VOLTERRA INTEGRAL AND
INTEGRODIFFERENTIAL EQUATIONS IN TWO
VARIABLES
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India
EMail: bgpachpatte@gmail.com
Received:
15 June, 2009
Accepted:
19 November, 2009
Communicated by:
W.S. Cheung
2000 AMS Sub. Class.:
34K10, 35R10.
Key words:
Volterra integral and integrodifferential equations, Banach fixed point theorem,
Bielecki type norm, integral inequalities, existence and uniqueness, estimates on
the solutions, approximate solutions.
vol. 10, iss. 4, art. 108, 2009
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Abstract:
The aim of this paper is to study the existence, uniqueness and other properties
of solutions of certain Volterra integral and integrodifferential equations in two
variables. The tools employed in the analysis are based on the applications of the
Banach fixed point theorem coupled with Bielecki type norm and certain integral
inequalities with explicit estimates.
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Contents
1
Introduction
3
2
Existence and Uniqueness
5
3
Estimates on the Solutions
10
4
Approximate Solutions
14
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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1.
Introduction
Let Rn denote the real n-dimensional Euclidean space with appropriate norm denoted by |·|. We denote by Ia = [a, ∞) , R+ = [0, ∞), the given subsets of R,
the set of real numbers, E = {(x, y, m, n) : a ≤ m ≤ x < ∞, b ≤ n ≤ y < ∞}
and ∆ = Ia × Ib . For x, y ∈ R, the partial derivatives of a function z(x, y) with
respect to x, y and xy are denoted by D1 z (x, y) , D2 z (x, y) and D2 D1 z (x, y) =
D1 D2 z (x, y) . Consider the Volterra integral and integrodifferential equations of the
forms:
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
(1.1)
u (x, y) = f (x, y, u (x, y) , (Ku) (x, y)) ,
and
(1.2)
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with the given initial boundary conditions
(1.3)
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D2 D1 u (x, y) = f (x, y, u (x, y) , (Ku) (x, y)) ,
u (x, 0) = σ (x) ,
u (0, y) = τ (y) ,
u (0, 0) = 0,
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for (x, y) ∈ ∆, where
Z
(1.4)
x
Z
(Ku) (x, y) =
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y
k (x, y, m, n, u (m, n)) dndm,
a
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b
k ∈ C (E×Rn ,Rn ) , f ∈ C (∆×Rn ×Rn ,Rn ) , σ ∈ C (Ia ,Rn ) , τ ∈ C (Ib ,Rn ) . By
a solution of equation (1.1) (or equations (1.2) – (1.3)) we mean a function u ∈
C (∆,Rn ) which satisfies the equation (1.1) (or equations (1.2) – (1.3)).
In general, existence theorems for equations of the above forms are proved by
the use of one of the three fundamental methods (see [1], [3] – [9], [12] – [16]): the
Close
method of successive approximations, the method based on the theory of nonexpansive and monotone mappings and on the theory exploiting the compactness of the
operator often by the use of the well known fixed point theorems. The aim of the
present paper is to study the existence, uniqueness and other properties of solutions
of equations (1.1) and (1.2) – (1.3) under various assumptions on the functions involved therein. The main tools employed in the analysis are based on applications
of the well known Banach fixed point theorem (see [3] – [5], [8]) coupled with a Bielecki type norm (see [2]) and the integral inequalities with explicit estimates given
in [11]. In fact, our approach here to the study of equations (1.1) and (1.2) – (1.3)
leads us to obtain new conditions on the qualitative properties of their solutions and
present some useful basic results for future reference, by using elementary analysis.
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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2.
Existence and Uniqueness
We first construct the appropriate metric space for our analysis. Let α > 0, β > 0
be constants and consider the space of continuous functions C (∆,Rn ) such that
|u(x,y)|
sup eα(x−a)+β(y−b)
< ∞ for u (x, y) ∈ C (∆,Rn ) and denote this special space by
(x,y)∈∆
Cα,β (∆,Rn ) with suitable metric
|u (x, y) − v (x, y)|
,
eα(x−a)+β(y−b)
(x,y)∈∆
d∞
α,β (u, v) = sup
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
and a norm defined by
|u|∞
α,β
= sup
(x,y)∈∆
|u (x, y)|
eα(x−a)+β(y−b)
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.
∞
The above definitions of d∞
α,β and |·|α,β are variants of Bielecki’s metric and norm
(see [2, 5]).
The following variant of the lemma proved in [5] holds.
Lemma 2.1. If α > 0, β > 0 are constants, then Cα,β (∆, Rn ) , |·|∞
α,β is a Banach
space.
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Our main results concerning the existence and uniqueness of solutions of equations (1.1) and (1.2) – (1.3) are given in the following theorems.
Theorem 2.2. Let α > 0, β > 0, L > 0, M ≥ 0, γ > 1 be constants with αβ = Lγ.
Suppose that the functions f, k in equation (1.1) satisfy the conditions
(2.1)
|f (x, y, u, v) − f (x, y, ū, v̄)| ≤ M [|u − ū| + |v − v̄|] ,
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|k (x, y, m, n, u) − k (x, y, m, n, v)| ≤ L |u − v| ,
(2.2)
and
(2.3)
1
d1 = sup
(x,y)∈∆
eα(x−a)+β(y−b)
|f (x, y, 0, (K0) (x, y))| < ∞.
If M 1 + γ1 < 1, then the equation (1.1) has a unique solution u ∈ Cα,β (∆,Rn ) .
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
Proof. Let u ∈ Cα,β (∆,Rn ) and define the operator T by
(T u) (x, y) = f (x, y, u (x, y) , (Ku) (x, y)) .
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Now we shall show that T maps Cα,β (∆,Rn ) into itself. From (2.4) and using the
hypotheses, we have
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(2.4)
|T u|∞
α,β ≤ sup
1
|f (x, y, 0, (K0) (x, y))|
eα(x−a)+β(y−b)
1
+ sup α(x−a)+β(y−b) |f (x, y, u (x, y) , (Ku) (x, y)) − f (x, y, 0, (K0) (x, y))|
(x,y)∈∆ e
Z xZ y
1
≤ d1 + sup α(x−a)+β(y−b) M |u (x, y)| +
L |u (m, n)| dndm
(x,y)∈∆ e
a
b
"
|u (x, y)|
= d1 + M sup α(x−a)+β(y−b)
(x,y)∈∆ e
#
Z xZ y
1
|u
(m,
n)|
+ sup α(x−a)+β(y−b)
Leα(m−a)+β(n−b) α(m−a)+β(n−b) dndm
e
(x,y)∈∆ e
a
b
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(x,y)∈∆
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"
≤ d1 + M |u|∞
α,β 1 + L sup
1
eα(x−a)+β(y−b)
y
α(m−a)+β(n−b)
×
e
dndm
a
b
L
∞
≤ d1 + M |u|α,β 1 +
αβ
1
∞
< ∞.
= d1 + M |u|α,β 1 +
γ
(x,y)∈∆
Z
xZ
This proves that the operator T maps Cα,β (∆,Rn ) into itself.
Now we verify that the operator T is a contraction map. Let u, v ∈ Cα,β (∆,Rn ) .
From (2.4) and using the hypotheses, we have
d∞
α,β
(T u, T v)
= sup
(x,y)∈∆
1
eα(x−a)+β(y−b)
|f (x, y, u (x, y) , (Ku) (x, y))
−f (x, y, v (x, y) , (Kv) (x, y))|
Z xZ y
1
≤ sup α(x−a)+β(y−b) M |u (x, y) − v (x, y)| +
L |u (m, n) − v (m, n)| dndm
(x,y)∈∆ e
a
b
"
|u (x, y) − v (x, y)|
= M sup
eα(x−a)+β(y−b)
(x,y)∈∆
#
Z xZ y
1
|u
(m,
n)
−
v
(m,
n)|
+L sup α(x−a)+β(y−b)
eα(m−a)+β(n−b)
dndm
e
eα(m−a)+β(n−b)
(x,y)∈∆
a
b
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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"
≤
M d∞
α,β
(u, v) 1 + L sup
(x,y)∈∆
Z
1
x
Z
×
eα(x−a)+β(y−b)
y
α(m−a)+β(n−b)
e
a
dndm
b
L
=
(u, v) 1 +
αβ
1
=M 1+
d∞
α,β (u, v) .
γ
1
Since M 1 + γ < 1, it follows from the Banach fixed point theorem (see [3] – [5],
[8]) that T has a unique fixed point in Cα,β (∆,Rn ) . The fixed point of T is however
a solution of equation (1.1). The proof is complete.
M d∞
α,β
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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Theorem 2.3. Let M, L, α, β, γ be as in Theorem 2.2. Suppose that the functions
f, k in equation (1.2) satisfy the conditions (2.1), (2.2) and
(2.5) d2 = sup
(x,y)∈∆
1
eα(x−a)+β(y−b)
Z
× σ (x) + τ (y) +
xZ
a
M
where σ, τ are as in (1.3). If αβ
1+
a unique solution u ∈ Cα,β (∆,Rn ) .
1
γ
b
y
f (s, t, 0, (K0) (s, t)) dtds < ∞,
< 1, then the equations (1.2) – (1.3) have
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Proof. Let u ∈ Cα,β (∆,Rn ) and define the operator S by
Z xZ y
(Su) (x, y) = σ (x) + τ (y) +
f (s, t, u (s, t) , (Ku) (s, t)) dtds,
a
b
for (x, y) ∈ ∆. The proof that S maps Cα,β (∆,Rn ) into itself and is a contraction
map can be completed by closely looking at the proof of Theorem 2.2 given above
with suitable modifications. Here, we leave the details to the reader.
Remark 1. We note that the problems of existence and uniqueness of solutions of
special forms of equations (1.1) and (1.2) – (1.3) have been studied under a variety
of hypotheses in [16]. In [7] the authors have obtained existence and uniqueness
of solutions to general integral-functional equations involving n variables by using
the comparative method (see also [1], [6], [12] – [15]). The approach here in the
treatment of existence and uniqueness problems for equations (1.1) and (1.2) – (1.3)
is fundamental and our results do not seem to be covered by the existing theorems.
Furthermore, the ideas used here can be extended to n dimensional versions of equations (1.1) and (1.2) – (1.3).
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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3.
Estimates on the Solutions
In this section we obtain estimates on the solutions of equations (1.1) and (1.2) –
(1.3) under some suitable assumptions on the functions involved therein.
We need the following versions of the inequalities given in [11, Remark 2.2.1, p.
66 and p. 86]. For similar results, see [10].
Lemma 3.1. Let u ∈ C (∆,R+ ), r, D1 r, D2 r, D2 D1 r ∈ C (E,R+ ) and c ≥ 0 be a
constant. If
Z xZ y
(3.1)
u (x, y) ≤ c +
r (x, y, ξ, η) u (ξ, η) dηdξ,
a
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
b
for (x, y) ∈ ∆, then
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Z
(3.2)
Volterra Integral and
Integrodifferential Equations
x
Z
u (x, y) ≤ c exp
y
A (s, t) dtds ,
a
for (x, y) ∈ ∆, where
Z
x
(3.3) A (x, y) = r (x, y, x, y) +
D1 r (x, y, ξ, y) dξ
a
Z y
Z xZ
+
D2 r (x, y, x, η) dη +
b
a
b
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y
D2 D1 r (x, y, ξ, η) dηdξ.
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b
Lemma 3.2. Let u, e, p ∈ C (∆,R+ ) and r, D1 r, D2 r, D2 D1 r ∈ C (E,R+ ). If
e(x, y) is nondecreasing in each variable (x, y) ∈ ∆ and
Z xZ y
(3.4) u (x, y) ≤ e(x, y) +
p (s, t)
a
b
Z sZ t
× u (s, t) +
r (s, t, m, n) u (m, n) dndm dtds,
a
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b
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for (x, y) ∈ ∆, then
Z
(3.5) u (x, y) ≤ e(x, y) 1 +
x
Z
y
p (s, t)
Z s Z t
× exp
[p (m, n) + A (m, n)] dndm dtds ,
a
b
a
b
for (x, y) ∈ ∆, where A(x, y) is defined by (3.3).
First, we shall give the following theorem concerning an estimate on the solution
of equation (1.1).
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
Theorem 3.3. Suppose that the functions f, k in equation (1.1) satisfy the conditions
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(3.6)
(3.7)
|f (x, y, u, v) − f (x, y, ū, v̄)| ≤ N [|u − ū| + |v − v̄|] ,
|k (x, y, m, n, u) − k (x, y, m, n, v)| ≤ r (x, y, m, n) |u − v| ,
where 0 ≤ N < 1 is a constant and r, D1 r, D2 r, D2 D1 r ∈ C (E,R+ ) . Let
(3.8)
c1 = sup |f (x, y, 0, (K0) (x, y))| < ∞.
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(x,y)∈∆
If u(x, y), (x, y) ∈ ∆ is any solution of equation (1.1), then
Z x Z y
c1
(3.9)
|u (x, y)| ≤
exp
B (s, t) dtds ,
1−N
a
b
for (x, y) ∈ ∆, where
(3.10)
B (x, y) =
N
A (x, y) ,
1−N
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in which A(x, y) is defined by (3.3).
Proof. By using the fact that u(x, y) is a solution of equation (1.1) and the hypotheses, we have
(3.11) |u (x, y)| ≤ |f (x, y, u(x, y), (Ku) (x, y)) − f (x, y, 0, (K0) (x, y))|
+ |f (x, y, 0, (K0) (x, y))|
Z xZ y
≤ c1 + N |u (x, y)| +
r (x, y, m, n) |u (m, n)| dndm .
a
b
B.G. Pachpatte
From (3.11) and using the assumption 0 ≤ N < 1, we observe that
Z xZ y
N
c1
(3.12) |u (x, y)| ≤
+
r (x, y, m, n) |u (m, n)| dndm.
1−N
1−N a b
Now a suitable application of Lemma 3.1 to (3.12) yields (3.9).
Next, we shall obtain an estimate on the solution of equations (1.2) – (1.3).
Theorem 3.4. Suppose that the function f in equation (1.2) satisfies the condition
(3.13)
|f (x, y, u, v) − f (x, y, ū, v̄)| ≤ p (x, y) [|u − ū| + |v − v̄|] ,
where p ∈ C (∆,R+ ) and the function k in equation (1.2) satisfies the condition
(3.7). Let
Z xZ y
(3.14) c2 = sup σ (x) + τ (y) +
f (s, t, 0, (K0) (s, t)) dtds < ∞.
(x,y)∈∆
a
b
If u(x, y), (x, y) ∈ ∆ is any solution of equations (1.2) – (1.3), then
|u (x, y)|
Z xZ
≤ c2 1 +
(3.15)
a
b
y
Z s Z
p (s, t) exp
t
[p (m, n) + A (m, n)] dndm dtds ,
a
b
Volterra Integral and
Integrodifferential Equations
vol. 10, iss. 4, art. 108, 2009
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for (x, y) ∈ ∆, where A(x, y) is defined by (3.3).
Proof. Using the fact that u(x, y) is a solution of equations (1.2) – (1.3) and the
hypotheses, we have
(3.16) |u (x, y)|
Z xZ y
≤ σ (x) + τ (y) +
f (s, t, 0, (K0) (s, t)) dtds
a
b
Z xZ y
+
|f (s, t, u (s, t) , (Ku) (s, t)) − f (s, t, 0, (K0) (s, t))| dtds
a
b
Z xZ y
Z sZ t
≤ c2 +
p (s, t) |u (s, t)|+
r (s, t, m, n) |u (m, n)| dndm dtds.
a
b
a
b
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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Contents
Now a suitable application of Lemma 3.2 to (3.16) yields (3.15).
Remark 2. We note that the results in Theorems 3.3 and 3.4 provide explicit estimates on the solutions of equations (1.1) and (1.2) – (1.3) and are obtained by
simple applications of the inequalities in Lemmas 3.1 and 3.2. If the estimates on
the right hand sides in (3.9) and (3.15) are bounded, then the solutions of equations
(1.1) and (1.2) – (1.3) are bounded.
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4.
Approximate Solutions
In this section we shall deal with the approximation of solutions of equations (1.1)
and (1.2) – (1.3). We obtain conditions under which we can estimate the error between the solutions and approximate solutions.
We call a function u ∈ C (∆,Rn ) an ε-approximate solution of equation (1.1) if
there exists a constant ε ≥ 0 such that
|u (x, y) − f (x, y, u (x, y) , (Ku) (x, y))| ≤ ε,
for all (x, y) ∈ ∆. Let u ∈ C (∆,Rn ), D2 D1 u exists and satisfies the inequality
|D2 D1 u (x, y) − f (x, y, u (x, y) , (Ku) (x, y))| ≤ ε,
for a given constant ε ≥ 0, where it is supposed that (1.3) holds. Then we call u(x, y)
an ε-approximate solution of equation (1.2) with (1.3).
The following theorems deal with estimates on the difference between the two
approximate solutions of equations (1.1) and (1.2) with (1.3).
Theorem 4.1. Suppose that the functions f and k in equation (1.1) satisfy the conditions (3.6) and (3.7). For i = 1, 2, let ui (x, y) be respectively εi -approximate
solutions of equation (1.1) on ∆. Then
Z x Z y
ε1 + ε2
(4.1)
|u1 (x, y) − u2 (x, y)| ≤
exp
B (s, t) dtds ,
1−N
a
b
for (x, y) ∈ ∆, where B(x, y) is given by (3.10).
Proof. Since ui (x, y) (i = 1, 2) for (x, y) ∈ ∆ are respectively εi -approximate
solutions to equation (1.1), we have
(4.2)
|ui (x, y) − f (x, y, ui (x, y) , (Kui ) (x, y))| ≤ εi .
Volterra Integral and
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B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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From (4.2) and using the elementary inequalities |v − z| ≤ |v| + |z| and |v| − |z| ≤
|v − z| , we observe that
(4.3)
ε1 + ε2 ≥ |u1 (x, y) − f (x, y, u1 (x, y) , (Ku1 ) (x, y))|
+ |u2 (x, y) − f (x, y, u2 (x, y) , (Ku2 ) (x, y))|
≥ |{u1 (x, y) − f (x, y, u1 (x, y) , (Ku1 ) (x, y))}
− {u2 (x, y) − f (x, y, u2 (x, y) , (Ku2 ) (x, y))}|
≥ |u1 (x, y) − u2 (x, y)| − |f (x, y, u1 (x, y) , (Ku1 ) (x, y))
−f (x, y, u2 (x, y) , (Ku2 ) (x, y))| .
Let w (x, y) = |u1 (x, y) − u2 (x, y)|, (x, y) ∈ ∆. From (4.3) and using the hypotheses, we observe that
Z xZ y
(4.4) w (x, y) ≤ ε1 + ε2 + N w (x, y) +
r (x, y, m, n) w (m, n) dndm .
a
b
From (4.4) and using the assumption that 0 ≤ N < 1, we observe that
Z xZ y
ε1 + ε2
N
(4.5)
w (x, y) ≤
+
r (x, y, m, n) w (m, n) dndm.
1−N
1−N a b
Now a suitable application of Lemma 3.1 to (4.5) yields (4.1).
Theorem 4.2. Suppose that the functions f and k in equation (1.2) satisfy the conditions (3.13) and (3.7). For i = 1, 2, let ui (x, y) be respectively εi -approximate
solutions of equation (1.2) on ∆ with
(4.6)
ui (x, 0) = αi (x) ,
ui (0, y) = βi (y) ,
ui (0, 0) = 0,
where αi ∈ C (Ia ,Rn ), βi ∈ C (Ib ,Rn ) such that
(4.7)
|α1 (x) − α2 (x) + β1 (y) − β2 (y)| ≤ δ,
Volterra Integral and
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vol. 10, iss. 4, art. 108, 2009
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where δ ≥ 0 is a constant. Then
(4.8)
Z xZ y
|u1 (x, y) − u2 (x, y)| ≤ e (x, y) 1 +
p (s, t)
a
b
Z s Z t
× exp
[p (m, n) + A (m, n)] dmdn dtds ,
a
b
Volterra Integral and
Integrodifferential Equations
for (x, y) ∈ ∆, where
(4.9)
e (x, y) = (ε1 + ε2 ) (x − a) (y − b) + δ.
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
Proof. Since ui (x, y) (i = 1, 2) for (x, y) ∈ ∆ are respectively εi -approximate
solutions of equation (1.2) with (4.6), we have
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|D2 D1 ui (x, y) − f (x, y, ui (x, y) , (Kui ) (x, y))| ≤ εi .
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(4.10)
First keeping x fixed in (4.10), setting y = t and integrating both sides over t from b
to y, then keeping y fixed in the resulting inequality and setting x = s and integrating
both sides over s from a to x and using (4.6), we observe that
εi (x − a) (y − b)
Z xZ y
≥
|D2 D1 ui (s, t) − f (s, t, ui (s, t) , (Kui ) (s, t))| dtds
a
b
Z x Z y
≥ {D2 D1 ui (s, t) − f (s, t, ui (s, t) , (Kui ) (s, t))} dtds
a b
Z xZ y
= ui (x, y) − [αi (x) + βi (y)] −
f (s, t, ui (s, t) , (Kui ) (s, t)) .
a
b
The rest of the proof can be completed by closely looking at the proof of Theorem
4.1 and using the inequality in Lemma 3.2. Here, we omit the details.
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Remark 3. When u1 (x, y) is a solution of equation (1.1) (respectively equations
(1.2) – (1.3)), then we have ε1 = 0 and from (4.1) (respectively (4.8)) we see that
u2 (x, y) → u1 (x, y) as ε2 → 0 (respectively ε2 → 0 and δ → 0). Furthermore,
if we put ε1 = ε2 = 0 in (4.1) (respectively ε1 = ε2 = 0, α1 (x) = α2 (x) ,
β1 (y) = β2 (y) , i.e. δ = 0 in (4.8)), then the uniqueness of solutions of equation
(1.1) (respectively equations (1.2) – (1.3)) is established.
Consider the equations (1.1), (1.2) – (1.3) together with the following Volterra
integral and integrodifferential equations
(4.11)
v (x, y) = f¯ (x, y, v (x, y) , (Kv) (x, y)) ,
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
and
(4.12)
D2 D1 v (x, y) = f¯ (x, y, v (x, y) , (Kv) (x, y)) ,
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with the given initial boundary conditions
v (0, y) = β̄ (y) ,
v (0, 0) = 0,
for (x, y) ∈ ∆, where K is given by (1.4) and f¯ ∈ C (∆×Rn ×Rn ,Rn ), ᾱ ∈
C (Ia ,Rn ) , β̄ ∈ C (Ib ,Rn ) .
The following theorems show the closeness of the solutions to equations (1.1),
(4.11) and (1.2) – (1.3), (4.12) – (4.13).
(4.13)
v (x, 0) = ᾱ (x) ,
Theorem 4.3. Suppose that the functions f, k in equation (1.1) satisfy the conditions
(3.6), (3.7) and there exists a constant ε ≥ 0, such that
f (x, y, u, w) − f¯ (x, y, u, w) ≤ ε,
(4.14)
where f, f¯ are as given in (1.1) and (4.11). Let u(x, y) and v(x, y) be respectively
the solutions of equations (1.1) and (4.11) for (x, y) ∈ ∆. Then
Z x Z y
ε
(4.15)
|u (x, y) − v (x, y)| ≤
exp
B (s, t) dtds ,
1−N
a
b
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for (x, y) ∈ ∆, where B(x, y) is given by (3.10).
Proof. Let z (x, y) = |u (x, y) − v (x, y)| for (x, y) ∈ ∆. Using the facts that u(x, y)
and v(x, y) are the solutions of equations (1.1) and (4.11) and the hypotheses, we
observe that
(4.16)
z (x, y)
≤ |f (x, y, u (x, y) , (Ku) (x, y)) − f (x, y, v (x, y) , (Kv) (x, y))|
+ f (x, y, v (x, y) , (Kv) (x, y)) − f¯ (x, y, v (x, y) , (Kv) (x, y))
Z xZ y
≤ ε + N z (x, y) +
r (s, t, m, n) z (m, n) dndm .
a
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
b
From (4.16) and using the assumption that 0 ≤ N < 1, we observe that
Z xZ y
ε
N
(4.17)
z (x, y) ≤
+
r (s, t, m, n) z (m, n) dndm.
1−N
1−N a b
Now a suitable application of Lemma 3.1 to (4.17) yields (4.15).
Theorem 4.4. Suppose that the functions f, k in equation (1.2) are as in Theorem
4.2 and there exist constants ε ≥ 0, δ ≥ 0 such that the condition (4.14) holds and
α (x) − ᾱ (x) + β (y) − β̄ (y) ≤ δ,
(4.18)
where α, β and ᾱ, β̄ are as in (1.3) and (4.13). Let u(x, y) and v(x, y) be respectively
the solutions of equations (1.2) – (1.3) and (4.12) – (4.13) for (x, y) ∈ ∆. Then
Z xZ y
(4.19) |u (x, y) − v (x, y)| ≤ ē (x, y) 1 +
p (s, t)
a
b
Z s Z t
× exp
[p (m, n) + A (m, n)] dndm dtds ,
a
b
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for (x, y) ∈ ∆, where
(4.20)
ē (x, y) = ε (x − a) (y − b) + δ,
and A(x, y) is given by (3.3).
The proof can be completed by rewriting the equivalent integral equations corresponding to the equations (1.2) – (1.3) and (4.12) – (4.13) and by following the
proof of Theorem 4.3 and using Lemma 3.2. We leave the details to the reader.
Remark 4. It is interesting to note that Theorem 4.3 (respectively Theorem 4.4) relates the solutions of equations (1.1) and (4.11) (respectively equations (1.2) – (1.3)
and (4.12) – (4.13)) in the sense that if f is close to f¯, (respectively f is close to
f¯, α is close to ᾱ, β is close to β̄), then the solutions of equations (1.1) and (4.11)
(respectively solutions of equations (1.2) – (1.3) and (4.12) – (4.13)) are also close
together.
Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
vol. 10, iss. 4, art. 108, 2009
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References
[1] P.R. BEESACK, Systems of multidimensional Volterra integral equations and
inequalities, Nonlinear Analysis TMA, 9 (1985), 1451–1486.
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Volterra Integral and
Integrodifferential Equations
B.G. Pachpatte
[3] C. CORDUNEANU, Integral Equations and Stability of Feedback Systems,
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vol. 10, iss. 4, art. 108, 2009
[4] M.A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1964.
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[6] M. KWAPISZ, Weighted norms and existence and uniqueness of Lp solutions
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Ekvac., 18 (1975), 107–162.
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[8] R.K. MILLER, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo
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Volterra Integral and
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