝𝐾^{ℎ𝑖3/𝑝}_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖≤_𝑙1𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}(𝑎(𝑠,𝑡)+𝑣(𝑠,𝑡))+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝑇_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖𝑝𝐾^{ℎ𝑖3/𝑝}ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}×𝑣(𝑠,𝑡)+𝐿_𝑖𝑠,𝑡,ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+𝑝𝐾^{ℎ𝑖3/𝑝}_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}(𝑎(𝜉,𝜂)+𝑣(𝜉,𝜂))+𝑝−𝑗^𝑖=𝐻(𝑚,𝑛)+_𝑙1𝑝𝐾^{𝑗𝑖4/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}+𝑣(𝑠,𝑡)_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑣(𝜉,𝜂)⎤⎥⎥⎦,(2.51) where 𝐻(𝑚,𝑛)= 𝐽(𝑚,𝑛)+∑^𝑙2_𝑖=1∑^{𝑀−1𝑠=𝑚}₀∑^{𝑁−1𝑡=𝑛}₀{𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)(ℎ_𝑖/𝑝)𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑣(𝑠,𝑡)+∑^{𝑠𝜉=𝑚}₀∑^{𝑡𝜂=𝑛}₀𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)(𝑗_𝑖/𝑝)𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑣(𝜉,𝜂)} and 𝐽(𝑚,𝑛),𝑑_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)1−𝜇(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 𝑤(𝑚,𝑛)∈℘₊(Ω), 𝑢,𝑎,𝑏_𝑖,𝑐_𝑖,𝑑_𝑖,𝑒_𝑖,𝑝,𝑞_𝑖,𝑟_𝑖,ℎ_𝑖,𝑗_𝑖 are defined as in Theorem 2.5. Furthermore, assume 𝑎(𝑚,𝑛) is nondecreasing in the first variable. 𝐿_𝑖,𝑇_𝑖,𝑖=1,2,…,𝑙₂, are defined as in Theorem 2.10. If, for (𝑚,𝑛)∈Ω, 𝑢(𝑚,𝑛) satisfies 𝑢^𝑝(𝑚,𝑛)≤𝑎(𝑚,𝑛)+^𝑚−1_𝑠_=𝑚0+𝑤(𝑠,𝑛)𝑢^𝑝(𝑚,𝑛)_𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎡⎢⎢⎣𝑏_𝑖(𝑠,𝑡,𝑚,𝑛)𝑢^𝑞𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0+𝑐_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑟𝑖(𝜉,𝜂)_𝑙2⎤⎥⎥⎦_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎡⎢⎢⎣𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖s,𝑡,𝑢^ℎ𝑖(𝑠,𝑡)+^𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖(𝜉,𝜂,𝑚,𝑛)𝑢^𝑗𝑖(𝜉,𝜂)⎤⎥⎥⎦,(2.53) then 𝑢(𝑚,𝑛)≤𝑎(𝑚,𝑛)+𝐽(𝑀,𝑁)1−𝜇(𝑀,𝑁)𝐶(𝑚,𝑛)𝑤(𝑚,𝑛)^1/𝑝,(2.54) provided that 𝜇(𝑀,𝑁)<1, where 𝐽(𝑚,𝑛)=^𝑙1_{𝑖=1𝑚−1}_𝑠=𝑚0^𝑛−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑏̃_𝑖(𝑠,𝑡,𝑚,𝑛)𝑞^𝑖𝑝𝐾^{(𝑞𝑖1−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−𝑞^𝑖+_𝑠𝑝𝐾^{𝑞𝑖1/𝑝}_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑐̃_𝑖(𝜉,𝜂,𝑚,𝑛)𝑟^𝑖𝑝𝐾^{(𝑟𝑖2−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−𝑟^𝑖+_𝑙2𝑝𝐾^{𝑟𝑖2/𝑝}⎫⎪⎬⎪⎭_{𝑖=1𝑀−1}_𝑠=𝑚0^𝑁−1_𝑡=𝑛0⎧⎪⎨⎪⎩𝑑_𝑖(𝑠,𝑡,𝑚,𝑛)𝐿_𝑖𝑠,𝑡,𝑤(𝑠,𝑡)^ℎ𝑖/𝑝ℎ𝑝^𝑖𝐾^{(ℎ𝑖3−𝑝)/𝑝}𝑎(𝑠,𝑡)+𝑝−ℎ^𝑖+𝑝𝐾^{ℎ𝑖3/𝑝}_𝑠_𝜉=𝑚0^𝑡_𝜂=𝑛0𝑒_𝑖̃(𝜉,𝜂,𝑚,𝑛)𝑗^𝑖𝑝𝐾^{(𝑗𝑖4−𝑝)/𝑝}𝑎(𝜉,𝜂)+𝑝−