A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

Abstract

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

1. Introduction

The Gronwall-Bellman inequality [1, 2] plays an important role in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases from literatures (see, e.g., [3–18]). In 1956, Bihari [3] discussed the integral inequality 𝑢(𝑡)≤𝑐+^𝑡₀𝑓(𝑠)𝜔(𝑢(𝑠))𝑑𝑠,𝑡≥0, (1) where 𝑐>0 is a constant, 𝑓 is a continuous and nonnegative function, and 𝜔 is a continuous and nondecreasing positive function. Replacing 𝑡 by a function 𝑏(𝑡) in (1), Lipovan [4] investigated the retarded integral inequality 𝑢(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝑠)𝜔(𝑢(𝑠+))𝑑𝑠_{𝑏𝑡𝑏(𝑡)0}𝑔(𝑠)𝜔(𝑢(𝑠))𝑑𝑠,𝑡₀≤𝑡<𝑡₁. (2) Their results were further generalized by Agarwal et al. [5] to the inequality 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1^𝑏𝑖_𝑏^(𝑡)_𝑖𝑡0𝑓_𝑖(𝑡,𝑠)𝜔_𝑖(𝑢(𝑠))𝑑𝑠,𝑡₀≤𝑡<𝑡₁, (3) where the constant 𝑐 is replaced with a function 𝑎(𝑡), 𝑏_𝑖’s are continuously differentiable and nondecreasing functions, and 𝜔_𝑖’s are continuous and nondecreasing positive functions such that 𝜔₁∝𝜔₂∝⋯∝𝜔_𝑛, (4) that is, each ratio 𝜔_𝑖+1/𝜔_𝑖 is also nondecreasing on 𝐴⊆ℝ⧵{0}, called in [6] that 𝜔_𝑖+1 is stronger nondecreasing than 𝜔_𝑖. On the basis of this work, Wang [7] considered the inequality of two variables 𝑢^𝑝+𝑥,𝑦≤𝑎𝑥,𝑦_{𝑛𝑖=1𝑏𝑖𝑏(𝑥)𝑖𝑥0𝑐𝑖𝑐(𝑦)𝑖𝑦0}𝑓_𝑖𝑥,𝑦,𝑠,𝑡𝜔_𝑖(𝑢(𝑠,𝑡))𝑑𝑡𝑑𝑠, (5) where the functions 𝑎, 𝑓_𝑖, and 𝜔_𝑖 are not required to be monotone, and those 𝜔_𝑖’s are not required to be stronger monotone than the one after the next as shown in (4). This inequality belongs to both the case of multivariables, to which great attentions [7–11] have been paid, and to the case that the left-hand side is a composition of the unknown function with a known function, in which Ou-Iang's idea [19] was applied [11–14]. He applied a technique of monotonization to construct a sequence of functions, made each function possess stronger monotonization than the previous one, and gave an estimate for the unknown function 𝑢.

On the other aspect, many problems in the control theory can be modeled in the form of differential equations with the maxima of the unknown function [20–22]. In connection with the development of the theory of differential equations with maxima (see, e.g., [20, 21, 23]) and partial differential equations with maxima [24, 25], a new type of integral inequalities with maxima is required, respectively. There have been given some results for integral inequalities containing the maxima of the unknown function [23, 26–28]. Concretely, in 2012, Bohner et al. [26] discussed the following system of integral inequalities: 𝜑(𝑢(𝑡))≤𝑎(𝑡)+^𝑚_𝑖=1^𝛼𝑖_𝛼^(𝑡)_𝑖𝑡0𝑓_𝑖(𝑠)𝑢^𝑝(𝑠)𝜔_𝑖(𝑢(𝑠+))𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑡)𝑗𝑡0}𝑓_𝑗(𝑠)𝑢^𝑝(𝑠)×𝜔_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑔(𝑢(𝜉𝑢(𝑡)(𝑡),))𝑑𝑠,≤𝜓 (6) where 𝑎, 𝑓_𝑖’s, 𝜔_𝑖’s, 𝜑, and 𝜓 are nonnegative continuous functions and 𝛼_𝑖’s are nonnegative continuously differentiable and nondecreasing functions. They required that 𝑎(𝑡)≥1, 𝜑 is 𝐶¹ on ℝ₊∶=[0,+∞) and increasing such that 𝜑(𝑡𝑥)≥𝑡𝜑(𝑥) for 0≤𝑡≤1, and 𝜔_𝑖 satisfies the following: (i) 𝜔_𝑖∈𝐶¹(ℝ₊,ℝ₊) is an increasing function, and (ii) 𝜔_𝑖(𝑡𝑥)≥𝑡𝜔_𝑖(𝑥) for all 0≤𝑡≤1 and 𝑥>0. Bainov and Hristova [23] considered the following system: 𝑢𝑥,𝑦≤𝑎𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑢^𝑝()+𝑠,𝑡𝑑𝑡𝑑𝑠_{𝛼𝑥𝛼(𝑥)0𝑦𝑦0}ℎ(𝑠,𝑡)max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢^𝑝()𝑢,𝜉,𝑡𝑑𝑡𝑑𝑠,𝑥,𝑦≤𝜓𝑥,𝑦 (7) where 𝑎(𝑥,𝑦) is nonnegative and nondecreasing in both of its arguments, 𝑓, ℎ, and 𝜓 are continuous and nonnegative functions, and 𝑝∈(0,1].

In this paper, we consider the system of integral inequalities as follows: 𝜑𝑢+𝑥,𝑦≤𝑎𝑥,𝑦_{𝑛𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0}𝑓_𝑖𝑥,𝑦,𝑠,𝑡𝜔_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑛+𝑚𝑗=𝑛+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑗𝛽(𝑦)𝑗𝑦0}𝑓_𝑗𝑥,𝑦,𝑠,𝑡×𝜔_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑔(𝑢(∈𝑥𝜉,𝑡))𝑑𝑡𝑑𝑠,𝑥,𝑦₀,𝑥₁×𝑦₀,𝑦₁,𝑢𝑥,𝑦≤𝜓𝑥,𝑦,𝑥,𝑦∈𝛼_∗𝑥₀−ℎ,𝑥₀×𝑦₀,𝑦₁, (8) where 𝑎, 𝑓_𝑖’s, 𝜔_𝑖’s, and 𝑔 are continuous and nonnegative functions, 𝛼_𝑖’s and 𝛽_𝑖’s are nonnegative continuously differentiable and nondecreasing functions, and 𝛼_∗(𝑥₀)∶=min_{1≤𝑖≤𝑚+𝑛}𝛼_𝑖(𝑥₀). As required in previous works [27–29], we suppose that 0≤𝛼_𝑖(𝑡)≤𝑡, 0≤𝛽_𝑖(𝑡)≤𝑡, ℎ>0 is constant. In this paper, we require neither monotonicity of 𝑎, 𝜔_𝑖's, 𝑓_𝑖's, and 𝑔 nor 𝑎(𝑥,𝑦)≥1. We monotonize those 𝜔_𝑖’s to make a sequence of functions in which each one possesses stronger monotonicity than the previous one so as to give an estimation for the unknown function. We can use our result to discuss inequalities (6) and (7), giving the stronger results under weaker conditions. We finally apply the obtained result to a boundary value problem of a partial differential equation with maxima for uniqueness.

2. Main Result

Consider system (8) of integral inequalities with 𝑥₀<𝑥₁ and 𝑦₀<𝑦₁ in ℝ₊∶=[0,∞). Let Λ∶=[𝑥₀,𝑥₁)×[𝑦₀,𝑦₁), Ω∶=[𝛼_∗(𝑥₀)−ℎ,𝑥₀]×[𝑦₀,𝑦₁). Suppose that(H₁)𝛼_𝑖∶[𝑥₀,𝑥₁)→ℝ₊(𝑖=1,2,…,𝑚+𝑛) and 𝛽_𝑖∶[𝑦₀,𝑦₁)→[𝑦₀,𝑦₁), 𝑖=1,2,…,𝑚+𝑛, are nondecreasing such that 𝛼_𝑖(𝑥)≤𝑥 on [𝑥₀,𝑥₁),  𝛽_𝑖(𝑦)≤𝑦 on [𝑦₀,𝑦₁) and 𝛽_𝑖(𝑦₀)=𝑦₀; (H₂) all 𝑓_𝑖’s (𝑖=1,2,…,𝑚+𝑛) are continuous and nonnegative functions on Λ×[𝛼_∗(𝑥₀),𝑥₁)×[𝑦₀,𝑦₁);(H₃)𝑔,𝜑∶ℝ₊→ℝ₊ and 𝜓∶[𝛼_∗(𝑥₀)−ℎ,𝑥₁)→ℝ₊ are continuous, and 𝜑 is strictly increasing such that lim_𝑡→∞𝜑(𝑡)=+∞; (H₄) all 𝜔_𝑖’s (𝑖=1,2,…,𝑚+𝑛) are continuous on ℝ₊ and positive on (0,+∞); (H₅)𝑎(𝑥,𝑦) is a continuous and nonnegative function on Λ.

For those 𝜔_𝑖’s given in (H₄), define 𝜔^∼_𝑖(𝑡), 𝑖=1,…,𝑚+𝑛, inductively by 𝜔^∼₁(𝑡)∶=max_{[]𝜏∈0,𝑡}𝜔₁(𝜏)_∼,𝜔_𝑖+1(𝑡)∶=max_{[]𝜏∈0,𝑡}𝜔_𝑖+1(𝜏)_∼𝜔_𝑖(𝜏)+𝜖_𝑖^∼𝜔_𝑖(𝑡), (9) for 𝑖=1,2,…,𝑚−1 and 𝜔^∼_𝑚+1(𝑡)∶=max_{[]𝜏∈0,𝑡}𝜔_𝑚+1max_{𝑠∈[0,𝜏]}𝑔(𝑠)_∼𝜔_𝑚(𝜏)+𝜖_𝑚^∼𝜔_𝑚(𝑡),_∼𝜔_𝑗+1(𝑡)∶=max_{[]𝜏∈0,𝑡}𝜔_𝑗+1max_{𝑠∈[0,𝜏]}𝑔(𝑠)_∼𝜔_𝑗(𝜏)+𝜖_𝑗^∼𝜔_𝑗(𝑡), (10) for 𝑗=𝑚+1,…,𝑚+𝑛, where 𝜔_𝑗(𝑡)∶=max_{𝜏∈[0,𝑡]}{𝜔_𝑗(𝜏)} for 𝑗=𝑚+1,…,𝑚+𝑛, 𝜖_𝑖∶=𝜀₁ if 𝜔^∼_𝑖(0)=0 or ∶=0 if 𝜔^∼_𝑖(0)≠0 for 𝑖=1,2,…,𝑚+𝑛−1, and 𝜀₁>0 be a given very small constant.

Theorem 1. Suppose that (𝐻₁)--(𝐻₅) hold, max_{𝑠∈[𝛼∗(𝑥0)−ℎ,𝑥0]}𝜓(𝑠,𝑦)≤𝜑⁻¹(𝑎(𝑥₀,𝑦)) for all 𝑦∈[𝑦₀,𝑦₁) and 𝑢∈𝐶(Ω,ℝ₊) satisfies the system (8) of integral inequalities. Then, 𝑢𝑥,𝑦≤𝜑⁻¹𝑊⁻¹_𝑚+𝑛Ω_𝑚+𝑛𝑥,𝑦, (11) for all (𝑥,𝑦)∈[𝑥₀,𝑋₁)×[𝑦₀,𝑌₁), where Ω_𝑖𝑥,𝑦∶=𝑊_𝑖𝑟_𝑖𝑥,𝑦+^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0max_{()∈𝑥𝜄,𝜉0×𝑦,𝑥0,𝑦}𝑓_𝑖(𝜄,𝜉,𝑠,𝑡)𝑑𝑡𝑑𝑠, (12)𝑊_𝑖⁻¹ is the inverse of the function 𝑊_𝑖(𝑢)∶=^𝑢_𝑢𝑖𝑑𝑠_∼𝜔_𝑖𝜑⁻¹(𝑠),𝑢≥𝑢_𝑖,𝑖=1,2,…,𝑚+𝑛, (13)𝑢_𝑖>0 is a given constant, 𝜔^∼_𝑖 is defined just before the theorem, and 𝑟_𝑖(𝑥,𝑦) is defined recursively by 𝑟₁𝑥,𝑦=max_{()∈𝑥𝜄,𝜉0×𝑦,𝑥0,𝑦}𝑎(),𝑟𝜄,𝜉_𝑖+1𝑥,𝑦=𝑊_𝑖⁻¹Ω_𝑖𝑥,𝑦, (14) for 𝑖=1,2,…,𝑚+𝑛−1, and 𝑋₁∈[𝑥₀,𝑥₁),𝑌₁∈[𝑦₀,𝑦₁) are chosen such that Ω_𝑖𝑋₁,𝑌₁≤^∞_𝑢𝑖𝑑𝑠_∼𝜔_𝑖𝜑⁻¹(𝑠), (15) for 𝑖=1,2,…,𝑚+𝑛.

For the special choice that 𝑛=𝑚=1, 𝜔₁(𝑠)=𝑠^𝑝, 𝜔₂(𝑠)=𝑠, 𝑓₁(𝑥,𝑦,𝑠,𝑡)=𝑓(𝑠,𝑡), 𝑓₂(𝑥,𝑦,𝑠,𝑡)=ℎ(𝑠,𝑡), 𝛼₁(𝑠)=𝑠, 𝛼₂(𝑠)=𝛼(𝑠), 𝑔(𝑠)=𝑠^𝑝, and 𝛽₁(𝑠)=𝛽₂(𝑠)=𝑠, where 𝛼 is a nonnegative continuously differentiable and nondecreasing function, Theorem 1 gives an estimate for the unknown 𝑢 in the system (7). we require neither the monotonicity of 𝑎 nor the monotonicity of 𝜔_𝑖. Obviously, Lemma 2 and Theorem 1 are applicable to more general forms than Corollary 2.3.4 in [23]. Even if 𝜔_𝑖(𝑠) is enlarged to max_{1≤𝑖≤𝑚+𝑛}𝜔_𝑖(𝑠) such that (8) is changed into the form of (2.1) in [29], where 𝑚=𝑛=1, our theorem gives a better estimate. For example, the system of inequalities 𝑢𝑥,𝑦≤3+4^𝑥₀^𝑦₀𝑡𝑠𝑢(𝑠,𝑡)+1𝑑𝑡𝑑𝑠+4^√𝑥₀^𝑦₀𝑡𝑠max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢()∈𝜉,𝑡+1𝑑𝑡𝑑𝑠,𝑥,𝑦0,𝑥₁×0,𝑦₁,𝑢𝑥,𝑦≤𝑥+3,𝑥,𝑦∈[−ℎ,0]×0,𝑦₁, (16) implies that 𝑢𝑥,𝑦≤3+4^𝑥₀^𝑦₀𝑡𝑠(𝑢(𝑠,𝑡)+1)𝑑𝑡𝑑𝑠+4^√𝑥₀^𝑦₀𝑡𝑠max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢()∈𝜉,𝑡+1𝑑𝑡𝑑𝑠,𝑥,𝑦0,𝑥₁×0,𝑦₁,𝑢𝑥,𝑦≤𝑥+3,𝑥,𝑦∈[−ℎ,0]×0,𝑦₁, (17) by enlarging √𝑠+1 to 𝑠+1. Applying Theorem 1, we obtain 𝑢≤𝑥𝑥,𝑦²𝑦²+4²4𝑒^𝑥𝑦2,𝑥,𝑦∈0,𝑥₁×0,𝑦₁. (18) On the other hand, Theorem 2.2 of [29] gives from (17) that 𝑢(𝑡)≤4𝑒^{(𝑥2𝑦2+𝑥𝑦2)},𝑥,𝑦∈0,𝑥₁×0,𝑦₁. (19) Clearly, (18) is sharper than (19) for large 𝑥 and 𝑦.

In order to prove Theorem 1, we need the following lemma.

Lemma 2. Suppose that (C1)𝛼_𝑖∶[𝑥₀,𝑥₁)→ℝ₊(𝑖=1,2,…,𝑚+𝑛) and 𝛽_𝑖∶[𝑦₀,𝑦₁)→ℝ₊(𝑖=1,2,…,𝑚+𝑛) are nondecreasing such that 𝛼_𝑖(𝑥)≤𝑥 on [𝛼_∗(𝑥₀),𝑥₁) and 𝛽_𝑖(𝑦)≤𝑦 on [𝑦₀,𝑦₁) and 𝛽_𝑖(𝑦₀)=𝑦₀;(C2)𝜓∈𝐶([𝛼_∗(𝑥₀)−ℎ,𝑥₁)×[𝑦₀,𝑦₁),ℝ₊), 𝑏_𝑖∈𝐶([𝛼_∗(𝑥₀),𝑥₁)×[𝑦₀,𝑦₁),ℝ₊) for 𝑖=1,2,…,𝑚+𝑛; (C3) all ℎ_𝑖’s (𝑖=1,2,…,𝑚+𝑛) are continuous and nondecreasing on ℝ₊ and positive on (0,+∞) such that ℎ₁∝ℎ₂∝…∝ℎ_𝑚+𝑛; (C4)𝑎(𝑥,𝑦) is continuously differentiable in 𝑥 and 𝑦, nonnegative on [𝛼_∗(𝑥₀),𝑥₁)×[𝑦₀,𝑦₁), and max_{𝑠∈[𝛼∗(𝑥0)−ℎ,𝑥1]}𝜓(𝑠,𝑦)≤𝑎(𝑥₀,𝑦) for all 𝑦∈[𝑦₀,𝑦₁).
If 𝑢∈𝐶([𝛼_∗(𝑥₀)−ℎ,𝑥₁)×[𝑦₀,𝑦₁),ℝ₊) satisfies the system of inequalities as follows: 𝑢+𝑥,𝑦≤𝑎𝑥,𝑦^𝑚_𝑖=1^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0𝑏_𝑖(𝑠,𝑡)ℎ_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑗𝛽(𝑦)𝑗𝑦0}ℎ_𝑗(𝑠,𝑡)ℎ_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢()𝑢,𝜉,𝑡𝑑𝑡𝑑𝑠,𝑥,𝑦∈Λ,𝑥,𝑦≤𝜓𝑥,𝑦𝑥,𝑦∈Ω, (20) then 𝑢𝑥,𝑦≤𝐻⁻¹_𝑚+𝑛𝐻_𝑚+𝑛𝛾_𝑚+𝑛+𝑥,𝑦_{𝛼𝑚+𝑛𝛼(𝑥)𝑚+𝑛𝑥0𝛽𝑚+𝑛𝛽(𝑦)𝑚+𝑛𝑦0}ℎ_𝑚+𝑛(𝑠,𝑡)𝑑𝑡𝑑𝑠, (21) for all (𝑥,𝑦)∈[𝑥₀,𝑋^∗)×[𝑦₀,𝑌^∗), where 𝐻_𝑖⁻¹ is the inverse of the function 𝐻_𝑖(𝑢)∶=^𝑢_𝑢𝑖𝑑𝑥𝜔_𝑖(𝑥),𝑢≥𝑢_𝑖,𝑖=1,2,…,𝑚+𝑛, (22)𝑢_𝑖>0 is a given constant, and 𝛾_𝑖(𝑥,𝑦) is defined recursively by 𝛾₁𝑥,𝑦∶=𝑎𝑥₀,𝑦₀+^𝑥_𝑥0𝑎_𝑥𝑡,𝑦𝑑𝑡+^𝑦_𝑦0𝑎_𝑦(𝛾𝑥,𝑠)𝑑𝑠,_𝑖+1𝑥,𝑦∶=𝐻_𝑖⁻¹𝐻_𝑖𝛾_𝑖𝑥,𝑦+^𝛼𝑖_𝛼^(𝑡)_𝑖𝑡0^𝛽𝑖_𝛽^(𝑡)_𝑖𝑡0𝑏_𝑖(𝑠,𝑡)𝑑𝑡𝑑𝑠 (23) for 𝑖=1,2,…,𝑚+𝑛−1, and 𝑥₀≤𝑋^∗<𝑥₁,𝑦₀≤𝑌^∗<𝑦₁ are chosen such that 𝐻_𝑖𝛾_𝑖𝑋^∗,𝑌^∗+^{𝛼𝑖(𝑋∗)}_{𝛼𝑖𝑥0}^{𝛽𝑖(𝑌∗)}_{𝛽𝑖𝑦0}𝑏_𝑖(𝑠,𝑡)𝑑𝑡𝑑𝑠≤^∞_𝑢𝑖𝑑𝑠ℎ_𝑖(𝑠), (24) for 𝑖=1,2,…,𝑚+𝑛.

Proof. From (23), we see that 𝛾₁(𝑥,𝑦) is nondecreasing on Λ, 𝑎(𝑥,𝑦)≤𝛾₁(𝑥,𝑦), and max_{𝑠∈[𝛼∗(𝑥0)−ℎ,𝑥1]}𝜓(𝑠,𝑦)≤𝑎(𝑥₀,𝑦)≤𝛾₁(𝑥₀,𝑦) for 𝑦∈[𝑦₀,𝑦₁). It implies from (20) that 𝑢𝑥,𝑦≤𝛾₁𝑥,𝑦+^𝑚_𝑖=1^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0𝑏_𝑖(𝑠,𝑡)ℎ_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑗𝛽(𝑦)𝑗𝑦0}𝑏_𝑗(𝑠,𝑡)×ℎ_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢(𝜉,𝑡)𝑑𝑡𝑑𝑠 (25) for all (𝑥,𝑦)∈Λ. Let𝑧𝛾𝑥,𝑦∶=¹𝑥,𝑦+∑^𝑚_𝑖=1∫^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0∫^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0𝑏_𝑖(𝑠,𝑡)ℎ_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=}∑_𝑚+1∫^𝛼𝑗_𝛼^(𝑥)_𝑗𝑥0∫^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0𝑏_𝑗(𝑠,𝑡)ℎ_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢()𝛾𝜉,𝑡𝑑𝑡𝑑𝑠,𝑥,𝑦∈Λ,₁𝑥₀,𝑦,𝑥,𝑦∈Ω. (26)Clearly, 𝑧(𝑥,𝑦) is nondecreasing in 𝑥. Then, we have 𝑢𝑥,𝑦≤𝑧𝑥,𝑦,𝑥,𝑦∈𝛼_∗𝑥₀−ℎ,𝜉×𝑦₀,𝜂,max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢𝜉,𝑦≤max_{[]𝜉∈𝑠−ℎ,𝑠}𝑧,∈𝛼𝜉,𝑦=𝑧𝑠,𝑦𝑠,𝑦_∗𝑥₀,𝑥₁×𝑦₀,𝑦₁. (27) From (25), (27), and (28) and the definition of 𝑧(𝑥,𝑦) on Λ, we get 𝑧𝑥,𝑦≤𝛾₁𝑥,𝑦+^𝑚_𝑖=1^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0𝑏_𝑖(𝑠,𝑡)ℎ_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0}𝑏_𝑗(𝑠,𝑡)ℎ_𝑗(𝑧(𝑠,𝑡))𝑑𝑡𝑑𝑠. (28) Applying Theorem  1 of [7] to the case that 𝑓_𝑖(𝑥,𝑦,𝑠,𝑡)=𝑏_𝑖(𝑠,𝑡), 𝑎(𝑥,𝑦)=𝛾₁(𝑥,𝑦), 𝑝=1, and 𝜔_𝑖(𝑡)=ℎ_𝑖(𝑡), 𝑖=1,2,…,𝑚+𝑛, we obtain (21) from (28). This completes the proof.

Proof of Theorem 1. First of all, we monotonize some given functions 𝑓_𝑖, 𝜔_𝑖, 𝑔, and 𝑎 in the system (8) of integral inequalities. Let 𝑔^∼(𝑡)∶=max_{[]𝜏∈0,𝑡}𝑔(𝜏),𝑡≥0,_∼𝑎𝑥,𝑦∶=max_{()∈𝑥𝜏,𝜉0×𝑦,𝑥0,𝑦}{𝑎(,∈𝑥𝜏,𝜉)}𝑥,𝑦₀,𝑥₁×𝑦₀,𝑦₁. (29) From (13), we see that the function 𝑊_𝑖 is strictly increasing, and therefore its inverse 𝑊_𝑖⁻¹ is well defined, continuous, and increasing in its domain. The sequence {𝜔^∼_𝑖(𝑡)}, defined by 𝜔_𝑖(𝑠), consists of nondecreasing nonnegative functions on ℝ₊ and satisfies 𝜔_𝑖(𝑡)≤𝜔^∼_𝑖(𝑡)𝜔,𝑖=1,2,…,𝑚,_𝑖(𝑡)≤𝜔_𝑖(𝑡)𝜔,𝑖=𝑚+1,…,𝑚+𝑛,_𝑖∼𝑔(𝑡)≤𝜔^∼_𝑖(𝑡),𝑖=𝑚+1,…,𝑚+𝑛. (30) Moreover, 𝜔^∼_𝑖∝𝜔^∼_𝑖+1,𝑖=1,2,…,𝑚+𝑛, (31) because the ratios 𝜔^∼_𝑖+1/𝜔^∼_𝑖, 𝑖=1,2,…,𝑚+𝑛, are all nondecreasing. Furthermore, let 𝑓^∼_𝑖𝑥,𝑦,𝑠,𝑡∶=max_{()∈𝑥𝜄,𝜉0×𝑦,𝑥0,𝑦}𝑓_𝑖(𝜄,𝜉,𝑠,𝑡), (32) which is nondecreasing in 𝑥 and 𝑦 for each fixed 𝑡 and 𝑠 and satisfies 𝑓^∼_𝑖(𝑥,𝑦,𝑠,𝑡)≥𝑓_𝑖(𝑥,𝑦,𝑠,𝑡)≥0 for all 𝑖=1,2,…,𝑚+𝑛. The monotonicity of 𝑔^∼ implies that max_{[]𝜉∈𝑠−ℎ,𝑠}𝑔𝑢𝜉,𝑦≤max_[]^∼_{𝜉∈𝑠−ℎ,𝑠}𝑔𝑢𝜉,𝑦≤𝑔^∼max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢𝜉,𝑦 (33) for (𝑠,𝑦)∈[𝛼_∗(𝑥₀)−ℎ,𝑥₁)×[𝑦₀,𝑦₁). From (8) and the definition of 𝑓^∼_𝑖(𝑥,𝑦,𝑠,𝑡), we obtain 𝜑𝑢𝑥,𝑦≤𝑎^∼+𝑥,𝑦_{𝑛𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑖𝑥,𝑦,𝑠,𝑡×𝜔_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑛+𝑚𝑗=𝑛+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑗𝛽(𝑦)𝑗𝑦0∼}𝑓_𝑗𝑥,𝑦,𝑠,𝑡×𝜔_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑔(𝑢(𝑢,𝜉,𝑡))𝑑𝑡𝑑𝑠,𝑥,𝑦∈Λ,𝑥,𝑦≤𝜓𝑥,𝑦𝑥,𝑦∈Ω. (34) Concerning (34), we consider the auxiliary system of inequalities 𝜑𝑢𝑥,𝑦≤𝑎^∼()+𝑋,𝑌_{𝑛𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑖(𝑋,𝑌,𝑠,𝑡)×𝜔_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚𝑗=1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑗(𝑋,𝑌,𝑠,𝑡)×𝜔_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑔(𝑢(∈𝑥𝜉,𝑡))𝑑𝑡𝑑𝑠,𝑥,𝑦₀,𝑋×𝑦₀,𝑢,,𝑌𝑥,𝑦≤𝜓𝑥,𝑦𝑥,𝑦∈Ω, (35) where 𝑥₀≤𝑋≤𝑋₁ and 𝑦₀≤𝑌≤𝑌₁ are chosen arbitrarily, and claim 𝑢𝑥,𝑦≤𝜑⁻¹𝑊Ω⁻¹_𝑚+𝑛^∼_𝑖𝑋,𝑌,𝑥,𝑦, (36) for all 𝑥₀≤𝑥≤min{𝑋,𝑋₂}, 𝑦₀≤𝑦≤min{𝑌,𝑌₂}, where Ω^∼_𝑖𝑋,𝑌,𝑥,𝑦∶=𝑊𝑟_𝑖^∼_𝑖+𝑋,𝑌,𝑥,𝑦_{𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠, (37)𝑖=1,2,…,𝑚+𝑛, 𝑟^∼_𝑖(𝑋,𝑌,𝑥,𝑦) is defined inductively by 𝑟^∼₁𝑋,𝑌,𝑥,𝑦∶=𝑎^∼(𝑋,𝑌_∼),𝑟_𝑖+1𝑋,𝑌,𝑥,𝑦∶=𝑊_𝑖⁻¹𝑊Ω_𝑖^∼_𝑖𝑋,𝑌,𝑥,𝑦, (38) for 𝑖=1,2,…,𝑚+𝑛−1, and 𝑋₂,𝑌₂ are chosen such that Ω^∼_𝑖𝑋,𝑌,𝑋₂,𝑌₂≤^∞_𝑢𝑖𝑑𝑠_∼𝜔_𝑖𝜑⁻¹(𝑠), (39) for 𝑖=1,2,…,𝑚+𝑛.
Notice that we may take 𝑋₂=𝑋₁ and 𝑌₂=𝑌₁. In fact, the monotonicity that 𝑟^∼_𝑖(𝑋,𝑌,𝑥,𝑦) and 𝑓^∼_𝑖(𝑋,𝑌,𝑥,𝑦) are both nondecreasing in 𝑋 and 𝑌 for fixed 𝑥, 𝑦. Furthermore, it is easy to check that 𝑟^∼_𝑖+1(𝑋,𝑌,𝑋,𝑌)=𝑟_𝑖(𝑋,𝑌), for 𝑖=1,2,…,𝑚+𝑛. If 𝑋₂, 𝑌₂ are replaced with 𝑋₁, 𝑌₁, respectively, on the left side of (39), we get from (15) that Ω^∼_𝑖𝑋,𝑌,𝑋₁,𝑌₁=𝑊𝑟_𝑖^∼_𝑖𝑋,𝑌,𝑋₁,𝑌₁+𝑓^{𝛼𝑖(𝑋1)}_{𝛼𝑖𝑥0}^{𝛽𝑖(𝑌1)}_{𝛽𝑖𝑦0}^∼(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠≤𝑊𝑟_𝑖^∼_𝑖𝑋₁,𝑌₁,𝑋₁,𝑌₁+𝑓^{𝛼𝑖(𝑋1)}_{𝛼𝑖𝑥0}^{𝛽𝑖(𝑌1)}_{𝛽𝑖𝑦0}^∼X₁,𝑌₁,𝑠,𝑡𝑑𝑡𝑑𝑠=𝑟_𝑖𝑋₁,𝑌₁+𝑓^{𝛼𝑖(𝑋1)}_{𝛼𝑖𝑥0}^{𝛽𝑖(𝑌1)}_{𝛽𝑖𝑦0}^∼𝑋₁,𝑌₁,𝑠,𝑡𝑑𝑡𝑑𝑠=Ω_𝑖𝑋₁,𝑌₁≤^∞_𝑢𝑖𝑑𝑠_∼𝜔_𝑖𝜑⁻¹(𝑠). (40) Thus, it means that we can take 𝑋₂=𝑋₁, 𝑌₂=𝑌₁.
Now, we prove (36) by induction. From (33), (35), and the definitions of 𝑔^∼(𝑡), 𝜔^∼_𝑖(𝑡), and 𝜔_𝑖(𝑡), we obtain 𝜑𝑢≤𝑥,𝑦_∼𝑎()+𝑋,𝑌_{𝑚𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝑦𝑦0∼}𝑓_𝑖(𝑋,𝑌,𝑠,𝑡)𝜔_𝑖(𝑢(+𝑠,𝑡))_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝑦𝑦0∼}𝑓_𝑗()×𝜔𝑋,𝑌,𝑠,𝑡_𝑗∼𝑔max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢()≤𝜉,𝑡𝑑𝑡𝑑𝑠_∼𝑎()+𝑋,𝑌_{𝑚𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑖(𝑋,𝑌,𝑠,𝑡)𝜔^∼_𝑖(𝑢(𝑠+))𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑗𝛽(𝑦)𝑗𝑦0∼}𝑓_𝑗()×𝑋,𝑌,𝑠,𝑡_∼𝜔_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢(𝜉)𝑑𝑡𝑑𝑠, (41) for all (𝑥,𝑦)∈[𝑥₀,𝑋)×[𝑦₀,𝑌), where 𝑥₀≤𝑋≤𝑋₁ and 𝑦₀≤𝑌≤𝑌₁ are chosen arbitrarily. Since max_{𝑠∈[𝛼∗(𝑥0)−ℎ,𝑥0]}𝜓(𝑠,𝑦)≤𝜑⁻¹(𝑎(𝑥₀,𝑦)) and 𝑎(𝑥₀,𝑦)≤𝑎^∼(𝑥₀,𝑦)≤𝑎^∼(𝑋,𝑌), we have max_{𝑠∈[𝛼∗(𝑥0)−ℎ,𝑥0]}𝜓(𝑠,𝑦)≤𝜑⁻¹(𝑎^∼(𝑋,𝑌)). Define a function 𝑧(𝑥,𝑦)∶[𝛼_∗(𝑥₀)−ℎ,𝑋)×[𝑦₀,𝑌)→ℝ₊ by𝑧=𝑥,𝑦_∼𝑎(𝑋,𝑌)+∑^𝑚_𝑖=1∫^𝛼𝑖_𝛼^(𝑥)_𝑖𝑥0∫𝑓^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0^∼_𝑖(𝑋,𝑌,𝑠,𝑡)𝜔^∼_𝑖(𝑢(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=}∑_𝑚+1∫^𝛼𝑗_𝛼^(𝑥)_𝑗𝑥0∫𝑓^𝛽𝑖_𝛽^(𝑦)_𝑖𝑦0^∼_𝑗(𝑠,𝑡)𝜔^∼_𝑗max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢(𝜉,𝑡)𝑑𝑡𝑑𝑠,𝑥,𝑦∈𝑥₀,𝑋×𝑦₀,𝑌_∼,𝑎(𝑋,𝑌),𝑥,𝑦∈𝛼_∗𝑥₀−ℎ,𝑥₀×𝑦₀,𝑌. (42)Clearly, 𝑧(𝑥,𝑦) is nondecreasing in 𝑥. By (41) and the definition of 𝑧(𝑥,𝑦), we have 𝑢𝑥,𝑦≤𝜑⁻¹𝑧,∈𝛼𝑥,𝑦𝑥,𝑦_∗𝑥₀−ℎ,𝑋×𝑦₀,𝑌. (43) Then noting that 𝑧(𝑥,𝑦) is nondecreasing and 𝜑(𝑡) is strictly increasing, from (43), we obtain max_{[]𝜉∈𝑠−ℎ,𝑠}𝑢𝜉,𝑦≤max_{[]𝜉∈𝑠−ℎ,𝑠}𝜑⁻¹𝑧𝜉,𝑦≤max_{[𝜉∈𝑠−ℎ,s]}𝜑⁻¹𝑧𝑠,𝑦≤𝜑⁻¹max_{[]𝜉∈𝑠−ℎ,𝑠}𝑧,∈𝛼𝜉,𝑦𝑠,𝑦_∗𝑥₀,𝑋×𝑦₀,𝑌. (44) It follows from (43), (44), and the definition of 𝑧(𝑥,𝑦) that 𝑧𝑥,𝑦≤𝑎^∼()+𝑋,𝑌_{𝑚𝑖=1𝛼𝑖𝛼(𝑥)𝑖𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑖()×𝑋,𝑌,𝑠,𝑡_∼𝜔_𝑖𝜑⁻¹(𝑧(+𝑠,𝑡))𝑑𝑡𝑑𝑠_{𝑚+𝑛𝑗=𝑚+1𝛼𝑗𝛼(𝑥)𝑗𝑥0𝛽𝑖𝛽(𝑦)𝑖𝑦0∼}𝑓_𝑖()×𝑋,𝑌,𝑠,𝑡_∼𝜔_𝑗𝜑⁻¹max_{[]𝜉∈𝑠−ℎ,𝑠}𝑧(∈𝑥𝜉,𝑡)𝑑𝑡𝑑𝑠,𝑥,𝑦₀,𝑋×𝑦₀,𝑧≤,𝑌𝑥,𝑦𝑎^∼(𝑋,𝑌),𝑥,𝑦∈𝛼_∗𝑥₀−ℎ,𝑥₀×𝑦₀,𝑌. (45) In order to demonstrate the basic condition of monotonicity, let ℎ(𝑡)∶=𝜑⁻¹(𝑡), which is clearly a continuous and nondecreasing function on ℝ₊. Thus, each 𝜔^∼_𝑖(ℎ(𝑡)) is continuous and nondecreasing on ℝ₊ and satisfies 𝜔^∼_𝑖(ℎ(𝑡))>0 for 𝑡>0. Moreover, since 𝜔^∼_𝑖(𝑡)∝𝜔^∼_𝑖+1(𝑡), 𝜔^∼_𝑖+1(ℎ(𝑡))/𝜔^∼_𝑖(ℎ(𝑡)) is also continuous and nondecreasing on ℝ₊ and positive on (0,+∞), implying that 𝜔^∼_𝑖(ℎ(𝑡))∝𝜔^∼_𝑖+1(ℎ(𝑡)), for 𝑖=1,2,…,𝑚+𝑛−1. By Lemma 2 and (45), 𝑧𝑥,𝑦≤𝑊⁻¹_𝑚+𝑛𝑊𝑟^∼_𝑚+𝑛+𝑋,𝑌,𝑥,𝑦_{𝛼𝑚+𝑛𝛼(𝑥)𝑚+𝑛𝑥0𝛽𝑚+𝑛𝛽(𝑦)𝑚+𝑛𝑦0∼}𝑓_𝑚+𝑛(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠, (46) for 𝑥₀≤𝑥<𝑋₂ and 𝑦₀≤𝑦<𝑌₂. It follows from (43) and (46) that 𝑢𝑥,𝑦≤𝜑⁻¹𝑊⁻¹_𝑚+𝑛𝑊𝑟^∼_𝑚+𝑛𝑋,𝑌,𝑥,𝑦+𝑓^{𝛼𝑚+𝑛}_𝛼^(𝑥)_{𝑚+𝑛𝑥0}^{𝛽𝑚+𝑛}_𝛽^(𝑦)_{𝑚+𝑛𝑦0}^∼_𝑚+𝑛(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠, (47) for 𝑥₀≤𝑥<𝑋₂ and 𝑦₀≤𝑦<𝑌₂. This proves the claimed (36).
Taking 𝑥=𝑋,𝑦=𝑌, 𝑋₂=𝑋₁, and 𝑌₂=𝑌₁ in (36), we have 𝑢(𝑋,𝑌)≤𝜑⁻¹𝑊⁻¹_𝑚+𝑛𝑊𝑟_𝑚+𝑛^∼_𝑚+𝑛()+𝑋,𝑌,𝑋,𝑌_{𝛼𝑚+𝑛𝛼(𝑋)𝑚+𝑛𝑥0𝛽𝑚+𝑛𝛽(𝑌)𝑚+𝑛𝑦0∼}𝑓_𝑚+𝑛(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠, (48) for all 𝑥₀≤𝑋<𝑋₁, 𝑦₀≤𝑌<𝑌₁. It is easy to verify 𝑟^∼_𝑚+𝑛(𝑋,𝑌,𝑋,𝑌)=𝑟_𝑚+𝑛(𝑋,𝑌). Thus, (48) can be written as 𝑢(𝑋,𝑌)≤𝜑⁻¹𝑊⁻¹_𝑚+𝑛𝑊_𝑚+𝑛𝑟_𝑚+𝑛()+𝑋,𝑌_{𝛼𝑚+𝑛𝛼(𝑋)𝑚+𝑛𝑥0𝛽𝑚+𝑛𝛽(𝑌)𝑚+𝑛𝑦0∼}𝑓_𝑚+𝑛(𝑋,𝑌,𝑠,𝑡)𝑑𝑡𝑑𝑠. (49) Since 𝑋,𝑌 are arbitrary, replacing 𝑋 and 𝑌 with 𝑥 and 𝑦, respectively, we get 𝑢𝑥,𝑦≤𝜑⁻¹𝑊⁻¹_𝑚+𝑛𝑊_𝑚+𝑛𝑟_𝑚+𝑛𝑥,𝑦+𝑓^{𝛼𝑚+𝑛}_𝛼^(𝑥)_{𝑚+𝑛𝑥0}^{𝛽𝑚+𝑛}_𝛽^(𝑦)_{𝑚+𝑛𝑦0}^∼_𝑚+𝑛𝑥,𝑦,𝑠,𝑡𝑑𝑡𝑑𝑠, (50) for all (𝑥,𝑦)∈[𝑥₀,𝑋₁)×[𝑦₀,𝑌₁). This completes the proof.

3. Applications

In this section, we apply our result to prove the boundedness of solutions for a differential equation with the maxima.

Consider a system of partial differential equations with maxima 𝜕²𝑧𝑥,𝑦𝜕𝑥𝜕𝑦=𝐹𝑥,𝑦,𝑧𝑥,𝑦,max_{𝛼(𝑥)(𝑥𝑠∈,𝛽)}𝜔𝑧,∈𝑥𝑠,𝑦𝑥,𝑦₀,𝑥₁×𝑦₀,𝑦₁𝑧𝑥,𝑦=𝜓𝑥,𝑦,𝑥,𝑦∈𝛽𝑥₀−ℎ,𝑥₀×𝑦₀,𝑦₁,𝑧𝑥,𝑦₀=𝑓(𝑥),𝑧𝑥₀,𝑦=𝑔𝑦,𝑥≥𝑥₀,𝑦≥𝑦₀, (51) where 𝐹∈𝐶([𝑥₀,𝑥₁)×[𝑦₀,𝑦₁)×ℝ²,ℝ), 𝜔∈𝐶([0,∞),ℝ₊), 𝛼,𝛽∈𝐶¹([𝑥₀,𝑥₁),ℝ₊) are nondecreasing such that 𝛼(𝑥)≤𝑥, 𝛽(𝑥)≤𝑥, and 0<𝛽(𝑥)−𝛼(𝑥)≤ℎ (ℎ is a positive constant) for 𝑥∈[𝑥₀,𝑥₁], 𝜓∈𝐶([𝛽(𝑥₀)−ℎ,𝑥₀]×[𝑦₀,𝑦₁), and 𝑓∈𝐶([𝑥₀,𝑥₁),ℝ), 𝑔∈𝐶([𝑦₀,𝑦₁),ℝ) satisfy that 𝑓(𝑥₀)=𝑔(𝑦₀) and 𝑔(𝑦)=𝜓(𝑥₀,𝑦), for all 𝑦∈[𝑦₀,𝑦₁).

Equation (51) is more general than the equation considered in Section 2.4 of [23]. The following result gives an estimate for its solutions.

Corollary 3. Suppose that functions 𝐹 and 𝜓 in (51) satisfy 𝐹𝑥,𝑦,𝑠,𝑡≤ℎ₁𝑥,𝑦𝜇₁(|𝑠|)+ℎ₂𝑥,𝑦𝜇₂𝑡,𝜓≤𝑔𝑦∈𝛽𝑥(||)𝑥,𝑦,∀𝑥,𝑦₀−ℎ,𝑥₀×𝑦₀,𝑦₁, (52) where ℎ_𝑖∈𝐶([𝑥₀,𝑥₁)×[𝑦₀,𝑦₁),ℝ₊) and 𝜇_𝑖∈𝐶(ℝ₊,(0,∞)), 𝑖=1,2. Then, any solution 𝑧(𝑥,𝑦) of (51) has the estimate 𝑧𝑥,𝑦≤𝑄₂⁻¹𝑄₂𝛾𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₂(𝑠,𝑡)𝑑𝑡𝑑𝑠, (53) for all (𝑥,𝑦)∈[𝑥₀,𝑋₁)×[𝑦₀,𝑌₁), where 𝛾𝑥,𝑦∶=𝑄₁⁻¹𝑄₁𝛾₁𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(),𝛾𝑠,𝑡𝑑𝑡𝑑𝑠₁𝑥,𝑦∶=max_{∈𝑥𝜉,𝜂0×𝑦,𝑥0,𝑦}𝑓(𝜉)+𝑔𝜂−𝑓𝑥₀,𝑄₂(𝑢)∶=^𝑢_𝑢2𝑑𝑠max𝜇_{𝜏∈[0,𝑠]}^∼𝜔₂^∼(𝜏)/max_{𝜏1∈[0,𝜏]}𝜇₁𝜏₁max_{𝜏∈[0,𝑠]}𝜇₁(𝜏,𝑄₁)(𝑢)∶=^𝑢_𝑢1𝑑𝑠max_{𝜏∈[0,𝑠]}𝜇₁(𝜏,_∼)𝜔(𝑡)∶=max_{[]𝑠∈0,𝑡}{𝜔(𝑠)},𝜇^∼₂(𝑡)∶=max_{[]𝑠∈0,𝑡}𝜇₂(𝑠), (54) and 𝑋₁, 𝑌₁ are given as in Theorem 1, and constants 𝑢₁>0, 𝑢₂>0 are given arbitrarily.

Proof. From (51), we obtain 𝑧(𝑥)𝑦𝑥𝑥,𝑦=𝑓+𝑔−𝑓₀+^𝑥_𝑥0^𝑦_𝑦0𝐹𝑠,𝑡,𝑧(𝑠,𝑡),max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝜔(𝑧(𝑧,∈∈𝛽𝑥𝜉,𝑡))𝑑𝑡𝑑𝑠,𝑥,𝑦∈Λ,𝑥,𝑦=𝜓𝑥,𝑦𝑥,𝑦𝑥,𝑦₀−ℎ,𝑥₀×𝑦₀,𝑦₁. (55) From (52) and (55), we get 𝑧≤𝑓(𝑥)𝑦𝑥𝑥,𝑦+𝑔−𝑓₀+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝜇₁𝑧(+(|𝑠,𝑡)|)𝑑𝑡𝑑𝑠_{𝑥𝑥0𝑦𝑦0}ℎ₂(𝑠,𝑡)𝜇₂max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝜔(𝑧(≤𝑓(𝑥)𝑦𝑥𝜉,𝑡))𝑑𝑡𝑑𝑠+𝑔−𝑓₀+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝜇₁𝑧(+(|𝑠,𝑡)|)𝑑𝑡𝑑𝑠_{𝑥𝑥0𝑦𝑦0}ℎ₂(𝑠,𝑡)𝜇^∼₂max_{𝛼(𝑠)(𝑠}^∼_{𝜉∈,𝛽)}𝜔𝑧(𝑧≤𝜓,∈𝛽𝑥𝜉,𝑡)𝑑𝑡𝑑𝑠,𝑥,𝑦∈Λ,𝑥,𝑦𝑥,𝑦𝑥,𝑦₀−ℎ,𝑥₀×𝑦₀,𝑦₁. (56) Set 𝑣(𝑥,𝑦)=|𝑧(𝑥,𝑦)| for (𝑥,𝑦)∈[𝛽(𝑥₀)−ℎ,𝑥₁)×[𝑦₀,𝑦₁). Noting that max_{𝜉∈[𝛼(𝑠),𝛽(𝑠)]}𝑧(𝜉,𝑦)≤max_{𝜉∈[𝛽(𝑠)−ℎ,𝛽(𝑠)]}𝑧(𝜉,𝑦), from (56), we get 𝑣𝑥,𝑦≤𝛾₁𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝜇₁(𝑣(𝑠,𝑡))𝑑𝑡𝑑𝑠+_𝛽𝑥^𝛽(𝑥)₀^𝑦_𝑦0ℎ₂𝛽⁻¹𝜂,𝑡𝛽⁻¹𝜂𝜇^′∼₂×max^∼_{𝜉∈𝜂−ℎ,𝜂}𝜔(𝑣(𝑣≤𝜓,∈𝛽𝑥𝜉,𝑡))𝑑𝑡𝑑𝜂𝑥,𝑦𝑥,𝑦𝑥,𝑦₀−ℎ,𝑥₀×𝑦₀,𝑦₁. (57) Applying Theorem 1 to specified 𝑚=𝑛=1, 𝜑(𝑢)=𝑢, 𝑓₁(𝑥,𝑦,𝑠,𝑡)=ℎ₁(𝑠,𝑡), 𝛼₁(𝑡)=𝑡, and 𝛼₂(𝑡)=𝛽(𝑡), 𝛽_𝑖(𝑡)=𝑡, 𝑖=1,2, 𝑓₂(𝑠,𝑡)=ℎ₂(𝛽⁻¹(𝑠),𝑡)(𝛽⁻¹(𝑠))^′, and 𝜔_𝑖(𝑢)=𝜇_𝑖(𝑢),𝑖=1,2, we obtain (53) from (57).

Next, we discuss the uniqueness of solutions for system (51).

Corollary 4. Suppose that 𝑔(𝑠)=𝑠 and 𝐹𝑥,𝑦,𝑠₁,𝑡₁−𝐹𝑥,𝑦,𝑠₂,𝑡₂≤ℎ₁𝑥,𝑦𝜇₁𝑥₁−𝑥₂+ℎ₂𝑥,𝑦𝜇₂𝑦₁−𝑦₂, (58) for all (𝑥,𝑦)∈[𝑥₀,𝑥₁]×[𝑦₀,𝑦₁] and all 𝑥_𝑖,𝑦_𝑖∈ℝ(𝑖=1,2), where ℎ_𝑖∈𝐶([𝑥₀,𝑥₁]×[𝑦₀,𝑦₁],ℝ₊) and 𝜇_𝑖∈𝐶([ℝ,ℝ₊) are both nondecreasing such that 𝜇_𝑖(0)=0, 𝜇_𝑖(𝑢)>0 for 𝑢>0, 𝜇₂/𝜇₁ is also nondecreasing, and ∫¹₀𝑑𝑠/𝜇_𝑖(𝑠)=+∞, 𝑖=1,2. Then, system (51) has at most one solution on [𝑥₀,𝑥₁)×[𝑦₀,𝑦₁).

Proof. 𝑔(𝑠)=𝑠. From (51), we get 𝜕²𝑧𝑥,𝑦𝜕𝑥𝜕𝑦=𝐹𝑥,𝑦,𝑧𝑥,𝑦,max_{𝛼(𝑥)(𝑥𝑠∈,𝛽)}𝑧,∈𝑥𝑠,𝑦𝑥,𝑦₀,𝑥₁×𝑦₀,𝑦₁,𝑧𝑥,𝑦=𝜓𝑥,𝑦,𝑥,𝑦∈𝛼𝑥₀−ℎ,𝑥₀×𝑦₀,𝑦₁,𝑧𝑥,𝑦₀=𝑓(𝑥),𝑧𝑥₀,𝑦=𝑔𝑦,𝑥≥𝑥₀,𝑦≥𝑦₀. (59) Assume that (59) has two different solutions 𝑢(𝑥,𝑦) and 𝑣(𝑥,𝑦). From the equivalent integral equation system (55), we have 𝑢≤𝑥,𝑦−𝑣𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}ℎ₁(𝑠,𝑡)𝜇₁𝑢()(+(|𝑠,𝑡−𝑣𝑠,𝑡)|)𝑑𝑡𝑑𝑠_{𝑥𝑥0𝑦𝑦0}ℎ₂(𝑠,𝑡)𝜇₂max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑢(𝜉,𝑡)−max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑣(𝜉,𝑡)𝑑𝑡𝑑𝑠≤^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝜇₁𝑢(𝑠)(𝑠+(|−𝑣)|)𝑑𝑡𝑑𝑠_{𝑥𝑥0𝑦𝑦0}ℎ₂(𝑠,𝑡)𝜇₂×max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑢(𝜉,𝑡)−𝑣(𝜉,𝑡)𝑑𝑠, (60) for all (𝑥,𝑦)∈[𝑥₀,𝑥₁]×[𝑦₀,𝑦₁]. The continuity of the function 𝑢(𝑥,𝑦) implies that for any fixed points 𝑠∈[𝑥₀,𝑥] and 𝑡∈[𝑦₀,𝑦] there exists a point 𝜂∈[𝛼(𝑠),𝛽(𝑠)] such that the inequality max_{𝜉∈[𝛼(𝑠),𝛽(𝑠)]}𝑢(𝜉,𝑡)=𝑢(𝜂,𝑡) holds, and therefore max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑢(𝜉,𝑡)−max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑣(=𝑢𝜉,𝑡)𝜂,𝑡−max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑣(≤𝑢𝜉,𝑡)𝜂,𝑡−𝑣𝜂,𝑡≤max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑢(𝜉,𝑡)−𝑣(𝜉,𝑡). (61) Hence, 𝑢≤𝑥,𝑦−𝑣𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}ℎ₁(𝑠,𝑡)𝜇₁𝑢+𝑠,𝑦−𝑣𝑠,𝑦𝑑𝑡𝑑𝑠_{𝑥𝑥0𝑦𝑦0}ℎ₂(𝑠,𝑡)𝜇₂max_{𝛼(𝑠)(𝑠𝜉∈,𝛽)}𝑢(𝜉,𝑡)−𝑣(𝜉,𝑡)𝑑𝑠. (62) Let 𝜙𝑢,∈𝛽𝑥𝑥,𝑦∶=𝑥,𝑦−𝑣𝑥,𝑦𝑥,𝑦₀−ℎ,𝑥₁×𝑦₀,𝑦₁. (63) Because max_{𝜉∈[𝛼(𝑠),𝛽(𝑠)]}𝑢(𝜉,𝑦)≤max_{𝜉∈[𝛽(𝑠)−ℎ,𝛽(𝑠)]}𝑢(𝜉,𝑦), from (62), we obtain 𝜙𝑥,𝑦≤^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝜇₁𝜙(𝑠,𝑡)𝑑𝑡𝑑𝑠+_𝛽𝑥^𝛽(𝑥)₀^𝑦_𝑦0ℎ₂𝛽⁻¹𝜂,𝑡𝛽⁻¹𝜂^′×𝜇₂max_{𝜉∈𝜂−ℎ,𝜂}𝜙()∈𝑥𝜉,𝑡𝑑𝑡𝑑𝜂,𝑥,𝑦₀,𝑥₁×𝑦₀,𝑦₁,𝜙𝑥,𝑦≤0,𝑥,𝑦∈𝛽𝑥₀−ℎ,𝑥₀×𝑦₀,𝑦₁. (64) Applying Theorem 1 to specified 𝑚=𝑛=1, 𝜑(𝑢)=𝑢, 𝑓₁(𝑠,𝑡)=ℎ₁(𝑠,𝑡), 𝛼₁(𝑡)=𝑡, 𝛼₂(𝑡)=𝛽(𝑡), 𝑓₂(𝑠,𝑡)=ℎ₂(𝛽⁻¹(𝑡),𝑠)(𝛽⁻¹(𝑡))^′, 𝑔(𝑡)=𝑡, 𝑎(𝑥,𝑦)=0, and 𝜔_𝑖(𝑡)=𝜇_i(𝑡),𝑖=1,2, from (64), we obtain 𝜙𝑥,𝑦≤𝑄₂⁻¹𝑄₂𝛾₂(𝑠,𝑡)+^𝑥_𝑥0^𝑦_𝑦0ℎ₂(𝑠,𝑡)𝑑𝑡𝑑𝑠, (65) for all (𝑥,𝑦)∈[𝑥₀,𝑋₁]×[𝑦₀,𝑌₁], where 𝑄₁(𝑢)∶=^𝑢₁𝑑𝑠𝜇₁(𝑠),𝑄₂(𝑢)∶=^𝑢₁𝑑𝑠𝜇₂(𝜏),𝑟₁𝑥,𝑦∶=0,𝑟₂𝑥,𝑦∶=𝑄₁⁻¹𝑄₁𝑟₁𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝑑𝑡𝑑𝑠. (66)(67)(68) By the definition of 𝑄_𝑖 and properties of 𝜇_𝑖, noting that ∫¹₀𝑑𝑠/𝜇_𝑖(𝑠)=+∞(𝑖=1,2), we obtain lim_𝑢→0+𝑄_𝑖(𝑢)=−∞,lim_{𝑢→−∞}𝑄_𝑖⁻¹(𝑢)=0,𝑖=1,2. (69) Since ∫^𝑥_𝑥0∫^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝑑𝑡𝑑𝑠 is finite on a finite interval, [𝑥₀,𝑥₁] and [𝑦₀,𝑦₁], by (67), we obtain 𝑄₁𝑟₁𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₁(𝑠,𝑡)𝑑𝑡𝑑𝑠=−∞. (70) Thus, we obtain 𝛾₂(𝑥,𝑦)=0 from (68), (69), and (70) immediately. Similarly, noting that ∫^𝑥_𝑥0∫^𝑦_𝑦0ℎ₂(𝑠,𝑡)𝑑𝑡𝑑𝑠 is finite on finite interval, [𝑥₀,𝑥₁] and [𝑦₀,𝑦₁], from (69), we obtain 𝑄₂𝑟₂𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0ℎ₂(𝑠,𝑡)𝑑𝑡𝑑𝑠=−∞. (71) Thus, we conclude from (65), (69), and (71) that |𝑢(𝑥,𝑦)−𝑣(𝑥,𝑦)|≤0, which implies that 𝑢(𝑥,𝑦)=𝑣(𝑥,𝑦), for all (𝑥,𝑦)∈[𝑥₀,𝑋₁)×[𝑦₀,𝑌₁), where 𝑋₁, 𝑌₁ are given as in Theorem 1. The uniqueness is proved.

Acknowledgments

The author is grateful to the reviewer for his valuable suggestions. This research was supported by Scientific Research Fund of Sichuan Provincial Education Department of China.