Properties of Some Partial Dynamic Equations on Time Scales

Abstract

The main objective of the paper is to study the properties of the solution of a certain partial dynamic equation on time scales. The tools employed are based on the application of the Banach fixed-point theorem and a certain integral inequality with explicit estimates on time scales.

1. Introduction

Recently, there has been a lot of interest in shown studying various properties of dynamic equations on time scales by various authors [1–11]. In this paper, we study some partial dynamic equations on time scales. Let ℝ^𝑛 denote the 𝑛 dimensional Euclidean space with appropriate norm |⋅|. In this, let ℝ denote the set of real numbers, ℤ the set of integers, and 𝕋 the arbitrary time scales. Let 𝕋₁ and 𝕋₂ be two time scales, and let Ω=𝕋₁×𝕋₂. Let 𝐶_rd denote the set of rd-continuous function. We assume basic understanding of time scales and notations. More information about time scales calculus can be found in [12–14]. The partial delta derivative of 𝑧(𝑥,𝑦) for (𝑥,𝑦)∈Ω with respect to 𝑥, 𝑦, and 𝑥𝑦 is denoted by 𝑧^Δ1(𝑥,𝑦), 𝑧^Δ2(𝑥,𝑦), and 𝑧^Δ1Δ2(𝑥,𝑦)=𝑧^Δ2Δ1(𝑥,𝑦).

Many physical systems can be modeled using dynamical systems on time scales. As response to the needs of diverse applications, many authors have studied qualitative properties of various equations on time scales [4–9, 11]. Motivated by the results in this paper, I consider the partial dynamic equation of the form 𝑢^Δ2Δ1𝑥,𝑦=𝑓𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦, (1) with the initial boundary conditions 𝑢𝑥,𝑥₀=𝛼(𝑥),𝑢𝑦₀,𝑦=𝛽𝑦,𝑢𝑥₀,𝑦₀=0, (2) for (𝑥,𝑦)∈Ω, where ()=𝐻𝑢𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑔𝑥,𝑦,𝑚,𝑛,𝑢(𝑚,𝑛),𝑢^Δ2Δ1𝑥,𝑦Δ𝑛Δ𝑚, (3)𝑓∈𝐶_rd(Ω×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛), 𝑔∈𝐶_rd(Ω×Ω×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛), 𝛼,𝛽∈𝐶_rd(𝕋,ℝ^𝑛), and (𝐻₀)(𝑥,𝑦)=∫^𝑥_𝑥0∫^𝑦_𝑦0𝑔(𝑥,𝑦,𝑚,𝑛,0,0)Δ𝑛Δ𝑚.

2. Preliminaries and Basic Inequality

We now give some basic definitions and notations about time scales. Define the jump operators 𝜎,𝜌∶𝕋→𝕋 by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}. (4) If 𝜎(𝑡)=𝑡 and 𝜎(𝑡)<𝑡, then the point 𝑡∈𝕋 is left dense and left scattered. If 𝜌(𝑡)=𝑡 and 𝜌(𝑡)<𝑡, then the point 𝑡∈𝕋 is right-dense and right-scattered. If 𝕋 has a right scattered minimum 𝑚, define 𝕋^𝑘∶=𝕋−𝑚; otherwise, 𝕋^𝑘∶=𝕋. If 𝕋 has a left-scattered maximum 𝑀, define 𝕋^𝑘∶=𝕋−𝑀; otherwise, 𝕋^𝑘∶=𝕋. The graininess function 𝜇∶𝕋→𝑅₊=[0,∞) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. We say that 𝑝∶𝕋→ℝ is regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋. For 𝑓∶𝕋→𝑅 and 𝑡∈𝕋^𝑘, the delta derivative of 𝑓 at 𝑡 denoted by 𝑓^Δ(𝑡) is the number (provided it exists) with the property that given any 𝜖>0, there is a neighborhood 𝑈 of 𝑡 such that 𝑓(𝜎(𝑡),𝜏)−𝑓(𝑠,𝜏)−𝑓^Δ(𝑡,𝜏)(𝜎(𝑡)−𝑠)≤𝜖|𝜎(𝑡)−𝑠|, (5) for all 𝑠∈𝑈. For 𝕋=𝑅, 𝑓^Δ(𝑡)=𝑓^′(𝑡) the usual derivative; for 𝕋=𝑍, the delta derivative is the forward difference operator, 𝑓^Δ(𝑡)=𝑓(𝑡+1)−𝑓(𝑡). A function 𝑓∶𝕋→𝑅 is right-dense continuous or rd-continuous provided it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) and at left-dense points in 𝕋 and its left-sided limit exists (finite) at left-dense points in 𝕋. If 𝕋=𝑅, then 𝑓 is rd-continuous if and only if 𝑓 is continuous. It is known [12, Theorem 1.74] that 𝑓 is right-dense continuous, there is a function 𝐹 such that 𝑓^Δ(𝑡)=𝑓(𝑡) and ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎), (6) where 𝑎,𝑏∈𝕋. Note that when 𝕋=𝑅, 𝜎(𝑡)=𝑡, 𝜇(𝑡)=0, 𝑓^Δ=𝑓^′, and ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∫^𝑏_𝑎𝑓(𝑡)d𝑡, while then 𝕋=𝑍, 𝜎(𝑡)=𝑡+1, 𝜇(𝑡)=1, 𝑓^Δ=Δ𝑓, and ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∑^𝑝−1_𝑡=𝑎𝑓(𝑡). We denote by ℜ the set of all regressive and rd-continuous functions and ℜ⁺={𝑝∈ℝ∶1+𝜇(𝑡)𝑝(𝑡)>0forall𝑡∈𝕋}. For 𝑝∈ℜ, we define (see [12, Theorem 2.35]) the exponential function 𝑒_𝑝(⋅,𝑡₀) on time scale 𝕋 as the unique solution to the scalar initial value problem 𝑥^Δ=𝑝(𝑡)𝑥,𝑥𝑡₀=1. (7) If 𝑝∈ℜ⁺, then 𝑒_𝑝(𝑡,𝑡₀)>0 for all 𝑡∈𝕋 (see [12, Theorem 2.41]). As usual, the set of rd-continuous functions is denoted by 𝐶_rd.

Denote by |𝑧(𝑥,𝑦)|_𝑊=|𝑧(𝑥,𝑦)|+|𝑧^Δ1Δ2(𝑥,𝑦)|. For (𝑡,𝑠)∈Ω; the notation 𝑎(𝑡,𝑠)=𝑂(𝑏(𝑡,𝑠)); then there exists a constant 𝑞>0 such that |𝑎(𝑡,𝑠)/𝑏(𝑡,𝑠)|≤𝑞 right hand neighbourhood. Let 𝐺 be the space with function (𝜙(𝑥,𝑦),𝜙^Δ1Δ2(𝑥,𝑦))∈𝑊 which are rd-continuous for (𝑥,𝑦)∈Ω and satisfy the condition Φ𝑥,𝑦_𝑊≤𝑂𝑒_𝜆𝑥,𝑦, (8) for (𝑥,𝑦)∈Ω, where 𝜆>0 is a constant. In space 𝐺, define the norm |Φ|_𝐺=Sup_{𝑥,𝑦∈Ω}Φ𝑥,𝑦_𝑊𝑒_Θ𝜆𝑥,𝑦. (9) The norm defined in (9) is clearly a Banach space. Then, (8) implies that there is a constant 𝑁≥0 such that Φ𝑥,𝑦_𝑊≤𝑁𝑒_𝜆𝑥,𝑦. (10) Using (10), we observe that Φ𝑥,𝑦_𝑊≤𝑁. (11) The solution of (1) and (2) means a function 𝑢(𝑥,𝑦)∈𝐶_rd(Ω,ℝ^𝑛) satisfying (1) and (2). It is easy to observe that solution 𝑢(𝑥,𝑦) of initial boundary value problem (1) and (2) satisfies following dynamic integral equations on time scales: 𝑢(𝑥)𝑦+𝑥,𝑦=𝛼+𝛽_{𝑥𝑥0𝑦𝑦0}𝑓(𝑠,𝑡,𝑢(𝑠,𝑡),(𝐻𝑢)(𝑠,𝑡))Δ𝑡Δ𝑠, (12) for (𝑥,𝑦)∈Ω.

I need the following lemma proved in [15].

Lemma 1. Let 𝑢,𝑎,𝑓∈𝐶_rd(Ω,ℝ₊) with 𝑎 and 𝑓 nondecreasing in each of the variables and 𝑔(𝑡,𝑡₁,𝑠,𝑠₂)∈𝐶_rd(Ω×Ω,ℝ₊), 𝑎₁<𝑠₁<𝑡₁, 𝑎₂<𝑠₂<𝑡₂. If 𝑢𝑡₁,𝑡₂≤𝑎𝑡₁,𝑡₂+𝑓𝑡₁,𝑡₂^𝑡1_𝑎1^𝑡2_𝑎2𝑔𝑡,𝑡₁,𝑠,𝑠₂×𝑢𝑠₁,𝑠₂Δ₁𝑠₁Δ₂𝑠₂, (13) for (𝑡₁,𝑡₂)∈Ω, then 𝑢𝑡₁,𝑡₂≤𝑎𝑡₁,𝑡₂𝑒_∫^𝑡_{2𝑎2𝑓(𝑡,𝑡2)𝑔(𝑡,𝑡2,𝑡1,𝑠2)Δ2𝑠2}𝑡₁,𝑎₁, (14) for (𝑡₁,𝑡₂)∈Ω.

Now I prove the following integral inequality which is used in our results.

Lemma 2. Let 𝑤(𝑥,𝑦),𝑎(𝑥,𝑦)∈𝐶_rd(Ω,ℝ₊), 𝑝(𝑥,𝑦,𝑠,𝑡), 𝑏(𝑥,𝑦,𝑠,𝑡), 𝑏^Δ1(𝑥,𝑦,𝑠,𝑡), 𝑏^Δ2(𝑥,𝑦,𝑠,𝑡), 𝑏^Δ1Δ2(𝑥,𝑦,𝑠,𝑡)∈𝐶_rd(Ω×Ω,ℝ₊), and let 𝑐≥0 be a constant. If 𝑤𝑥,𝑦≤𝑐+^𝑥_𝑥0^𝑦_𝑦0𝑎()𝑤()𝑤()+𝑠,𝑡𝑠,𝑡+𝑏𝑥,𝑦,𝑠,𝑡𝑠,𝑡_{𝑠𝑥0𝑡𝑦0}𝑝()()𝑠,𝑡,𝑚,𝑛×𝑤𝑚,𝑛Δ𝑛Δ𝑚Δ𝑡Δ𝑠, (15) for (𝑥,𝑦)∈Ω, then 𝑤𝑥,𝑦≤𝑐𝑒_{𝑄(𝑥,𝑦)}𝑥,𝑥₀,𝑄𝑥,𝑦=^𝑦_𝑦0[𝑎(𝑠,𝑡)+𝐴(𝑠,𝑡)]Δ𝑡, (16)(17) for (𝑥,𝑦)∈Ω, where 𝐴𝑥,𝑦=𝑏𝑥,𝑦,𝑥,𝑦+^𝑥_𝑥0𝑏^Δ1+𝑥,𝑦,𝑚,𝑦Δ𝑚_𝑦𝑦0𝑏^Δ1+𝑥,𝑦,𝑥,𝑛Δ𝑛_{𝑥𝑥0𝑦𝑦0}𝑏^Δ1Δ2+𝑥,𝑦,𝑚,𝑛Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑝𝑥,𝑦,𝑚,𝑛Δ𝑛Δ𝑚. (18)

Proof. Define a function 𝑧(𝑥,𝑦) by the right hand side of (15), then 𝑧(𝑥,𝑥₀)=𝑧(𝑦₀,𝑦)=𝑐, 𝑤(𝑥,𝑦)≤𝑧(𝑥,𝑦), and 𝑧(𝑥,𝑦) is nondecreasing in 𝑥 and 𝑦: 𝑧^Δ1Δ2𝑤𝑤𝑥,𝑦=𝑎𝑥,𝑦𝑥,𝑦+𝑏𝑥,𝑦,𝑥,𝑦𝑥,𝑦+^𝑥_𝑥0𝑏^Δ1𝑤+𝑥,𝑦,𝑚,𝑦𝑚,𝑦Δ𝑚_𝑦𝑦0𝑏^Δ2𝑤()+𝑥,𝑦,𝑥,𝑛𝑥,𝑛Δ𝑛_{𝑥𝑥0𝑦𝑦0}𝑏^Δ2Δ1𝑤()+𝑥,𝑦,𝑚,𝑛𝑚,𝑛Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑝𝑤()𝑧𝑧+𝑥,𝑦,𝑚,𝑛𝑚,𝑛Δ𝑛Δ𝑚≤𝑎𝑥,𝑦𝑥,𝑦+𝑏𝑥,𝑦,𝑥,𝑦𝑥,𝑦_𝑥𝑥0𝑏^Δ1𝑧+𝑥,𝑦,𝑚,𝑦𝑚,𝑦Δ𝑚_𝑦𝑦0𝑏^Δ2𝑧()+𝑥,𝑦,𝑥,𝑛𝑥,𝑛Δ𝑛_{𝑥𝑥0𝑦𝑦0}𝑏^Δ2Δ1𝑧()+𝑥,𝑦,𝑚,𝑛𝑚,𝑛Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑝𝑧()≤𝑎+𝑥,𝑦,𝑚,𝑛𝑚,𝑛Δ𝑛Δ𝑚𝑥,𝑦+𝑏𝑥,𝑦,𝑥,𝑦_𝑥𝑥0𝑏^Δ1+𝑥,𝑦,𝑚,𝑦Δ𝑚_𝑦𝑦0𝑏^Δ2+𝑥,𝑦,𝑥,𝑛Δ𝑛_{𝑥𝑥0𝑦𝑦0}𝑏^Δ2Δ1+𝑥,𝑦,𝑚,𝑛Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑝𝑧=𝑎𝑧,𝑥,𝑦,𝑚,𝑛Δ𝑛Δ𝑚𝑥,𝑦𝑥,𝑦+𝐴𝑥,𝑦𝑥,𝑦 (19) where 𝐴(𝑥,𝑦) is given by (18). Now, by application of Lemma 1, we get 𝑧𝑥,𝑦≤𝑐𝑒_∫^𝑦_{𝑦0[𝑎(𝑠,𝑡)+𝐴(𝑠,𝑡)]Δ2𝑡}𝑥,𝑥₀. (20) Using (20) in 𝑤(𝑥,𝑦)≤𝑧(𝑥,𝑦), we get the required inequality (16).

3. Existence and Uniqueness

Now our main results are as follows.

Theorem 3. Suppose that(i) the function 𝑓,𝑔 in (1) satisfies the conditions 𝑓𝑥,𝑦,𝑢,𝑣,𝑤−𝑓𝑥,𝑦,𝑢,𝑣,𝑤≤𝑘𝑥,𝑦[|𝑢−𝑢|+|𝑣−𝑣|+|𝑤−𝑤,𝑔|]𝑥,𝑦,𝑚,𝑛,𝑢,𝑣−𝑔𝑥,𝑦,𝑚,𝑛,𝑢,𝑣≤ℎ𝑥,𝑦,𝑚,𝑛[|𝑢−𝑢|+|𝑣−𝑣|], (21)
where 𝑘(𝑥,𝑦)∈𝐶_rd(Ω,ℝ₊), ℎ(𝑥,𝑦,𝑚,𝑛)∈𝐶_rd(Ω×Ω,ℝ₊).(ii) for 𝜆 as in (10),(a) there exists a nonnegative constant 𝛾 such that 𝛾<1, 𝐿𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝐿(𝑠,𝑡)Δ𝑡Δ𝑠≤𝛾𝑒_𝜆𝑥,𝑦, (22) where 𝐿𝑥,𝑦=𝑘𝑥,𝑦𝑒_𝜆+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}ℎ𝑥,𝑦,𝑚,𝑛×𝑒_𝜆(𝑚,𝑛)Δ𝑛Δ𝑚, (23)(b) there exists a nonnegative constant 𝜂 such that |𝛼(𝑥+𝛽𝑦+𝑓)|𝑥,𝑦,𝑥₀,𝑦₀,()+𝐻0𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑥₀,𝑦₀,(𝐻0)(𝑠,𝑡)Δ𝑡Δ𝑠≤𝜂𝑒_𝜆𝑥,𝑦; (24)
then, the initial boundary value problem (1) and (2) has a unique solution on Ω.

Proof. Let 𝑢(𝑥,𝑦)∈𝐺, and define the operator 𝑆 by ()(𝑥)𝑦+𝑆𝑢𝑥,𝑦=𝛼+𝛽_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢^Δ2Δ1(),(𝑠,𝑡𝐻𝑢)(𝑠,𝑡)Δ𝑡Δ𝑠. (25) By delta differentiating (25) with respect to 𝑥 and 𝑦, we get (𝑆𝑢)^Δ2Δ1𝑥,𝑦=𝑓𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦. (26) Now, we show that 𝑆𝑢 maps 𝐺 into itself and 𝑆𝑢 is rd-continuous on Ω.
From (25) and (26) and hypothesis, we have (𝑆𝑢)𝑥,𝑦+(𝑆𝑢)^Δ2Δ1≤|𝛼(𝑥+𝛽𝑦+𝑥,𝑦)|_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢^Δ2Δ1(𝑠,𝑡),(𝐻𝑢)(𝑠,𝑡)−𝑓𝑠,𝑡,𝑥₀,𝑦₀,()+𝐻0)(𝑠,𝑡Δ𝑡Δ𝑠_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑥₀,𝑦₀,()+𝑓𝐻0)(𝑠,𝑡Δ𝑡Δ𝑠𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦−𝑓𝑥,𝑦,𝑥₀,𝑦₀,()+𝑓𝐻0𝑥,𝑦𝑥,𝑦,𝑥₀,𝑦₀,(𝐻0)𝑥,𝑦≤𝜂𝑒_𝜆𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑘(𝑠,𝑡)|𝑢(𝑠,𝑡)|_𝐺+^𝑠_𝑥0^𝑡_𝑦0ℎ(𝑠,𝑡,𝑚,𝑛)|𝑢(𝑚,𝑛)|_𝐺}𝑢×Δ𝑛Δ𝑚Δ𝑡Δ𝑠+𝑘𝑥,𝑦𝑥,𝑦₁+^𝑥_𝑥0^𝑦_𝑦0ℎ𝑥,𝑦,𝑚,𝑛|𝑢(𝑚,𝑛)|_𝐺×Δ𝑛Δ𝑚≤𝜂𝑒_𝜆𝑥,𝑦+|𝑢|_𝐺𝑘𝑥,𝑦𝑒_𝜆+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}ℎ𝑥,𝑦,𝑚,𝑛×𝑒_𝜆()+𝑚,𝑛Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑘(𝑠,𝑡)𝑒_𝜆()+𝑠,𝑡_{𝑠𝑥0𝑡𝑦0}ℎ(𝑠,𝑡,𝑚,𝑛)×𝑒_𝜆(𝑚,𝑛)Δ𝑛Δ𝑚×Δ𝑡Δ𝑠≤𝜂𝑒_𝜆𝑥,𝑦+𝑁𝐿𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝐿()≤𝑒𝑠,𝑡Δ𝑡Δ𝑠𝜂+𝑁𝛾_𝜆𝑥,𝑦. (27) From (27), it follows that 𝑆𝑢∈𝐺, thus proving that 𝑆 maps 𝐺 into itself.
Now, we verify that 𝑆 is a contraction map. Let 𝑢(𝑥,𝑦),𝑣(𝑥,𝑦)∈𝐺. From (25) and (26) and using the hypotheses, we have (𝑆𝑢)𝑥,𝑦−(𝑆𝑣)𝑥,𝑦+(𝑆𝑢)^Δ2Δ1−()≤𝑥,𝑦𝑆𝑣𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢^Δ2Δ1(),()()𝑠,𝑡𝐻𝑢)(𝑠,𝑡−𝑓𝑠,𝑡,𝑣𝑠,𝑡,𝑣^Δ2Δ1(),()+𝑓𝑠,𝑡𝐻𝑣)(𝑠,𝑡×Δ𝑡Δ𝑠𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦−𝑓𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1,()≤𝑥,𝑦𝐻𝑣𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑘(𝑠,𝑡)|𝑢(𝑠,𝑡)−𝑣(𝑠,𝑡)|_𝐺+^𝑠_𝑥0^𝑡_𝑦0ℎ()×|𝑢()(𝑠,𝑡,𝑚,𝑛𝑚,𝑛−𝑣𝑚,𝑛)|_𝐺𝑢Δ𝑛Δ𝑚×Δ𝑡Δ𝑠+𝑘𝑥,𝑦𝑥,𝑦−𝑣𝑥,𝑦_𝐺+^𝑥_𝑥0^𝑦_𝑦0ℎ𝑥,𝑦,𝑚,𝑛|𝑢(𝑚,𝑛)−𝑣(𝑚,𝑛)|_𝐺≤||×Δ𝑛Δ𝑚𝑢−𝑣_𝐺𝑘𝑥,𝑦𝑒_𝜆+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}ℎ𝑥,𝑦,𝑚,𝑛𝑒_𝜆()+𝑚,𝑛×Δ𝑛Δ𝑚_{𝑥𝑥0𝑦𝑦0}𝑘(𝑠,𝑡)𝑒_𝜆()+𝑠,𝑡_{𝑠𝑥0𝑡𝑦0}ℎ(𝑠,𝑡,𝑚,𝑛)×𝑒_𝜆()=||𝑚,𝑛Δ𝑛Δ𝑚×Δ𝑡Δ𝑠𝑢−𝑣_𝑠𝐿𝑥,𝑦+^𝑥_𝑥0^𝑦_𝑦0𝐿()||𝑠,𝑡Δ𝑡Δ𝑠≤𝛾𝑢−𝑣_𝐺𝑒_𝜆𝑥,𝑦. (28) From (28), we have |𝑆𝑢−𝑆𝑣|_𝐺≤𝛾|𝑢−𝑣|_𝐺. (29)
Since 𝛾<1, it follows from Banach fixed-point theorem that 𝑆 has unique fixed point in 𝐺. The fixed point of 𝐺 is the solution of (1) and (2). This completes the proof.

Theorem 4. Suppose that function 𝑓,𝑔 in (1) satisfies the condition 𝑓𝑥,𝑦,𝑢,𝑣,𝑤−𝑓𝑥,𝑦,𝑢,𝑣,𝑤≤𝑑[|𝑢−𝑢|+|𝑣−𝑣|+|𝑤−𝑤,𝑔|]𝑥,𝑦,𝑚,𝑛,𝑢,𝑣−𝑔𝑥,𝑦,𝑚,𝑛,𝑢,𝑣≤𝑝𝑥,𝑦,𝑚,𝑛[|𝑢−𝑢|+|𝑣−𝑣|], (30) where 𝑑 is a nonnegative constant such that 𝑑<1 and 𝑝(𝑥,𝑦,𝑚,𝑛), 𝑝^Δ1(𝑥,𝑦,𝑚,𝑛), 𝑝^Δ2(𝑥,𝑦,𝑚,𝑛), 𝑝^Δ2Δ1(𝑥,𝑦,𝑚,𝑛), ∈𝐶_rd(Ω×Ω,ℝ^𝑛). Then, initial boundary value problem (1) and (2) has at most one solution on Ω.

Proof. Let 𝑢₁(𝑥,𝑦) and 𝑢₂(𝑥,𝑦) be any two solutions of (1) and (2) and 𝑤(𝑥,𝑦)=|𝑢₁(𝑥,𝑦)−𝑢₂(𝑥,𝑦)|+|𝑢^Δ2Δ1₁(𝑥,𝑦)−𝑢^Δ2Δ1₂(𝑥,𝑦)|; then by hypotheses we have 𝑤≤𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑢₁(𝑠,𝑡),𝑢^Δ2Δ1₁(𝑠,𝑡),𝐻𝑢₁(𝑠,𝑡)−𝑓𝑠,𝑡,𝑢₂(𝑠,𝑡),𝑢^Δ2Δ1₂(𝑠,𝑡),𝐻𝑢₂()+𝑓𝑠,𝑡×Δ𝑡Δ𝑠𝑥,𝑦,𝑢₁𝑥,𝑦,𝑢^Δ2Δ1₁𝑥,𝑦,𝐻𝑢₁𝑥,𝑦−𝑓𝑥,𝑦,𝑢₂𝑥,𝑦,𝑢^Δ2Δ1₂𝑥,𝑦,𝐻𝑢₂≤𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑑𝑢₁(𝑠,𝑡)−𝑢₂(+𝑢𝑠,𝑡)_Δ2Δ11(𝑠,𝑡)−𝑢^Δ2Δ1₂(+𝑠,𝑡)_{𝑠𝑥0𝑡𝑦0}𝑝()×𝑢𝑠,𝑡,𝑚,𝑛₁(𝑚,𝑛)−𝑢₂(+𝑢𝑚,𝑛)_Δ2Δ11(𝑚,𝑛)−𝑢^Δ2Δ1₂(𝑢𝑚,𝑛)×Δ𝑛Δ𝑚Δ𝑡Δ𝑠+𝑑₁𝑥,𝑦−𝑢₂+𝑢𝑥,𝑦_Δ2Δ11𝑥,𝑦−𝑢^Δ2Δ1₂+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑝×𝑢𝑥,𝑦,𝑚,𝑛₁(𝑚,𝑛)−𝑢₂(+𝑢𝑚,𝑛)_Δ2Δ11𝑥,𝑦−𝑢^Δ2Δ1₂𝑥,𝑦Δ𝑛Δ𝑚. (31) From (31), we have 𝑤≤1𝑥,𝑦1−𝑑^𝑥_𝑥0^𝑦_𝑦0()𝑤()+𝑑𝑤𝑠,𝑡+𝑝𝑥,𝑦,𝑠,𝑡𝑠,𝑡_{𝑠𝑥0𝑡𝑦0}𝑝(𝑠,𝑡,𝑚,𝑛)𝑤(𝑚,𝑛)Δ𝑛Δ𝑚×Δ𝑡Δ𝑠. (32)
Now, a suitable application of Lemma 2 to (32) yields 𝑢₁𝑥,𝑦−𝑢₂𝑥,𝑦+𝑢^Δ2Δ1₁𝑥,𝑦−𝑢^Δ2Δ1₂𝑥,𝑦≤0, (33) which implies that 𝑢₁(𝑥,𝑦)=𝑢₂(𝑥,𝑦) for (𝑥,𝑦)∈Ω. Therefore, there is at most one solution of (1) and (2) on Ω.

4. Properties of Solutions

The following theorem contains estimates of solutions of (1) and (2).

Theorem 5. Assume that 𝑓𝑢|+|𝑣|+|𝑤,𝑔𝑢|+|𝑣,𝛽(𝑥+𝜏𝑦𝑥,𝑦,𝑢,𝑣,𝑤≤𝛾[||]𝑥,𝑦,𝑚,𝑛,𝑢,𝑣≤𝑞𝑥,𝑦,𝑚,𝑛[||])≤𝜂, (34)(35)(36) where 𝛾,𝜂 are nonnegative constants such that 𝜂<1 and 𝑞(𝑥,𝑦,𝑚,𝑛), q^Δ1(𝑥,𝑦,𝑚,𝑛), 𝑞^Δ2(𝑥,𝑦,𝑚,𝑛), 𝑞^Δ2Δ1(𝑥,𝑦,𝑚,𝑛), ∈𝐶_rd(Ω×Ω,ℝ^𝑛). If 𝑢(𝑥,𝑦) for (𝑥,𝑦)∈Ω is any solution of (1) and (2); then 𝑢𝑥,𝑦+𝑢^Δ2Δ1≤𝜂𝑥,𝑦𝑒1−𝛾^{𝑄(𝑥,𝑦)}𝑥,𝑥₀, (37) where 𝑄𝑥,𝑦=^𝑦_𝑦0𝛾+1−𝛾𝐴(𝑠,𝑡)Δ𝑡, (38) for (𝑥,𝑦)∈Ω, 𝐴=1𝑥,𝑦𝑞+1−𝛾𝑥,𝑦,𝑚,𝑛_𝑥𝑥0𝑞^Δ1𝑥,𝑦,𝑚,𝑦Δ𝑚+^𝑦_𝑦0𝑞^Δ2+𝑥,𝑦,𝑥,𝑛Δ𝑛_{𝑥𝑥0𝑦𝑦0}𝑞^Δ2Δ1𝑥,𝑦,𝑚,𝑛Δ𝑛Δ𝑚. (39)

Proof. Since 𝑢(𝑥,𝑦) is a solution of (1) and (2) and by hypothesis, we have 𝑢𝑥,𝑦+𝑢^Δ2Δ1(𝑥)𝑦+𝑥,𝑦≤𝛽+𝜏_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢^Δ2Δ1(),()+𝑓𝑠,𝑡𝐻𝑢)(𝑠,𝑡×Δ𝑡Δ𝑠𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦≤𝜂+^𝑥_𝑥0^𝑦_𝑦0𝛾|𝑢(𝑠,𝑡)|+𝑢^Δ2Δ1(+𝑠,𝑡)_{𝑠𝑥0𝑡𝑦0}𝑞()×𝑢(+𝑢𝑠,𝑡,𝑚,𝑛|𝑚,𝑛)|^Δ2Δ1(𝑢+𝑢𝑚,𝑛)×Δ𝑛Δ𝑚Δ𝑡Δ𝑠+𝛾𝑥,𝑦^Δ2Δ1+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑞𝑥,𝑦,𝑚,𝑛|𝑢(𝑚,𝑛)|+𝑢^Δ2Δ1(𝑚,𝑛)×Δ𝑛Δ𝑚. (40) From (40), we have 𝑢𝑥,𝑦+𝑢^Δ2Δ1≤𝜂𝑥,𝑦+11−𝛾1−𝛾^𝑥_𝑥0^𝑦_𝑦0𝛾|𝑢(𝑠,𝑡)|+𝑢^Δ2Δ1(𝑢(+𝑢𝑠,𝑡)+𝑞𝑥,𝑦,𝑠,𝑡|𝑠,𝑡)|_Δ2Δ1(+𝑠,𝑡)_{𝑠𝑥0𝑡𝑦0}𝑞()×𝑢(+𝑢𝑠,𝑡,𝑚,𝑛|𝑚,𝑛)|^Δ2Δ1(𝑚,𝑛)×Δ𝑛Δ𝑚Δ𝑡Δ𝑠. (41) Now, a suitable application of Lemma 2 to (41) yields (37).

Remark 6. If the estimate obtained in (37) is bounded, then solution 𝑢(𝑥,𝑦) of (1) and (2) and also 𝑢^Δ2Δ1(𝑥,𝑦) is bounded on Ω.

The following result deals with the continuous dependence of solution of (1) and (2).

Theorem 7. Assume that the functions 𝑓,𝑔 in (1) and (2) satisfy the conditions (30). Let 𝑢₁(𝑥,𝑦) and 𝑢₂(𝑥,𝑦) be the solutions of (1) with the given initial boundary conditions 𝑢₁𝑥,𝑥₀=𝛽₁(𝑥),𝑢₁𝑦₀,𝑦=𝜏₁𝑦,𝑢₁𝑥₀,𝑦₀𝑢=0,₂𝑥,𝑥₀=𝛽₂(𝑥),𝑢₂𝑦₀,𝑦=𝜏₂𝑦,𝑢₁𝑥₀,𝑦₀=0, (42)(43) respectively where 𝛽₁,𝛽₂,𝜏₁,𝜏₂∈𝐶_rd(𝕋,ℝ^𝑛), 𝛽₁(𝑥)+𝜏₁𝑦−𝛽₂(𝑥)−𝜏₂𝑦≤𝜇, (44) where 𝜇≥0 is a constant. Then 𝑢₁𝑥,𝑦−𝑢₂𝑥,𝑦+𝑢^Δ2Δ1₁𝑥,𝑦−𝑢^Δ2Δ1₂≤𝜇𝑥,𝑦𝑒1−𝑑^{𝑝(𝑥,𝑦)}𝑥,𝑥₀, (45) for (𝑥,𝑦)∈Ω, where 𝑝𝑥,𝑦=^𝑦_𝑦0𝑑(1−𝑑+𝐵𝑠,𝑡)Δ𝑡Δ𝑠, (46) where 𝐵(𝑠,𝑡) is defined by the right hand side of (18) replacing 𝑏(𝑥,𝑦,𝑚,𝑛) and 𝑝(𝑥,𝑦,𝑚,𝑛) by (1/(1−𝑑))𝑝(𝑥,𝑦,𝑚,𝑛).

Proof. Let 𝑤(𝑥,𝑦)=|𝑢₁(𝑥,𝑦)−𝑢₂(𝑥,𝑦)|+|𝑢^Δ2Δ1₁(𝑥,𝑦)−𝑢^Δ2Δ1₂(𝑥,𝑦)| for (𝑥,𝑦)∈Ω. We have 𝑤≤𝛽𝑥,𝑦₁(𝑥)+𝜏₁𝑦−𝛽₂(𝑥)−𝜏₂𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓𝑠,𝑡,𝑢₁(𝑠,𝑡),𝑢^Δ2Δ1₁(𝑠,𝑡),𝐻𝑢₁(𝑠,𝑡)−𝑓𝑠,𝑡,𝑢₂(𝑠,𝑡),𝑢^Δ2Δ1₂(𝑠,𝑡),𝐻𝑢₂()+𝑓𝑠,𝑡×Δ𝑡Δ𝑠𝑥,𝑦,𝑢₁𝑥,𝑦,𝑢^Δ2Δ1₁𝑥,𝑦,𝐻𝑢₁𝑥,𝑦−𝑓𝑥,𝑦,𝑢₂𝑥,𝑦,𝑢^Δ2Δ1₂𝑥,𝑦,𝐻𝑢₂𝑥,𝑦+𝜇+^𝑥_𝑥0^𝑦_𝑦0𝑑𝑢₁(𝑠,𝑡)−𝑢₂(+𝑢𝑠,𝑡)_Δ2Δ11(𝑠,𝑡)−𝑢^Δ2Δ1₂(+𝑠,𝑡)_{𝑠𝑥0𝑡𝑦0}𝑝()×𝑢𝑠,𝑡,𝑚,𝑛₁(𝑚,𝑛)−𝑢₂(+𝑢𝑚,𝑛)_Δ2Δ11(𝑚,𝑛)−𝑢^Δ2Δ1₂(𝑢𝑚,𝑛)×Δ𝑛Δ𝑚Δ𝑡Δ𝑠+𝑑₁𝑥,𝑦−𝑢₂+𝑢𝑥,𝑦_Δ2Δ11𝑥,𝑦−𝑢^Δ2Δ1₂+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑝×𝑢𝑥,𝑦,𝑚,𝑛₁(𝑚,𝑛)−𝑢₂(+𝑢𝑚,𝑛)_Δ2Δ11(𝑚,𝑛)−𝑢^Δ2Δ1₂(𝑚,𝑛)Δ𝑛Δ𝑚. (47) From (47), we have 𝑤≤𝜇𝑥,𝑦+11−𝑑1−𝑑×^𝑥_𝑥0^𝑦_𝑦0𝑤()𝑤()+𝑠,𝑡+𝑝𝑥,𝑦,𝑠,𝑡𝑠,𝑡_{𝑠𝑥0𝑡𝑦0}𝑝(𝑠,𝑡,𝑚,𝑛)𝑤(𝑚,𝑛)Δ𝑛Δ𝑚×Δ𝑡Δ𝑠. (48)
Now, a suitable application of Lemma 2 to (48) yields the bound (45) which shows the dependency of solution of (1) on the initial boundary conditions.
Now, we consider initial boundary value problem (1) and (2) and the corresponding initial boundary value problem 𝑣^Δ2Δ1𝑥,𝑦=𝐺𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1𝑥,𝑦,𝐻𝑣𝑥,𝑦, (49) with the initial boundary conditions 𝑣𝑥,𝑥₀=𝛽(𝑥),𝑣𝑦₀,𝑦=𝜏𝑦,𝑣𝑥₀,𝑦₀=0, (50) where =𝐻𝑣𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝐽𝑥,𝑦,𝑚,𝑛,𝑣(𝑚,𝑛),𝑣^Δ2Δ1(𝑚,𝑛)Δ𝑛Δ𝑚, (51)𝐺∈𝐶_rd(Ω×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛), 𝐽∈𝐶_rd(Ω×Ω×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛), and 𝛽,𝜏∈𝐶_rd(𝕋,ℝ^𝑛).
Now, we present the result which deals with continuous dependence of solution of initial boundary value problem (1) and (2) on the functions involved therein.

Theorem 8. Assume that the functions 𝑓,𝑔 in (1) satisfy the conditions (30) and 𝛽(𝑥)+𝜏𝑦−𝛽(𝑥)−𝜏𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣^Δ2Δ1,(()𝑥,𝑦𝐻𝑣)(𝑠,𝑡)−𝐺𝑠,𝑡,𝑣𝑠,𝑡,𝑣^Δ2Δ1(𝑠,𝑡),(+𝑓𝐻𝑣𝑠,𝑡)×Δ𝑡Δ𝑠𝑥,𝑣,𝑣𝑥,𝑦,𝑣^Δ2Δ1𝑥,𝑦,(𝐻𝑣)𝑥,𝑦−𝐺𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1𝑥,𝑦,𝐻𝑣𝑥,𝑦≤𝜖, (52) where 𝑓,𝑔,𝛽,𝜏 and 𝐺,𝐽,𝛽,𝜏 are as in initial boundary value problem (1) and (2) and initial boundary value problem (49), (50), 𝜖≥0 is a constant, and 𝑣(𝑥,𝑦) is a solution of initial boundary value problem (49) and (50). Then, solution (1) and (2) depends continuously on the functions involved therein.

Proof. Let 𝑤(𝑥,𝑦)=|𝑢(𝑥,𝑦)−𝑣(𝑥,𝑦)|+|𝑢^Δ2Δ1(𝑥,𝑦)−𝑣^Δ2Δ1(𝑥,𝑦)| for (𝑥,𝑦)∈Ω. We have 𝑤≤𝛽(𝑥)𝑦−𝑥,𝑦+𝜏𝛽(𝑥)−𝜏𝑦+^𝑥_𝑥0^𝑦_𝑦0𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢^Δ2Δ1(),()()𝑠,𝑡𝐻𝑢)(𝑠,𝑡−𝑓𝑠,𝑡,𝑣𝑠,𝑡,𝑣^Δ2Δ1(),()+𝑠,𝑡𝐻𝑣)(𝑠,𝑡×Δ𝑡Δ𝑠_{𝑥𝑥0𝑦𝑦0}𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣^Δ2Δ1,(()𝑥,𝑦𝐻𝑣)(𝑠,𝑡)−𝐺𝑠,𝑡,𝑣𝑠,𝑡,𝑣^Δ2Δ1(𝑠,𝑡),(+𝑓𝐻𝑣𝑠,𝑡)×Δ𝑡Δ𝑠𝑥,𝑦,𝑢𝑥,𝑦,𝑢^Δ2Δ1𝑥,𝑦,(𝐻𝑢)𝑥,𝑦−𝑓𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1,()+𝑓𝑥,𝑦𝐻𝑣𝑥,𝑦𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1𝑥,𝑦,(𝐻𝑣)𝑥,𝑦−𝐺𝑥,𝑦,𝑣𝑥,𝑦,𝑣^Δ2Δ1(𝑠,𝑡),𝐻𝑣(𝑠,𝑡)≤𝜖+^𝑥_𝑥0^𝑦_𝑦0𝑑𝑢()(+𝑢|𝑠,𝑡−𝑣𝑠,𝑡)|_Δ2Δ1(𝑠,𝑡)−𝑣^Δ2Δ1(+𝑠,𝑡)_{𝑥𝑥0𝑦𝑦0}𝑝()×𝑢()(+𝑢𝑠,𝑡,𝑚,𝑛|𝑚,𝑛−𝑣𝑚,𝑛)|_Δ2Δ1(𝑠,𝑡)−𝑣^Δ2Δ1(𝑢+𝑢𝑠,𝑡)×Δ𝑛Δ𝑚Δ𝑡Δ𝑠+𝑑𝑥,𝑦−𝑣𝑥,𝑦_Δ2Δ1𝑥,𝑦−𝑣^Δ2Δ1+𝑥,𝑦_{𝑥𝑥0𝑦𝑦0}𝑝×𝑢()(+𝑢𝑥,𝑦,𝑚,𝑛|𝑚,𝑛−𝑣𝑚,𝑛)|_Δ2Δ1(𝑚,𝑛)−𝑣^Δ2Δ1(𝑚,𝑛)×Δ𝑛Δ𝑚. (53) From (53), we have 𝑤≤𝜖𝑥,𝑦+11−𝑑1−𝑑^𝑥_𝑥0^𝑦_𝑦0𝑤(𝑠,𝑡)+𝑝𝑥,𝑦,𝑠,𝑡𝑤(𝑠,𝑡)+^𝑠_𝑥0^𝑡_𝑦0𝑝(𝑠,𝑡,𝑚,𝑛)𝑤(𝑚,𝑛)Δ𝑛Δ𝑚×Δ𝑡Δ𝑠. (54) Now, a suitable application of Lemma 2 yields 𝑤≤𝜖𝑥,𝑦𝑒1−𝑑^{𝑝(𝑥,𝑦)}𝑥,𝑥₀, (55) for (𝑥,𝑦)∈Ω, where 𝑝(𝑥,𝑦) is given by (43). From (35), it follows that solution of initial boundary value problem (1) and (2) depends continuously on the functions involved therein.

Remark 9. In this paper, we have studied the existence and uniqueness of solution of (1) using Banach fixed-point theorem. Here, we have offered simple and concise proofs of properties of solutions of (1). I believeed results given will serve as a model for further investigations.

Acknowledgment

The author is grateful to the anonymous referee whose comments and suggestions helped the author to improve the text.