Estimates of Certain Integral Inequalities on Time Scales

Abstract

The main objective of this paper is to establish explicit bounds on certain integral inequalities on time scales, which can be used as tools in the study of certain classes of integral equations on time scales. Some applications of our results are also given.

1. Introduction

Recently, many authors have studied various aspects of dynamic inequalities on time scales using various techniques [1–4]. In this paper, we obtain explicit bounds on certain integral inequalities on time scales. In the present paper we offer some fundamental dynamic integral inequalities on time scales which can be used as tools for handling the qualitative behavior of solutions of certain dynamic equations on time scales. Excellent information about introduction to time scales can be found in [5, 6]. In what follows, ℝ denotes the set of real numbers, ℤ the set of integers, and 𝕋 denotes arbitrary time scales. Now following [3, 4] we give some basic definitions about calculus on time scales in two variables.

We say that 𝑓∶𝕋→ℝ is rd-continuous provided 𝑓 is continuous at each right-dense point of 𝕋 and has a finite left sided limit at each left-dense point of 𝕋 and will be denoted by 𝐶_rd. Let 𝕋₁ and 𝕋₂ be two time scales with at least two points and consider the time scales intervals 𝕋₁=[𝑥₀,∞)∩𝕋₁ and 𝕋₂=[𝑦₀,∞)∩𝕋₂ for 𝑥₀∈𝕋₁ and 𝑦₀∈𝕋₂ and Ω=𝕋₁×𝕋₂. Let 𝜎₁,  𝜌₁,  Δ₁ and 𝜎₂,  𝜌₂,  Δ₂ denote the forward jump operators, backward jump operators and the delta differentiation operator, respectively, on 𝕋₁ and 𝕋₂. Let 𝑎<𝑏 are points in 𝕋₁, 𝑐<𝑑 are points in 𝕋₂, [𝑎,𝑏) is the half closed bounded interval in 𝕋₁, and [𝑐,𝑑) is the half closed bounded interval in 𝕋₂.

We say that a real-valued function 𝑓 on 𝕋₁×𝕋₂ at (𝑡₁,𝑡₂)∈𝕋₁×𝕋₂ has a Δ₁ partial derivative 𝑓^Δ1(𝑡₁,𝑡₂) with respect to 𝑡₁ if for each 𝜖>0 there exists a neighborhood 𝑈_𝑡1 of 𝑡₁ such that 𝑓𝜎₁𝑡₁,𝑡₂−𝑓𝑠,𝑡₂−𝑓^Δ1𝑡₁,𝑡₂𝜎₁𝑡₁𝜎−𝑠≤𝜀₁𝑡₁−𝑠, (1) for all 𝑠∈𝑈_𝑡1. We say that 𝑓 on 𝕋₁×𝕋₂ at (𝑡₁,𝑡₂)∈𝕋₁×𝕋₂ has a Δ₂ partial derivative 𝑓^Δ2(𝑡₁,𝑡₂) with respect to 𝑡₂ if for each 𝜂>0 there exists a neighborhood 𝑈_𝑡2 of 𝑡₂ such that 𝑓𝑡₁,𝜎₂𝑡₂−𝑓𝑡₁,𝑙−𝑓^Δ2𝑡₁,𝑡₂𝜎₂𝑡₂𝜎−𝑙≤𝜂₂𝑡₂−𝑙, (2) for all 𝑙∈𝑈_𝑡2. The function 𝑓 is called rd-continuous in 𝑡₂ if for every 𝛼₁∈𝕋₁, the function 𝑓(𝛼₁,𝑡₂) is rd-continuous on 𝕋₂. The function 𝑓 is called rd-continuous in 𝑡₁ if for every 𝛼₂∈𝕋₂ the function 𝑓(𝑡₁,𝛼₂) is rd-continuous on 𝕋₁.

We require that the following lemma proved in [1, 2] holds.

Lemma 1. Let 𝑢, 𝑓∈𝐶_𝑟𝑑(𝕋,ℝ), 𝑘(𝑡,𝑠), 𝑘^Δ1(𝑡,𝑠)∈𝐶_𝑟𝑑(Ω,ℝ) and 𝑐 nonnegative constants. If 𝑢(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠_𝑠0𝑘(𝑠,𝜏)𝑢(𝜏)Δ𝜏Δ𝑠, (3) for 𝑡∈𝕋, then 𝑢(𝑡)≤𝑐1+^𝑡_𝑡0𝑓(𝑠)𝑒_𝑓+𝐴𝑠,𝑠₀Δ𝑠, (4) for 𝑡∈𝕋, where 𝐴(𝑡)=𝑘(𝑡,𝜎(𝑡))+^𝑡_𝑡0𝑘^Δ1𝑡,𝜂Δ𝜂, (5) for 𝑡∈𝕋.

Lemma 2 (see [1, Lemma]). Let 𝑢,  𝑎,  𝑓∈𝐶^′_rd(𝕋₁×𝕋₂,ℝ₊) and 𝑎 is nondecreasing in each of its variables. If 𝑢𝑥,𝑦≤𝑎𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑢(𝑠,𝑡)Δ𝑡Δ𝑠, (6) for (𝑥,𝑦)∈𝕋₁×𝕋₂, then 𝑢𝑥,𝑦≤𝑎𝑥,𝑦+𝑒_∫^𝑦_{𝑦0𝑓(𝑥,𝑡)Δ𝑡}𝑥,𝑥₀, (7) for (𝑥,𝑦)∈𝕋₁×𝕋₂.

2. Main Results

Our main results are given in the following theorems.

Theorem 3. Let 𝑢,  𝑓,    𝑎∈𝐶_rd(𝕋,𝑅₊), 𝑘(𝑡,𝑠),   𝑘^Δ(𝑡,𝑠)∈𝐶_rd(Ω,𝑅₊). If 𝑢(𝑡)≤𝑎(𝑡)+^𝑡_𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠_𝑠0𝑘(𝑠,𝜏)𝑢(𝜏)Δ𝜏Δ𝑠, (8) for 𝑡∈𝕋, then 𝑢(𝑡)≤𝑎(𝑡)+ℎ(𝑡)1+^𝑡_𝑡0𝑓(𝑠)𝑒_𝑓+𝐴𝑠,𝑠₀Δ𝑠, (9) for 𝑡∈𝕋, where ℎ(𝑡)=^𝑡_𝑡0𝑓(𝑠)𝑎(𝑠)+^𝑠_𝑠0𝑘(𝑠,𝜏)𝑎(𝜏)Δ𝜏Δ𝑠, (10) for 𝑡∈𝕋 and 𝐴(𝑡) defined by (5).

Proof. Define a function 𝑧(𝑡) by 𝑧(𝑡)=^𝑡_𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠_𝑠0𝑘(𝑠,𝜏)𝑢(𝜏)Δ𝜏Δ𝑠, (11) then from (8), 𝑢(𝑡)≤𝑎(𝑡)+𝑧(𝑡), and using this in (11) we get 𝑧(𝑡)≤^𝑡_𝑡0𝑓(𝑠)𝑎(𝑠)+𝑧(𝑠)+^𝑠_𝑠0𝑘(𝑎(𝜏)(𝜏(𝑡)+𝑠,𝜏)(+𝑧))Δ𝜏Δ𝑠=ℎ^𝑡_𝑡0𝑓(𝑠)𝑧(𝑠)+^𝑠_𝑠0𝑘(𝑠,𝜏)𝑧(𝜏)Δ𝜏Δ𝑠, (12) where ℎ(𝑡) is defined by (10). Clearly ℎ(𝑡) is nonnegative, rd-continuous and nondecreasing in 𝑡, 𝑡∈𝕋 for 𝑎(𝑡) nondecreasing. First we assume that ℎ(𝑡)>0 for 𝑡∈𝕋. From (12), we have 𝑧(𝑡)ℎ(𝑡)≤1+^𝑡_𝑡0𝑓(𝑠𝑧(𝑠))ℎ(𝑡)+^𝑠_𝑠0𝑘()𝑧(𝜏)𝑠,𝜏ℎ(𝜏)Δ𝜏Δ𝑠. (13) Now as an application of Lemma 1 we have 𝑧(𝑡)ℎ(𝑡)≤1+^𝑡_𝑡0𝑓(𝑠)𝑒_𝑓+𝐴𝑠,𝑠₀Δ𝑠. (14) The desired inequality (9) follows from (14) and the fact that 𝑢(𝑡)≤𝑎(𝑡)+𝑧(𝑡). If ℎ(𝑡)≥0, we carry out the above procedure with ℎ(𝑡)+𝜖 instead of ℎ(𝑡), where 𝜖>0 is an arbitrary small constants, and then subsequently pass to the limit as 𝜖→0 to obtain (9). The following theorem deals with two independent versions of inequalities established in Theorem 3.

Theorem 4. Let 𝑢,𝑓∈𝐶_rd(Ω,𝑅₊), 𝑘(𝑥,𝑦,𝑠,𝑡), 𝑘^Δ1(𝑥,𝑦,𝑠,𝑡), 𝑘^Δ2(𝑥,𝑦,𝑠,𝑡), 𝑘^Δ2Δ1(𝑥,𝑦,𝑠,𝑡)∈  𝐶_rd(Ω×Ω,𝑅₊), and 𝑐 be a nonnegative constants. If 𝑢𝑥,𝑦≤𝑐+^𝑥_𝑥0^𝑦_𝑦0𝑓()×𝑢()+𝑠,𝑡𝑠,𝑡^𝑠_𝑠0^𝑡_𝑡0𝑘(𝑠,𝑡,𝜏,𝜉)𝑢(𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠, (15) for (𝑥,𝑦)∈Ω, then 𝑢𝑥,𝑦≤𝑐1+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑒_∫^𝑡_{𝑡0[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝜉Δ𝜏}𝑥,𝑥₀Δ𝑡Δ𝑠, (16) where 𝐴𝑥,𝑦=𝑘𝑥,𝑦,𝜎(𝑥),𝜎𝑦+^𝑥_𝑥0𝑘^Δ1𝑦+𝑥,𝑦,𝜏,𝜎Δ𝜏_𝑦𝑦0𝑘^Δ1(𝑥)+𝑥,𝑦,𝜎,𝜂Δ𝜂_{𝑥𝑥0𝑦𝑦0}𝑘^Δ2Δ1𝑥,𝑦,𝜏,𝜂Δ𝜂Δ𝜏, (17) for (𝑥,𝑦)∈Ω.

Proof. Let 𝑐>0 and define a function 𝑧(𝑥,𝑦) by the right-hand side of (15); then 𝑧(𝑥₀,𝑦)=𝑧(𝑥,𝑦₀)=𝑐, 𝑢(𝑥,𝑦)≤𝑧(𝑥,𝑦), and 𝑧^Δ2Δ1𝑢+𝑥,𝑦=𝑓𝑥,𝑦𝑥,𝑦^𝑥_𝑥0^𝑦_𝑦0𝑘𝑢()𝑧+𝑥,𝑦,𝜏,𝜉𝜏,𝜉Δ𝜉Δ𝜏≤𝑓𝑥,𝑦𝑥,𝑦^𝑥_𝑥0^𝑦_𝑦0𝑘𝑥,𝑦,𝜏,𝜉𝑧(𝜏,𝜉)Δ𝜉Δ𝜏. (18) Define a function 𝑣(𝑥,𝑦) by 𝑣𝑥,𝑦=𝑧𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑘𝑥,𝑦,𝜏,𝜉𝑧(𝜏,𝜉)Δ𝜉Δ𝜏, (19) then 𝑣(𝑥₀,𝑦)=𝑧(𝑥₀,𝑦)=𝑐, 𝑣(𝑥,𝑦₀)=𝑧(𝑥,𝑦₀)=𝑐, 𝑧(𝑥,𝑦)≤𝑣(𝑥,𝑦), 𝑧^Δ1Δ2(𝑥,𝑦)≤𝑓(𝑥,𝑦)𝑣(𝑥,𝑦), 𝑣(𝑥,𝑦) is nondecreasing for (𝑥,𝑦)∈Ω, and 𝑣^Δ1Δ2𝑥,𝑦=𝑧^Δ1Δ2(𝑥)𝑦𝑧+𝑥,𝑦+𝑘𝑥,𝑦,𝜎,𝜎𝑥,𝑦_𝑥𝑥0𝑘^Δ1𝑦𝑧+𝑥,𝑦,𝜏,𝜎𝜏,𝑦Δ𝜏_𝑦𝑦0𝑘^Δ2(𝑥)𝑧()+𝑥,𝑦,𝜎,𝜉𝑥,𝜉Δ𝜉_{𝑥𝑥0𝑦𝑦0}𝑘^Δ2Δ1𝑧()𝑣(𝑥)𝑦𝑣+𝑥,𝑦,𝜏,𝜉𝜏,𝜉Δ𝜉Δ𝜏≤𝑓𝑥,𝑦𝑥,𝑦+𝑘𝑥,𝑦,𝜎,𝜎𝑥,𝑦_𝑥𝑥0𝑘^Δ1𝑥,𝑦,𝜏,𝜎𝑦𝑣𝜏,𝑦Δ𝜏+^𝑦_𝑦0𝑘^Δ2(𝑥)𝑣()+𝑥,𝑦,𝜎,𝜉𝑥,𝜉Δ𝜉_{𝑥𝑥0𝑦𝑦0}𝑘^Δ2Δ1𝑣()≤𝑓𝑣,𝑥,𝑦,𝜏,𝜉𝜏,𝜉Δ𝜉Δ𝜏𝑥,𝑦+𝐴𝑥,𝑦𝑥,𝑦 (20) where 𝐴(𝑥,𝑦) is defined by (17).
Using the fact that 𝑧(𝑥,𝑦)>0,   𝑧^Δ1(𝑥,𝑦)≥0,   𝑧^Δ2(𝑥,𝑦)≥0, and we have 𝑣^Δ2Δ1𝑥,𝑦𝑣≤𝑓+𝑣𝑥,𝑦𝑥,𝑦+𝐴𝑥,𝑦^Δ1𝑥,𝑦𝑣^Δ2𝑥,𝑦𝑣𝑣𝑦.𝑥,𝑦𝑥,𝜎 (21)
Now keeping 𝑥 fixed and delta integrating with respect to 𝑡 from 𝑦₀ to 𝑦, we obtain estimates 𝑣^Δ1𝑥,𝑦𝑣≤𝑥,𝑦^𝑦_𝑦0𝑓(𝑥,𝜉)+𝐴𝑥,𝑦Δ₂𝜉, (22) keeping 𝑦 fixed in (22) and delta integrating 𝑠 from 𝑥₀ to 𝑥, we get 𝑣𝑥,𝑦≤𝑐+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)𝑣𝑥,𝑦Δ₂𝜉Δ₁𝜏. (23) Then Lemma 2 implies 𝑣𝑥,𝑦≤𝑐+𝑒_∫^𝑦_{𝑦0[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝜉}𝑥,𝑥₀. (24) Using (24) in (18) and integrating the resulting inequality from 0 to 𝑦 and then from 0 to 𝑥 for (𝑥,𝑦)∈Ω, we get 𝑧𝑥,𝑦≤𝑐1+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝜏)𝑒_∫^𝑡_{𝑡0[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝜉Δ𝜏}𝑠,𝑠₀Δ𝑡Δ𝑠. (25)
Using (25) in 𝑢(𝑥,𝑦)≤𝑧(𝑥,𝑦), we get the required inequality in (16).

Theorem 5. Let 𝑢,   𝑓,   𝑎∈𝐶_rd(Ω×Ω,𝑅₊) and 𝑘(𝑥,𝑦,𝑠,𝑡),   𝑘^Δ1(𝑥,𝑦,𝑠,𝑡),   𝑘^Δ2(𝑥,𝑦,𝑠,𝑡),  𝑘^Δ2Δ1(𝑥,𝑦,𝑠,𝑡)∈𝐶_rd(Ω,𝑅₊). If 𝑢+𝑥,𝑦≤𝑎𝑥,𝑦^𝑥_𝑥0^𝑦_𝑦0𝑓()×𝑢()+𝑠,𝑡𝑠,𝑡^𝑠_𝑠0^𝑡_𝑡0𝑘(𝑠,𝑡,𝜏,𝜉)𝑢(𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠, (26) For (𝑥,𝑦)∈Ω, then 𝑢+𝑥,𝑦≤𝑎𝑥,𝑦ℎ×𝑥,𝑦1+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑒_∫^𝑡_{𝑡0[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝜉Δ𝜏}𝑥,𝑥₀Δ𝑡Δ𝑠, (27) where ℎ=𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓()×𝑎()𝑠,𝑡𝑠,𝑡+^𝑠_𝑠0^𝑡_𝑡0𝑘(𝑠,𝑡,𝜎(𝑥),𝜉)𝑎(𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠, (28) for (𝑥,𝑦)∈Ω and 𝐴(𝑥,𝑦) as defined in (17).

Proof. Define a function 𝑧(𝑥,𝑦) by the right-hand side of (26) 𝑧𝑥,𝑦≤^𝑥_𝑥0^𝑦_𝑦0𝑓()×𝑢()+𝑠,𝑡𝑠,𝑡^𝑠_𝑠0^𝑡_𝑡0𝑘(𝑠,𝑡,𝜏,𝜉)𝑢(𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠, (29) then from (26) we have 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝑧(𝑥,𝑦), and using this in (28) we get 𝑧≤𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓(𝑠,𝑡)𝑎(𝑠,𝑡)+𝑧(𝑠,𝑡)+^𝑠_𝑠0^𝑡_𝑡0𝑘()×𝑎()(≤𝑠,𝑡,𝜏,𝜉𝜏,𝜉+𝑧𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠ℎ𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑧(𝑠,𝑡)+^𝑠_𝑠0^𝑡_𝑡0𝑘()()𝑠,𝑡,𝜏,𝜉×𝑧𝜏,𝜉Δ𝜉Δ𝜏Δ𝑡Δ𝑠, (30) where ℎ(𝑥,𝑦) is defined by (28). Here ℎ(𝑥,𝑦) nondecreasing for (𝑥,𝑦)∈Ω, then from (30) we have 𝑧𝑥,𝑦ℎ𝑥,𝑦≤1+^𝑥_𝑥0^𝑦_𝑦0𝑓()×𝑧()𝑠,𝑡𝑠,𝑡ℎ(𝑠,𝑡)+^𝑠_𝑠0^𝑡_𝑡0𝑘()𝑧()𝑠,𝑡,𝜏,𝜉𝜏,𝜉ℎ(𝜏,𝜉)Δ𝜉Δ𝜏Δ𝑡Δ𝑠. (31) Now as an application of inequality in Theorem 3 to (31) we get 𝑧𝑥,𝑦ℎ≤𝑥,𝑦1+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑒_{[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝜉Δ𝜏}𝑥,𝑥₀Δ𝑡Δ𝑠. (32) The required inequality in (27) follows from (32) and using the fact that 𝑢(𝑥,𝑦)≤𝑎(𝑥,𝑦)+𝑧(𝑥,𝑦).

3. Applications

In this section, we give application of inequality in Theorem 4 to study certain properties of solution of nonlinear partial integrodifferential equation on time scales, 𝑢^Δ2Δ1,𝑥,𝑦=𝐹𝑥,𝑦,𝑢𝑥,𝑦^𝑥_𝑥0^𝑦_𝑦0ℎ𝑥,𝑦,𝜏,𝜉,𝑢(𝜏,𝜉)Δ𝜉Δ𝜏, (33) with the initial boundary condition 𝑢𝑥,𝑦₀=𝛼₁(𝑥),𝑢𝑥₀,𝑦=𝛼₂𝑦,𝛼₁(0)=𝛼₂(0)=0, (34) where 𝑢∈𝐶_rd(Ω,𝑅₊), ℎ∈𝐶_rd(Ω×𝑅,𝑅₊), and 𝐹∈𝐶_rd(Ω×Ω,𝑅₊).

The following theorem deals with the estimate on the solution of (33)-(34).

Theorem 6. Assume that ℎ()|𝑢(,𝐹𝑢|+|𝑣,𝛼𝑥,𝑦,𝑠,𝑡,𝑢𝑠,𝑡≤𝑘𝑥,𝑦,𝑠,𝑡𝑠,𝑡)|𝑥,𝑦,𝑢,𝑣≤𝑓𝑥,𝑦[||]₁(𝑥)+𝛼₂𝑦≤𝑐, (35) where 𝑘, 𝑓, and 𝑐 are as defined in (9). If 𝑢(𝑥,𝑦), (𝑥,𝑦)∈Ω is any solution of (33)-(34), then 𝑢𝑥,𝑦≤𝑐1+^𝑥_𝑥0^𝑦_𝑦0𝑓(𝑠,𝑡)𝑒_∫^𝑡_{𝑡0[𝑓(𝜏,𝜉)+𝐴(𝜏,𝜉)]Δ𝑡}𝑠,𝑠₀Δ𝑡Δ𝑠 (36) for (𝑥,𝑦)∈Ω, where 𝐴(𝑥,𝑦) is defined by (17).

Proof. The solution 𝑢(𝑥,𝑦) of (33)-(34) can be written as 𝑢𝑥,𝑦=𝛼₁(𝑥)+𝛼₂𝑦+^𝑥_𝑥0^𝑦_𝑦0𝐹𝑠,𝑡,𝑢(𝑠,𝑡_{𝑠𝑠0𝑡𝑡0}),ℎ(𝑠,𝑡,𝜏,𝜉,𝑢(𝜏,𝜉))Δ𝜉Δ𝜏Δ𝑡Δ𝑠. (37) Using (35) in (37), we have 𝑢𝑥,𝑦≤𝑐+^𝑥_𝑥0^𝑦_𝑦0𝑓()×𝑢(+𝑠,𝑡|𝑠,𝑡)|^𝑠_𝑠0^𝑡_𝑡0𝑘𝑠,𝑡,𝜏,𝜉,𝑢(𝜏,𝜉)×Δ𝜉Δ𝜏Δ𝑡Δ𝑠. (38) Now an application of inequality in Theorem 4 yields the estimate in (36).

In the next result, we give uniqueness of solutions of (33)-(34).

Theorem 7. Suppose that the functions ℎ, 𝐹 in (33) satisfy the conditions ℎ𝑥,𝑦,𝑠,𝑡,𝑢₁−ℎ𝑥,𝑦,𝑠,𝑡,𝑢₂≤𝑘𝑥,𝑦,𝑠,𝑡𝑢₁−𝑢₂,𝐹𝑥,𝑦,𝑢₁,𝑢₂−𝐹𝑥,𝑦,𝑣₁,𝑣₂≤𝑓𝑥,𝑦𝑢₁−𝑣₁+𝑢₂−𝑣₂, (39) where 𝑘 and 𝑓 are as in Theorem 4. Then the problem (33)-(34) has at most one solution on Ω.

Proof. Let 𝑢₁(𝑥,𝑦) and 𝑢₂(𝑥,𝑦) be two solutions of (33)-(34) on Ω, then we have 𝑢₁𝑥,𝑦−𝑢₂=𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝐹𝑠,𝑡,𝑢₁(𝑠,𝑡),^𝑠_𝑠0^𝑡_𝑡0ℎ𝑠,𝑡,𝜏,𝜉,𝑢₁(𝜏,𝜉)Δ𝜉Δ𝜏−𝐹𝑠,𝑡,𝑢₂(𝑠,𝑡),^𝑠_𝑠0^𝑡_𝑡0ℎ𝑠,𝑡,𝜏,𝜉,𝑢₂(𝜏,𝜉)×Δ𝜉Δ𝜏Δ𝑡Δ𝑠. (40) From (39) and (40), we have 𝑢₁𝑥,𝑦−𝑢₂≤𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓(𝑠,𝑡)𝑢₁(𝑠,𝑡)−𝑢₂(𝑠,𝑡)+^𝑠_𝑠0^𝑡_𝑡0𝑘(𝑠,𝑡,𝜏,𝜉)𝑢₁(𝜏,𝜉)−𝑢₂(𝜏,𝜉)×Δ𝜉Δ𝜏Δ𝑡Δ𝑠. (41) As an application of inequality in Theorem 4 with 𝑐=0 yields |𝑢₁(𝑥,𝑦)−𝑢₂(𝑥,𝑦)|≤0, therefore 𝑢₁(𝑥,𝑦)=𝑢₂(𝑥,𝑦); that is, there is at most one solution of (33)-(34) on Ω.