# Properties of Some Partial Dynamic Equations on Time Scales

Authors: Deepak B. Pachpatte
Journal: International Journal of Partial Differential Equations
 (2013)

---

## 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.


---

## Body

## 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 ℝn 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 Crd 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 zΔ1(x,y), zΔ2(x,y), and zΔ1Δ2(x,y)=zΔ2Δ1(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
(1)uΔ2Δ1(x,y)=f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y)),
with the initial boundary conditions
(2)u(x,x0)=α(x),u(y0,y)=β(y),u(x0,y0)=0,
for (x,y)∈Ω, where
(3)(Hu)(x,y)=∫x0x‍∫y0y‍g(x,y,m,n,u(m,n),uΔ2Δ1(x,y))ΔnΔm,f∈Crd(Ω×ℝn×ℝn×ℝn,ℝn), g∈Crd(Ω×Ω×ℝn×ℝn,ℝn), α,β∈Crd(𝕋,ℝn), and (H0)(x,y)=∫x0x‍∫y0y‍g(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 operatorsσ,ρ:𝕋→𝕋 by
(4)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t}.
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 𝕋k:=𝕋-m; otherwise, 𝕋k:=𝕋. If 𝕋 has a left-scattered maximum M, define 𝕋k:=𝕋-M; otherwise, 𝕋k:=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∈𝕋k, the delta derivative of f at t denoted by fΔ(t) is the number (provided it exists) with the property that given any ϵ>0, there is a neighborhood U of t such that
(5)|f(σ(t),τ)-f(s,τ)-fΔ(t,τ)(σ(t)-s)|≤ϵ|σ(t)-s|,
for all s∈U. For 𝕋=R, fΔ(t)=f′(t) the usual derivative; for 𝕋=Z, the delta derivative is the forward difference operator, fΔ(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 fΔ(t)=f(t) and
(6)∫ab‍f(t)Δt=F(b)-F(a),
where a,b∈𝕋. Note that when 𝕋=R, σ(t)=t, μ(t)=0, fΔ=f′, and ∫ab‍f(t)Δt=∫ab‍f(t)dt, while then 𝕋=Z, σ(t)=t+1, μ(t)=1, fΔ=Δf, and ∫ab‍f(t)Δt=∑t=ap-1f(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 ep(·,t0) on time scale 𝕋 as the unique solution to the scalar initial value problem
(7)xΔ=p(t)x,x(t0)=1.
If p∈ℜ+, then ep(t,t0)>0 for all t∈𝕋 (see [12, Theorem 2.41]). As usual, the set of rd-continuous functions is denoted by Crd.Denote by|z(x,y)|W=|z(x,y)|+|zΔ1Δ2(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),ϕΔ1Δ2(x,y))∈W which are rd-continuous for (x,y)∈Ω and satisfy the condition
(8)|Φ(x,y)|W≤O(eλ(x,y)),
for (x,y)∈Ω, where λ>0 is a constant. In space G, define the norm
(9)|Φ|G=Sup(x,y)∈Ω[|Φ(x,y)|WeΘλ(x,y)].
The norm defined in (9) is clearly a Banach space. Then, (8) implies that there is a constant N≥0 such that
(10)|Φ(x,y)|W≤Neλ(x,y).
Using (10), we observe that
(11)|Φ(x,y)|W≤N.
The solution of (1) and (2) means a function u(x,y)∈Crd(Ω,ℝn) 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:
(12)u(x,y)=α(x)+β(y)+∫x0x‍∫y0y‍f(s,t,u(s,t),(Hu)(s,t))ΔtΔs,
for (x,y)∈Ω.I need the following lemma proved in [15].Lemma 1.
Letu,a,f∈Crd(Ω,ℝ+) with a and f nondecreasing in each of the variables and g(t,t1,s,s2)∈Crd(Ω×Ω,ℝ+), a1<s1<t1, a2<s2<t2. If
(13)u(t1,t2)≤a(t1,t2)+f(t1,t2)∫a1t1‍∫a2t2‍g(t,t1,s,s2)×u(s1,s2)Δ1s1Δ2s2,
for (t1,t2)∈Ω, then
(14)u(t1,t2)≤a(t1,t2)e∫a2t2‍f(t,t2)g(t,t2,t1,s2)Δ2s2(t1,a1),
for (t1,t2)∈Ω.Now I prove the following integral inequality which is used in our results.Lemma 2.
Letw(x,y),a(x,y)∈Crd(Ω,ℝ+), p(x,y,s,t), b(x,y,s,t), bΔ1(x,y,s,t), bΔ2(x,y,s,t), bΔ1Δ2(x,y,s,t)∈Crd(Ω×Ω,ℝ+), and let c≥0 be a constant. If
(15)w(x,y)≤c+∫x0x‍∫y0y‍{∫x0s‍a(s,t)w(s,t)+b(x,y,s,t)w(s,t)+∫x0s‍∫y0t‍p(s,t,m,n)×w(m,n)ΔnΔm∫x0s‍}ΔtΔs,
for (x,y)∈Ω, then
(16)w(x,y)≤ceQ(x,y)(x,x0),(17)Q(x,y)=∫y0y‍[a(s,t)+A(s,t)]Δt,
for (x,y)∈Ω, where
(18)A(x,y)=b(x,y,x,y)+∫x0x‍bΔ1(x,y,m,y)Δm+∫y0y‍bΔ1(x,y,x,n)Δn+∫x0x‍∫y0y‍bΔ1Δ2(x,y,m,n)ΔnΔm+∫x0x‍∫y0y‍p(x,y,m,n)ΔnΔm.Proof.
Define a functionz(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:
(19)zΔ1Δ2(x,y)=a(x,y)w(x,y)+b(x,y,x,y)w(x,y)+∫x0x‍bΔ1(x,y,m,y)w(m,y)Δm+∫y0y‍bΔ2(x,y,x,n)w(x,n)Δn+∫x0x‍∫y0y‍bΔ2Δ1(x,y,m,n)w(m,n)ΔnΔm+∫x0x‍∫y0y‍p(x,y,m,n)w(m,n)ΔnΔm≤a(x,y)z(x,y)+b(x,y,x,y)z(x,y)+∫x0x‍bΔ1(x,y,m,y)z(m,y)Δm+∫y0y‍bΔ2(x,y,x,n)z(x,n)Δn+∫x0x‍∫y0y‍bΔ2Δ1(x,y,m,n)z(m,n)ΔnΔm+∫x0x‍∫y0y‍p(x,y,m,n)z(m,n)ΔnΔm≤[∫y0y‍a(x,y)+b(x,y,x,y)+∫x0x‍bΔ1(x,y,m,y)Δm+∫y0y‍bΔ2(x,y,x,n)Δn+∫x0x‍∫y0y‍bΔ2Δ1(x,y,m,n)ΔnΔm+∫x0x‍∫y0y‍p(x,y,m,n)ΔnΔm]z(x,y)=[a(x,y)+A(x,y)]z(x,y),
where A(x,y) is given by (18). Now, by application of Lemma 1, we get
(20)z(x,y)≤ce∫y0y‍[a(s,t)+A(s,t)]Δ2t(x,x0).
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 functionf,g in (1) satisfies the conditions
(21)|f(x,y,u,v,w)-f(x,y,u¯,v¯,w¯)|≤k(x,y)[|u-u¯|+|v-v¯|+|w-w¯|],|g(x,y,m,n,u,v)-g(x,y,m,n,u¯,v¯)|≤h(x,y,m,n)[|u-u¯|+|v-v¯|],
wherek(x,y)∈Crd(Ω,ℝ+), h(x,y,m,n)∈Crd(Ω×Ω,ℝ+).(ii)
forλ as in (10),
(a)
there exists a nonnegative constantγ such that γ<1,
(22)L(x,y)+∫x0x‍∫y0y‍L(s,t)ΔtΔs≤γeλ(x,y),where(23)L(x,y)=k(x,y)eλ(x,y)+∫x0x‍∫y0y‍h(x,y,m,n)×eλ(m,n)ΔnΔm,(b)
there exists a nonnegative constantη such that
(24)|α(x)|+|β(y)|+|f(x,y,x0,y0,(H0)(x,y))|+∫x0x‍∫y0y‍|f(s,t,x0,y0,(H0)(s,t))|ΔtΔs≤ηeλ(x,y);
then, the initial boundary value problem (1) and (2) has a unique solution on Ω.Proof.
Letu(x,y)∈G, and define the operator S by
(25)(Su)(x,y)=α(x)+β(y)+∫x0x‍∫y0y‍f(s,t,u(s,t),uΔ2Δ1(s,t),(Hu)(s,t)uΔ2Δ1)ΔtΔs.
By delta differentiating (25) with respect to x and y, we get
(26)(Su)Δ2Δ1(x,y)=f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y)).
Now, we show that Su maps G into itself and Su is rd-continuous on Ω.
From (25) and (26) and hypothesis, we have
(27)|(Su)(x,y)|+|(Su)Δ2Δ1(x,y)|≤|α(x)|+|β(y)|+∫x0x‍∫y0y‍|f(s,t,u(s,t),uΔ2Δ1(s,t),(Hu)(s,t))-f(s,t,x0,y0,(H0)(s,t))|ΔtΔs+∫x0x‍∫y0y‍|f(s,t,x0,y0,(H0)(s,t))|ΔtΔs+|f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y))-f(x,y,x0,y0,(H0)(x,y))|+|f(x,y,x0,y0,(H0)(x,y))|≤ηeλ(x,y)+∫x0x‍∫y0y‍{k(s,t)|u(s,t)|G+∫x0s‍∫y0t‍h(s,t,m,n)|u(m,n)|G×ΔnΔm}ΔtΔs+k(x,y)|u(x,y)|1+∫x0x‍∫y0y‍h(x,y,m,n)|u(m,n)|G×ΔnΔm≤ηeλ(x,y)+|u|G{∫x0x‍k(x,y)eλ(x,y)+∫x0x‍∫y0y‍h(x,y,m,n)×eλ(m,n)ΔnΔm+∫x0x‍∫y0y‍{∫x0x‍k(s,t)eλ(s,t)+∫x0s‍∫y0t‍h(s,t,m,n)×eλ(m,n)ΔnΔm∫x0x‍}×ΔtΔs∫x0x‍}≤ηeλ(x,y)+N{L(x,y)+∫x0x‍∫y0y‍L(s,t)ΔtΔs}≤[η+Nγ]eλ(x,y).
From (27), it follows that Su∈G, thus proving that S maps G into itself.
Now, we verify thatS is a contraction map. Let u(x,y),v(x,y)∈G. From (25) and (26) and using the hypotheses, we have
(28)|(Su)(x,y)-(Sv)(x,y)|+|(Su)Δ2Δ1(x,y)-(Sv)(x,y)|≤∫x0x‍∫y0y‍|f(s,t,u(s,t),uΔ2Δ1(s,t),(Hu)(s,t))-f(s,t,v(s,t),vΔ2Δ1(s,t),(Hv)(s,t))|×ΔtΔs+|f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y))-f(x,y,v(x,y),vΔ2Δ1(x,y),(Hv)(x,y))|≤∫x0x‍∫y0y‍{∫x0s‍k(s,t)|u(s,t)-v(s,t)|G+∫x0s‍∫y0t‍h(s,t,m,n)×|u(m,n)-v(m,n)|GΔnΔm∫x0s‍}×ΔtΔs+k(x,y)|u(x,y)-v(x,y)|G+∫x0x‍∫y0y‍h(x,y,m,n)|u(m,n)-v(m,n)|G×ΔnΔm≤|u-v|G{∫x0s‍k(x,y)eλ(x,y)+∫x0x‍∫y0y‍h(x,y,m,n)eλ(m,n)×ΔnΔm+∫x0x‍∫y0y‍{∫x0s‍k(s,t)eλ(s,t)+∫x0s‍∫y0t‍h(s,t,m,n)×eλ(m,n)ΔnΔm∫x0s‍}×ΔtΔs∫x0s‍}=|u-v|s{L(x,y)+∫x0x‍∫y0y‍L(s,t)ΔtΔs}≤γ|u-v|Geλ(x,y).
From (28), we have
(29)|Su-Sv|G≤γ|u-v|G.
Sinceγ<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 functionf,g in (1) satisfies the condition
(30)|f(x,y,u,v,w)-f(x,y,u¯,v¯,w¯)|≤d[|u-u¯|+|v-v¯|+|w-w¯|],|g(x,y,m,n,u,v)-g(x,y,m,n,u¯,v¯)|≤p(x,y,m,n)[|u-u¯|+|v-v¯|],
where d is a nonnegative constant such that d<1 and p(x,y,m,n), pΔ1(x,y,m,n), pΔ2(x,y,m,n), pΔ2Δ1(x,y,m,n), ∈Crd(Ω×Ω,ℝn). Then, initial boundary value problem (1) and (2) has at most one solution on Ω.Proof.
Letu1(x,y) and u2(x,y) be any two solutions of (1) and (2) and w(x,y)=|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|; then by hypotheses we have
(31)w(x,y)≤∫x0x‍∫y0y‍|f(s,t,u1(s,t),u1Δ2Δ1(s,t),(Hu1)(s,t))-f(s,t,u2(s,t),u2Δ2Δ1(s,t),(Hu2)(s,t))|×ΔtΔs+|f(x,y,u1(x,y),u1Δ2Δ1(x,y),(Hu1)(x,y))-f(x,y,u2(x,y),u2Δ2Δ1(x,y),(Hu2)(x,y))|≤∫x0x‍∫y0y{d∫x0s‍[u2Δ2Δ1|u1(s,t)-u2(s,t)|‍+|u1Δ2Δ1(s,t)-u2Δ2Δ1(s,t)|]+∫x0s‍∫y0t‍p(s,t,m,n)×[u1Δ2Δ1|u1(m,n)-u2(m,n)|+|u1Δ2Δ1(m,n)-u2Δ2Δ1(m,n)|]×ΔnΔm∫x0s‍u2Δ2Δ1}ΔtΔs+d[u1Δ2Δ1|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|]+∫x0x‍∫y0y‍p(x,y,m,n)×[u2Δ2Δ1|u1(m,n)-u2(m,n)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|]ΔnΔm.
From (31), we have
(32)w(x,y)≤11-d∫x0x‍∫y0y‍[∫y0t‍dw(s,t)+p(x,y,s,t)w(s,t)+∫x0s‍∫y0t‍p(s,t,m,n)w(m,n)ΔnΔm]×ΔtΔs.
Now, a suitable application of Lemma2 to (32) yields
(33)|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|≤0,
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(34)|f(x,y,u,v,w)|≤γ[|u|+|v|+|w|],(35)|g(x,y,m,n,u,v)|≤q(x,y,m,n)[|u|+|v|],(36)|β(x)|+|τ(y)|≤η,
where γ,η are nonnegative constants such that η<1 and q(x,y,m,n), qΔ1(x,y,m,n), qΔ2(x,y,m,n), qΔ2Δ1(x,y,m,n), ∈Crd(Ω×Ω,ℝn). If u(x,y) for (x,y)∈Ω is any solution of (1) and (2); then
(37)|u(x,y)|+|uΔ2Δ1(x,y)|≤η1-γeQ¯(x,y)(x,x0),
where
(38)Q¯(x,y)=∫y0y‍[γ1-γ+A¯(s,t)]Δt,
for (x,y)∈Ω,
(39)A¯(x,y)=11-γq(x,y,m,n)+∫x0x‍qΔ1(x,y,m,y)Δm+∫y0y‍qΔ2(x,y,x,n)Δn+∫x0x‍∫y0y‍qΔ2Δ1(x,y,m,n)ΔnΔm.Proof.
Sinceu(x,y) is a solution of (1) and (2) and by hypothesis, we have
(40)|u(x,y)|+|uΔ2Δ1(x,y)|≤β(x)+τ(y)+∫x0x‍∫y0y‍|f(s,t,u(s,t),uΔ2Δ1(s,t),(Hu)(s,t))|×ΔtΔs+|f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y))|≤η+∫x0x‍∫y0y‍{∫x0s‍γ[|u(s,t)|+|uΔ2Δ1(s,t)|]+∫x0s‍∫y0tq(s,t,m,n)‍×[|u(m,n)|+|uΔ2Δ1(m,n)|]×ΔnΔmuΔ2Δ1∫x0s‍}ΔtΔs+γ[|u(x,y)|+|uΔ2Δ1(x,y)|]+∫x0x‍∫y0y‍q(x,y,m,n)[|u(m,n)|+|uΔ2Δ1(m,n)|]×ΔnΔm.
From (40), we have
(41)|u(x,y)|+|uΔ2Δ1(x,y)|≤η1-γ+11-γ∫x0x‍∫y0y‍{γ[|u(s,t)|+|uΔ2Δ1(s,t)|]+q(x,y,s,t)[|uΔ2Δ1(s,t)||u(s,t)|+|uΔ2Δ1(s,t)|]+∫x0s‍∫y0t‍q(s,t,m,n)×[|u(m,n)|+|uΔ2Δ1(m,n)|]×ΔnΔm[|u(s,t)|+|uΔ2Δ1(s,t)|]}ΔtΔs.
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 uΔ2Δ1(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 functionsf,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
(42)u1(x,x0)=β1(x),u1(y0,y)=τ1(y),u1(x0,y0)=0,(43)u2(x,x0)=β2(x),u2(y0,y)=τ2(y),u1(x0,y0)=0,
respectively where β1,β2,τ1,τ2∈Crd(𝕋,ℝn),
(44)|β1(x)+τ1(y)-β2(x)-τ2(y)|≤μ,
where μ≥0 is a constant. Then
(45)|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|≤μ1-dep¯(x,y)(x,x0),
for (x,y)∈Ω, where
(46)p¯(x,y)=∫y0y‍[d1-d+B(s,t)]ΔtΔs,
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.
Letw(x,y)=|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)| for (x,y)∈Ω. We have
(47)w(x,y)≤|β1(x)+τ1(y)-β2(x)-τ2(y)|+∫x0x‍∫y0y‍|f(s,t,u1(s,t),u1Δ2Δ1(s,t),H(u1)(s,t))-f(s,t,u2(s,t),u2Δ2Δ1(s,t),H(u2)(s,t))|×ΔtΔs+|f(x,y,u1(x,y),u1Δ2Δ1(x,y),H(u1)(x,y))-f(x,y,u2(x,y),u2Δ2Δ1(x,y),H(u2)(x,y))|+μ+∫x0x‍∫y0y‍{∫x0s‍d[u1Δ2Δ1|u1(s,t)-u2(s,t)|+|u1Δ2Δ1(s,t)-u2Δ2Δ1(s,t)|]+∫x0s‍∫y0t‍p(s,t,m,n)×[u1Δ2Δ1|u1(m,n)-u2(m,n)|+|u1Δ2Δ1(m,n)-u2Δ2Δ1(m,n)|]×ΔnΔmu1Δ2Δ1∫x0s‍}ΔtΔs+d[u1Δ2Δ1|u1(x,y)-u2(x,y)|+|u1Δ2Δ1(x,y)-u2Δ2Δ1(x,y)|]+∫x0x‍∫y0y‍p(x,y,m,n)×[u1Δ2Δ1|u1(m,n)-u2(m,n)|+|u1Δ2Δ1(m,n)-u2Δ2Δ1(m,n)|]ΔnΔm.
From (47), we have
(48)w(x,y)≤μ1-d+11-d×∫x0x‍∫y0y‍{∫x0s‍w(s,t)+p(x,y,s,t)w(s,t)+∫x0s‍∫y0t‍p(s,t,m,n)w(m,n)ΔnΔm}×ΔtΔs.
Now, a suitable application of Lemma2 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
(49)vΔ2Δ1(x,y)=G(x,y,v(x,y),vΔ2Δ1(x,y),(H¯v(x,y))),
with the initial boundary conditions
(50)v(x,x0)=β¯(x),v(y0,y)=τ¯(y),v(x0,y0)=0,
where
(51)(H¯v)(x,y)=∫x0x‍∫y0y‍J(x,y,m,n,v(m,n),vΔ2Δ1(m,n))ΔnΔm,G∈Crd(Ω×ℝn×ℝn×ℝn,ℝn), J∈Crd(Ω×Ω×ℝn×ℝn,ℝn), and β¯,τ¯∈Crd(𝕋,ℝn).
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 functionsf,g in (1) satisfy the conditions (30) and
(52)|β(x)+τ(y)-β¯(x)-τ¯(y)|+∫x0x‍∫y0y‍|f(s,t,v(s,t),vΔ2Δ1(x,y),(Hv)(s,t))-G(s,t,v(s,t),vΔ2Δ1(s,t),(H¯v)(s,t))|×ΔtΔs+|f(x,v,v(x,y),vΔ2Δ1(x,y),(Hv)(x,y))-G(x,y,v(x,y),vΔ2Δ1(x,y),(H¯v)(x,y))|≤ϵ,
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.
Letw(x,y)=|u(x,y)-v(x,y)|+|uΔ2Δ1(x,y)-vΔ2Δ1(x,y)| for (x,y)∈Ω. We have
(53)w(x,y)≤|β(x)+τ(y)-β¯(x)-τ¯(y)|+∫x0x‍∫y0y‍|f(s,t,u(s,t),uΔ2Δ1(s,t),(Hu)(s,t))-f(s,t,v(s,t),vΔ2Δ1(s,t),(Hv)(s,t))|×ΔtΔs+∫x0x‍∫y0y‍|f(s,t,v(s,t),vΔ2Δ1(x,y),(Hv)(s,t))-G(s,t,v(s,t),vΔ2Δ1(s,t),(H¯v)(s,t))|×ΔtΔs+|f(x,y,u(x,y),uΔ2Δ1(x,y),(Hu)(x,y))-f(x,y,v(x,y),vΔ2Δ1(x,y),(Hv)(x,y))|+|f(x,y,v(x,y),vΔ2Δ1(x,y),(Hv)(x,y))-G(x,y,v(x,y),vΔ2Δ1(s,t),(H¯v)(s,t))|≤ϵ+∫x0x‍∫y0y‍{d[uΔ2Δ1|u(s,t)-v(s,t)|+|uΔ2Δ1(s,t)-vΔ2Δ1(s,t)|]+∫x0x‍∫y0y‍p(s,t,m,n)×[uΔ2Δ1|u(m,n)-v(m,n)|+|uΔ2Δ1(s,t)-vΔ2Δ1(s,t)|]×ΔnΔmuΔ2Δ1}ΔtΔs+d[uΔ2Δ1|u(x,y)-v(x,y)|+|uΔ2Δ1(x,y)-vΔ2Δ1(x,y)|]+∫x0x‍∫y0y‍p(x,y,m,n)×[uΔ2Δ1|u(m,n)-v(m,n)|+|uΔ2Δ1(m,n)-vΔ2Δ1(m,n)|]×ΔnΔm.
From (53), we have
(54)w(x,y)≤ϵ1-d+11-d∫x0x‍∫y0y‍{∫y0t‍w(s,t)+p(x,y,s,t)w(s,t)+∫x0s‍∫y0t‍p(s,t,m,n)w(m,n)ΔnΔm}×ΔtΔs.
Now, a suitable application of Lemma 2 yields
(55)w(x,y)≤ϵ1-dep¯(x,y)(x,x0),
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.



---

*Source: 345697-2013-07-01.xml*