ON A CERTAIN VOLTERRA-FREDHOLM TYPE
INTEGRAL EQUATION
Volterra-Fredholm Type
Integral Equation
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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EMail: bgpachpatte@gmail.com
Received:
10 February, 2008
Accepted:
18 October, 2008
Communicated by:
C.P. Niculescu
2000 AMS Sub. Class.:
34K10, 35R10.
Key words:
Volterra-Fredholm type, integral equation, Banach fixed point theorem, integral
inequality, Bielecki type norm, existence and uniqueness, continuous dependence.
Abstract:
The aim of this paper is to study the existence, uniqueness and other properties of
solutions of a certain Volterra-Fredholm type integral equation. The main tools
employed in the analysis are based on applications of the Banach fixed point
theorem and a certain integral inequality with explicit estimate.
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Contents
1
Introduction
3
2
Existence and Uniqueness
4
3
Bounds on Solutions
12
4
Continuous Dependence
16
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
Title Page
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1.
Introduction
Consider the following Volterra-Fredholm type integral equation
Z t
Z b
0
(1.1) x (t) = f (t) +
g (t, s, x (s) , x (s)) ds +
h (t, s, x (s) , x0 (s)) ds,
a
a
for −∞ < a ≤ t ≤ b < ∞, where x, f, g, h are in Rn , the n-dimensional Euclidean
space with appropriate norm denoted by |·|. Let R and 0 denote the set of real numbers and the derivative of a function. We denote by I = [a, b] , R+ = [0, ∞) the
given subsets of R and assume that f ∈ C (I, Rn ), g, h ∈ C (I 2 × Rn × Rn , Rn )
and are continuously differentiable with respect to t on the respective domains of
their definitions.
The literature provides a good deal of information related to the special versions
of equation (1.1), see [3, 5, 6, 8, 12] and the references cited therein. Recently, in
[1] the authors studied a Fredholm type equation similar to equation (1.1) for g = 0
using Perov’s fixed point theorem, the method of successive approximations and
the trapezoidal quadature rule. The purpose of this paper is to study the existence,
uniqueness and other properties of solutions of equation (1.1) under various assumptions on the functions involved and their derivatives. The well known Banach fixed
point theorem (see [5, p. 37]) coupled with a Bielecki type norm (see [2]) and an
integral inequality with an explicit estimate given in [10, p. 44] are used to establish
the results.
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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2.
Existence and Uniqueness
By a solution of equation (1.1) we mean a continuous function x(t) for t ∈ I which
is continuously differentiable with respect to t and satisfies the equation (1.1). For
every continuous function u(t) in Rn together with its continuous first derivative
u0 (t) for t ∈ I we denote by |u (t)|1 = |u (t)| + |u0 (t)| . Let S be a space of those
continuous functions u(t) in Rn together with the continuous first derivative u0 (t) in
Rn for t ∈ I which fulfil the condition
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
|u (t)|1 = O (exp (λ (t − a))) ,
vol. 9, iss. 4, art. 116, 2008
for t ∈ I, where λ is a positive constant. In the space S we define the norm (see
[2, 4, 7, 9, 11])
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(2.1)
(2.2)
Contents
|u|S = sup {|u (t)|1 exp (λ (t − a))} .
t∈I
It is easy to see that S with its norm defined in (2.2) is a Banach space. We note that
the condition (2.1) implies that there exists a nonnegative constant N such that
|u (t)|1 ≤ N exp (λ (t − a)) .
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Using this fact in (2.2) we observe that
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(2.3)
|u|S ≤ N.
We need the following special version of the integral inequality given in [10,
Theorem 1.5.2, part (b2 ), p. 44]. We shall state it in the following lemma for completeness.
Close
Lemma 2.1. Let u (t) ∈ C (I, R+ ) , k (t, s) , r (t, s) ∈ C (I 2 , R+ ) be nondecreasing
in t ∈ I for each s ∈ I and
Z t
Z b
u (t) ≤ c +
k (t, s) u (s) ds +
r (t, s) u (s) ds,
a
a
for t ∈ I where c ≥ 0 is a constant. If
Z s
Z b
d (t) =
r (t, s) exp
k (s, σ) dσ ds < 1,
a
a
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
for t ∈ I, then
u (t) ≤
c
exp
1 − d (t)
Z
t
k (t, s) ds ,
a
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for t ∈ I.
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The following theorem ensures the existence of a unique solution to equation
(1.1).
Theorem 2.2. Assume that
(i) the functions g, h in equation (1.1) and their derivatives with respect to t satisfy
the conditions
(2.4)
|g (t, s, u, v) − g (t, s, ū, v̄)| ≤ p1 (t, s) [|u − ū| + |v − v̄|] ,
(2.5)
∂
∂
g (t, s, u, v) − g (t, s, ū, v̄) ≤ p2 (t, s) [|u − ū| + |v − v̄|] ,
∂t
∂t
(2.6)
|h (t, s, u, v) − h (t, s, ū, v̄)| ≤ q1 (t, s) [|u − ū| + |v − v̄|] ,
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(2.7)
∂
∂
h (t, s, u, v) − h (t, s, ū, v̄) ≤ q2 (t, s) [|u − ū| + |v − v̄|] ,
∂t
∂t
where pi (t, s) , qi (t, s) ∈ C (I 2 , R+ ) (i = 1, 2),
(ii) for λ as in (2.1)
Volterra-Fredholm Type
Integral Equation
(a) there exists a nonnegative constant α such that α < 1 and
Z
t
(2.8) p1 (t, t) exp (λ (t − a)) +
p (t, s) exp (λ (s − a)) ds
a
Z b
+
q (t, s) exp (λ (s − a)) ds ≤ α exp (λ (t − a)) ,
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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a
for t ∈ I, where p (t, s) = p1 (t, s)+p2 (t, s) , q (t, s) = q1 (t, s)+q2 (t, s) ,
(b) there exists a nonnegative constant β such that
(2.9)
|f (t)| + |f 0 (t)| + |g (t, t, 0)|
Z t
∂
+
|g (t, s, 0, 0)| + g (t, s, 0, 0) ds
∂t
a
Z b
∂
+
|h (t, s, 0, 0)| + h (t, s, 0, 0) ds ≤ β exp (λ (t − a)) ,
∂t
a
where f, g, h are the functions given in equation (1.1).
Then the equation (1.1) has a unique solution x(t) in S on I .
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Proof. Let x (t) ∈ S and define the operator
Z t
Z b
0
(2.10) (T x) (t) = f (t) +
g (t, s, x (s) , x (s)) ds +
h (t, s, x (s) , x0 (s))ds.
a
a
Differentiating both sides of (2.10) with respect to t we have
Z t
∂
0
0
0
g (t, s, x (s) , x0 (s)) ds
(2.11) (T x) (t) = f (t) + g (t, t, x (t) , x (t)) +
∂t
a
Z b
∂
+
h (t, s, x (s) , x0 (s))ds.
a ∂t
Now we show that T x maps S into itself. Evidently, T x, (T x)0 are continuous on I
and T x, (T x)0 ∈ Rn . We verify that (2.1) is fulfilled. From (2.10), (2.11), using the
hypotheses and (2.3) we have
(2.12)
|(T x) (t)|1
= |(T x) (t)| + (T x)0 (t)
≤ |f (t)| + |f 0 (t)| + |g (t, t, x (t) , x0 (t)) − g (t, t, 0, 0)| + |g (t, t, 0, 0)|
Z t
Z t
0
+
|g (t, s, x (s) , x (s)) − g (t, s, 0, 0)| ds +
|g (t, s, 0, 0)| ds
a
a
Z t
∂
∂
0
+
∂t g (t, s, x (s) , x (s)) − ∂t g (t, s, 0, 0) ds
a
Z t
∂
g (t, s, 0, 0) ds
+
∂t
a
Z b
Z b
0
+
|h (t, s, x (s) , x (s)) − h (t, s, 0, 0)| ds +
|h (t, s, 0, 0)| ds
a
a
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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Z b
∂
∂
h (t, s, x (s) , x0 (s)) − h (t, s, 0, 0) ds
+
∂t
∂t
a
Z b
∂
h (t, s, 0, 0) ds
+
∂t
a
≤ β exp (λ (t − a)) + p1 (t, t) |x (t)|1
Z t
Z b
+
p (t, s) |x (s)|1 ds +
q (t, s) |x (s)|1 ds
a
a
≤ β exp (λ (t − a)) + |x|S p1 (t, t) exp (λ (t − a))
Z t
Z b
+
p (t, s) exp (λ (s − a)) ds +
q (t, s) exp (λ (s − a)) ds
a
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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a
≤ β exp (λ (t − a)) + |x|S α exp (λ (t − a))
≤ [β + N α] exp (λ (t − a)) .
From (2.12) it follows that T x ∈ S. This proves that T maps S into itself.
Now, we verify that the operator T is a contraction map. Let x (t) , y (t) ∈ S.
From (2.10), (2.11) and using the hypotheses we have
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(2.13)
|(T x) (t) − (T y) (t)|1
= |(T x) (t) − (T y) (t)| + (T x)0 (t) − (T y)0 (t)
≤ |g (t, t, x (t) , x0 (t)) − g (t, t, y (t) , y 0 (t))|
Z t
+
|g (t, s, x (s) , x0 (s)) − g (t, s, y (s) , y 0 (s))| ds
Za t ∂
∂
g (t, s, x (s) , x0 (s)) − g (t, s, y (s) , y 0 (s)) ds
+
∂t
a ∂t
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Z
b
|h (t, s, x (s) , x0 (s)) − h (t, s, y (s) , y 0 (s))| ds
a
Z b
∂
∂
0
0
+
∂t h (t, s, x (s) , x (s)) − ∂t h (t, s, y (s) , y (s)) ds
a
Z t
≤ p1 (t, t) |x (t) − y (t)|1 +
p (t, s) |x (s) − y (s)|1 ds
a
Z b
+
q (t, s) |x (s) − y (s)|1 ds
a
Z t
≤ |x − y|S p1 (t, t) exp (λ (t − a)) +
p (t, s) exp (λ (s − a)) ds
a
Z b
+
q (t, s) exp (λ (s − a)) ds
+
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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a
≤ |x − y|S α exp (λ (t − a)) .
From (2.13) we obtain
|T x − T y|S ≤ α |x − y|S .
Since α < 1, it follows from the Banach fixed point theorem (see [5, p. 37]) that T
has a unique fixed point in S. The fixed point of T is however a solution of equation
(1.1). The proof is complete.
Remark 1. We note that in 1956 Bielecki [2] first used the norm defined in (2.2) for
proving global existence and uniqueness of solutions of ordinary differential equations. It is now used very frequently to obtain global existence and uniqueness results
for wide classes of differential and integral equations. For developments related to
the topic, see [4] and the references cited therein.
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The following theorem holds concerning the uniqueness of solutions of equation
(1.1) in Rn without the existence part.
Theorem 2.3. Assume that the functions g, h in equation (1.1) and their derivatives
with respect to t satisfy the conditions (2.4) – (2.7). Further assume that the functions
pi (t, s) , qi (t, s) (i = 1, 2) in (2.4) – (2.7) are nondecreasing in t ∈ I for each
s ∈ I,
p1 (t, t) ≤ d,
(2.14)
for t ∈ I, where d ≥ 0 is a constant such that d < 1,
Z s
Z b
1
1
(2.15)
e (t) =
q (t, s) exp
p (s, σ) dσ ds < 1,
a 1−d
a 1−d
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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where
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Then the equation (1.1) has at most one solution on I .
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Proof. Let x(t) and y(t) be two solutions of equation (1.1) and
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p (t, s) = p1 (t, s) + p2 (t, s) ,
q (t, s) = q1 (t, s) + q2 (t, s) .
w (t) = |x (t) − y (t)| + |x0 (t) − y 0 (t)| .
Then by hypotheses we have
Z t
(2.16)
w (t) ≤
|g (t, s, x (s) , x0 (s)) − g (t, s, y (s) , y 0 (s))| ds
a
Z b
+
|h (t, s, x (s) , x0 (s)) − h (t, s, y (s) , y 0 (s))| ds
a
+ |g (t, t, x (t) , x0 (t)) − g (t, t, y (t) , y 0 (t))|
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Z t
∂
∂
0
0
g (t, s, x (s) , x (s)) − g (t, s, y (s) , y (s)) ds
+
∂t
a ∂t
Z b
∂
∂
0
0
+
∂t h (t, s, x (s) , x (s)) − ∂t h (t, s, y (s) , y (s)) ds
a
Z
≤
t
p1 (t, s) [|x (s) − y (s)| + |x0 (s) − y 0 (s)|]ds
a
Z b
+
q1 (t, s) [|x (s) − y (s)| + |x0 (s) − y 0 (s)|]ds
a
+ p1 (t, t) [|x (t) − y (t)| + |x0 (t) − y 0 (t)|]
Z t
+
p2 (t, s) [|x (s) − y (s)| + |x0 (s) − y 0 (s)|]ds
a
Z b
+
q2 (t, s) [|x (s) − y (s)| + |x0 (s) − y 0 (s)|]ds.
a
Using (2.14) in (2.16) we observe that
Z t
Z b
1
1
p (t, s) w (s) ds+
q (t, s) w (s) ds.
(2.17)
w (t) ≤
1−d a
1−d a
Now a suitable application of Lemma 2.1 to (2.17) yields
|x (t) − y (t)| + |x0 (t) − y 0 (t)| ≤ 0,
and hence x(t) = y(t), which proves the uniqueness of solutions of equation (1.1)
on I .
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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3.
Bounds on Solutions
In this section we obtain estimates on the solutions of equation (1.1) under some
suitable conditions on the functions involved and their derivatives.
The following theorem concerning an estimate on the solution of equation (1.1)
holds.
Theorem 3.1. Assume that the functions f, g, h in equation (1.1) and their derivatives with respect to t satisfy the conditions
0
|f (t)| + |f (t)| ≤ c̄,
|g (t, s, u, v)| ≤ m1 (t, s) [|u| + |v|] ,
∂
g (t, s, u, v) ≤ m2 (t, s) [|u| + |v|] ,
∂t
(3.1)
(3.2)
(3.3)
(3.5)
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where c̄ ≥ 0 is a constant and for i = 1, 2, mi (t, s) , ni (t, s) ∈ C (I 2 , R+ ) and they
are nondecreasing in t ∈ I for each s ∈ I. Further assume that
¯
m1 (t, t) ≤ d,
(3.6)
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
|h (t, s, u, v)| ≤ n1 (t, s) [|u| + |v|] ,
∂
h (t, s, u, v) ≤ n2 (t, s) [|u| + |v|] ,
∂t
(3.4)
Volterra-Fredholm Type
Integral Equation
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Z
b
s
1
¯m (s, σ) dσ ds < 1,
a
a 1−d
for t ∈ I where d¯ ≥ 0 is a constant such that d¯ < 1 and
(3.7)
ē (t) =
1
n (t, s) exp
1 − d¯
Z
m (t, s) = m1 (t, s) + m2 (t, s) ,
n (t, s) = n1 (t, s) + n2 (t, s) .
If x (t) , t ∈ I is any solution of equation (1.1), then
Z t
c̄
1
0
(3.8)
|x (t)| + |x (t)| ≤
exp
m (t, s) ds ,
1 − ē (t)
1 − d¯
a
for t ∈ I.
Proof. Let u (t) = |x (t)| + |x0 (t)| for t ∈ I. Using the fact that x(t) is a solution of
equation (1.1) and the hypotheses we have
Z t
0
(3.9) u (t) ≤ |f (t)| + |f (t)| +
|g (t, s, x (s) , x0 (s))| ds
a
Z b
+
|h (t, s, x (s) , x0 (s))| ds + |g (t, t, x (t) , x0 (t))|
a
Z b
Z t
∂
∂
0
0
+
∂t g (t, s, x (s) , x (s)) ds +
∂t h (t, s, x (s) , x (s)) ds
a
a
Z t
Z b
≤ c̄ +
m1 (t, s) u (s) ds +
n1 (t, s) u (s) ds
a
a
Z t
Z b
+ m1 (t, t) u (t) +
m2 (t, s) u (s) ds +
n2 (t, s) u (s) ds.
a
a
Using (3.6) in (3.9) we observe that
Z t
Z b
c̄
1
1
(3.10) u (t) ≤
+
m (t, s) u (s) ds +
n (t, s) u (s) ds.
1 − d¯ 1 − d¯ a
1 − d¯ a
Now an application of Lemma 2.1 to (3.10) yields (3.8).
Remark 2. We note that the estimate obtained in (3.8) yields not only the bound on
the solution of equation (1.1) but also the bound on its derivative. If the estimate on
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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the right hand side in (3.8) is bounded, then the solution of equation (1.1) and its
derivative is bounded on I .
Now we shall obtain an estimate on the solution of equation (1.1) assuming that
the functions g, h and their derivatives with respect to t satisfy Lipschitz type conditions.
Theorem 3.2. Assume that the hypotheses of Theorem 2.3 hold. Suppose that
Z t
Z b
0
(3.11)
|g (t, s, f (s)) , f (s)| ds +
|h (t, s, f (s) , f 0 (s))| ds
a
Za t ∂
0
0
ds
+ |g (t, t, f (t), f (t))| +
g
(t,
s,
f
(s),
f
(s))
∂t
a
Z b
∂
0
h (t, s, f (s), f (s)) ds ≤ D,
+
∂t
a
for t ∈ I, where D ≥ 0 is a constant. If x (t) , t ∈ I is any solution of equation (1.1),
then
(3.12)
|x (t) − f (t)| + |x0 (t) − f 0 (t)|
Z t
D
1
≤
exp
p (t, s) ds ,
1−d
1 − e (t)
a
for t ∈ I.
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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0
0
Proof. Let u (t) = |x (t) − f (t)| + |x (t) − f (t)| for t ∈ I. Using the fact that x(t)
is a solution of equation (1.1) and the hypotheses we have
Z t
(3.13)
u (t) ≤
|g (t, s, x (s) , x0 (s)) − g (t, s, f (s) , f 0 (s))|ds
a
Z t
+
|g (t, s, f (s) , f 0 (s))|ds
a
Close
Z
b
|h (t, s, x (s) , x0 (s)) − h (t, s, f (s) , f 0 (s))|ds
+
a
Z
+
b
|h (t, s, f (s) , f 0 (s))|ds
a
+ |g (t, t, x (t) , x0 (t)) − g (t, t, f (t) , f 0 (t))|
+ |g (t, t, f (t) , f 0 (t))|
Z t
∂
∂
0
0
+
∂t g (t, s, x (s) , x (s)) − ∂t g (t, s, f (s) , f (s))ds
a
Z t
∂
0
g (t, s, f (s) , f (s))ds
+
∂t
a
Z b
∂
∂
0
0
ds
+
h
(t,
s,
x
(s)
,
x
(s))
−
h
(t,
s,
f
(s)
,
f
(s))
∂t
∂t
a
Z b
∂
h (t, s, f (s) , f 0 (s))ds
+
∂t
a
Z t
Z b
≤D+
p1 (t, s) u (s) ds +
q1 (t, s) u (s) ds
a
a
Z t
Z b
+ p1 (t, t) u(t) +
p2 (t, s) u (s) ds +
q2 (t, s) u (s) ds.
a
a
Using (2.14) in (3.13) we observe that
Z t
Z b
D
1
1
(3.14) u (t) ≤
+
p (t, s) u (s) ds +
q (t, s) u (s) ds.
1−d 1−d a
1−d a
Now an application of Lemma 2.1 to (3.14) yields (3.12).
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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4.
Continuous Dependence
In this section we shall deal with continuous dependence of solutions of equation
(1.1) on the functions involved therein and also the continuous dependence of solutions of equations of the form (1.1) on parameters.
Consider the equation (1.1) and the following Volterra-Fredholm type integral
equation
Z t
Z b
0
(4.1) y (t) = F (t) +
G (t, s, y (s) , y (s)) ds +
H (t, s, y (s) , y 0 (s)) ds,
a
a
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
for t ∈ I, where y, F, G, H are in Rn . We assume that F ∈ C (I, Rn ) , G, H ∈
C(I 2 × Rn × Rn , Rn ) and are continuously differentiable with respect to t on the
respective domains of their definitions.
The following theorem deals with the continuous dependence of solutions of
equation (1.1) on the functions involved therein.
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Theorem 4.1. Assume that the hypotheses of Theorem 2.3 hold. Suppose that
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(4.2)
|f (t) − F (t)| + |f 0 (t) − F 0 (t)|
+ |g (t, t, y (t) , y 0 (t)) − G (t, t, y (t) , y 0 (t))|
Z t
+
|g (t, s, y (s) , y 0 (s)) − G (t, s, y (s) , y 0 (s))| ds
Za t ∂
∂
0
0
g (t, s, y (s) , y (s)) − G (t, s, y (s) , y (s)) ds
+
∂t
∂t
a
Z b
+
|h (t, s, y (s) , y 0 (s)) − H (t, s, y (s) , y 0 (s))| ds
a
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Z b
∂
∂
h (t, s, y (s) , y 0 (s)) − H (t, s, y (s) , y 0 (s)) ds
+
∂t
∂t
a
≤ ε,
where f, g, h and F, G, H are the functions involved in equations (1.1) and (4.1),
y(t) is a solution of equation (4.1) and ε > 0 is an arbitrary small constant. Then
the solution x (t) , t ∈ I of equation (1.1) depends continuously on the functions
involved on the right hand side of equation (1.1).
Proof. Let z (t) = |x (t) − y (t)| + |x0 (t) − y 0 (t)| for t ∈ I. Using the facts that x(t)
and y(t) are the solutions of equations (1.1) and (4.1) and the hypotheses we have
(4.3)
z (t) ≤ |f (t) − F (t)| + |f 0 (t) − F 0 (t)|
+ |g (t, t, x (t) , x0 (t)) − g (t, t, y (t) , y 0 (t))|
+ |g (t, t, y (t) , y 0 (t)) − G (t, t, y (t) , y 0 (t))|
Z t
+
|g (t, s, x (s) , x0 (s)) − g (t, s, y (s) , y 0 (s))| ds
Za t
+
|g (t, s, y (s) , y 0 (s)) − G (t, s, y (s) , y 0 (s))| ds
a
Z b
+
|h (t, s, x (s) , x0 (s)) − h (t, s, y (s) , y 0 (s))| ds
a
Z b
+
|h (t, s, y (s) , y 0 (s)) − H (t, s, y (s) , y 0 (s))| ds
Za t ∂
∂
0
0
g (t, s, x (s) , x (s)) − g (t, s, y (s) , y (s)) ds
+
∂t
∂t
a
Z t
∂
∂
g (t, s, y (s) , y 0 (s)) − G (t, s, y (s) , y 0 (s)) ds
+
∂t
a ∂t
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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Z b
∂
∂
h (t, s, x (s) , x0 (s)) − h (t, s, y (s) , y 0 (s)) ds
+
∂t
∂t
a
Z b
∂
∂
0
0
h (t, s, y (s) , y (s)) − H (t, s, y (s) , y (s)) ds
+
∂t
∂t
a
Z t
Z b
≤ ε + p1 (t, t) z (t) +
p (t, s) z (s) ds +
q (t, s) z (s) ds.
a
a
Using (2.14) in (4.3) we observe that
Z t
Z b
ε
1
1
(4.4)
z (t) ≤
+
p (t, s) z (s) ds +
q (t, s) z (s) ds.
1−d 1−d a
1−d a
Now an application of Lemma 2.1 to (4.4) yields
(4.5)
0
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
Title Page
0
|x (t) − y (t)| + |x (t) − y (t)|
Z t
ε
1
≤
exp
p (t, s) ds ,
1−d
1 − e (t)
a
for t ∈ I. From (4.5) it follows that the solutions of equation (1.1) depend continuously on the functions involved on the right hand side of equation (1.1).
Next, we consider the following Volterra-Fredholm type integral equations
Z t
Z b
0
(4.6) z (t) = f (t) +
g (t, s, z (s) , z (s) , µ) ds +
h (t, s, z (s) , z 0 (s) , µ) ds,
a
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a
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and
Z
(4.7)
Volterra-Fredholm Type
Integral Equation
z (t) = f (t) +
a
t
g (t, s, z (s) , z 0 (s) , µ0 ) ds
Z b
+
h (t, s, z (s) , z 0 (s) , µ0 ) ds,
a
for t ∈ I, where z, f, g, h are in Rn and µ, µ0 are real parameters. We assume that
f ∈ C (I, Rn ); g, h ∈ C (I 2 × Rn × Rn × R, Rn ) and are continuously differentiable with respect to t on the respective domains of their definitions.
Finally, we present the following theorem which deals with the continuous dependency of solutions of equations (4.6) and (4.7) on parameters.
Theorem 4.2. Assume that the functions g, h in equations (4.6) and (4.7) and their
derivatives with respect to t satisfy the conditions
(4.8)
|g (t, s, u, v, µ) − g (t, s, ū, v̄, µ)| ≤ k1 (t, s) [|u − ū| + |v − v̄|] ,
(4.9)
|g (t, s, u, v, µ) − g (t, s, u, v, µ0 )| ≤ δ1 (t, s) |µ − µ0 | ,
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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(4.10)
|h (t, s, u, v, µ) − h (t, s, ū, v̄, µ)| ≤ r1 (t, s) [|u − ū| + |v − v̄|] ,
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(4.11)
|h (t, s, u, v, µ) − h (t, s, u, v, µ0 )| ≤ γ1 (t, s) |µ − µ0 | ,
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(4.12)
∂
∂
g (t, s, u, v, µ) − g (t, s, ū, v̄, µ) ≤ k2 (t, s) [|u − ū| + |v − v̄|] ,
∂t
∂t
(4.13)
∂
∂
g (t, s, u, v, µ) − g (t, s, u, v, µ0 ) ≤ δ2 (t, s) |µ − µ0 | ,
∂t
∂t
Close
(4.14)
∂
h (t, s, u, v, µ) − ∂ h (t, s, ū, v̄, µ) ≤ r2 (t, s) [|u − ū| + |v − v̄|] ,
∂t
∂t
(4.15)
∂
∂
h (t, s, u, v, µ) − h (t, s, u, v, µ0 ) ≤ γ2 (t, s) |µ − µ0 | ,
∂t
∂t
where ki (t, s) , ri (t, s) ∈ C (I 2 , R+ ) (i = 1, 2) are nondecreasing in t ∈ I, for
each s ∈ I and δi (t, s) , γi (t, s) ∈ C (I 2 , R+ ) (i = 1, 2) . Further, assume that
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
k1 (t, t) ≤ λ,
(4.16)
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Z
e0 (t) =
(4.17)
a
Z
(4.18)
δ1 (t, t) +
b
1
r̄ (t, s) exp
1−λ
Z
s
1
k̄ (s, σ)dσ ds < 1,
1−λ
a
t
Z
[γ1 (t, s) + γ2 (t, s)]ds ≤ M,
[δ1 (t, s) + δ2 (t, s)]ds +
a
b
a
for t ∈ I where λ, M are nonnegative constants such that λ < 1 and
k̄ (t, s) = k1 (t, s) + k2 (t, s) ,
r̄ (t, s) = r1 (t, s) + r2 (t, s) .
Let z1 (t) and z2 (t) be the solutions of equations (4.6) and (4.7) respectively. Then
(4.19)
|z1 (t) − z2 (t)| + |z10 (t) − z20 (t)|
Z t
1
|µ − µ0 | M
exp
k̄ (t, s) ds ,
≤
1−λ
1 − e0 (t)
a
for t ∈ I.
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Proof. Let u (t) = |z1 (t) − z2 (t)| + |z10 (t) − z20 (t)| for t ∈ I. Using the facts that
z1 (t) and z2 (t) are the solutions of the equations (4.6) and (4.7) and the hypotheses
we have
Z t
(4.20) u (t) ≤ |g (t, s, z1 (s) , z10 (s) , µ) − g (t, s, z2 (s) , z20 (s) , µ)| ds
a
Z t
+ |g (t, s, z2 (s) , z20 (s) , µ) − g (t, s, z2 (s) , z20 (s) , µ0 )| ds
a
Z b
+ |h (t, s, z1 (s) , z10 (s) , µ) − h (t, s, z2 (s) , z20 (s) , µ)| ds
a
Z b
+ |h (t, s, z2 (s) , z20 (s) , µ) − h (t, s, z2 (s) , z20 (s) , µ0 )| ds
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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a
+|g (t, t, z1 (t) , z10 (t) , µ) − g (t, t, z2 (t) , z20 (t) , µ)|
+|g (t, t, z2 (t) , z20 (t) , µ) − g (t, t, z2 (t) , z20 (t) , µ0 )|
Z t
∂
∂
0
0
+ g (t, s, z1 (s) , z1 (s) , µ) − g (t, s, z2 (s) , z2 (s) , µ) ds
∂t
a ∂t
Z t
∂
∂
0
0
+ g (t, s, z2 (s) , z2 (s) , µ) − g (t, s, z2 (s) , z2 (s) , µ0 ) ds
∂t
a ∂t
Z b
∂
∂
0
0
+ h (t, s, z1 (s) , z1 (s) , µ) − h (t, s, z2 (s) , z2 (s) , µ) ds
∂t
a ∂t
Z b
∂
∂
0
0
+ h (t, s, z2 (s) , z2 (s) , µ) − h (t, s, z2 (s) , z2 (s) , µ0 ) ds
∂t
a ∂t
Z t
Z t
≤
k1 (t, s) u (s) ds +
δ1 (t, s) |µ − µ0 | ds
a
a
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Z
+
b
Z
a
b
γ1 (t, s) |µ − µ0 | ds
r1 (t, s) u (s) ds +
a
+ k1 (t, t) u (t) + δ1 (t, t) |µ − µ0 |
Z t
Z t
+
k2 (t, s) u (s) ds +
δ2 (t, s) |µ − µ0 | ds
a
a
Z b
Z b
r2 (t, s) u (s) ds +
γ2 (t, s) |µ − µ0 | ds.
+
a
a
Using (4.16), (4.18) in (4.20) we observe that
Z t
Z b
1
1
|µ − µ0 | M
+
k̄ (t, s) u (s) ds+
r̄ (t, s) u (s) ds.
(4.21) u (t) ≤
1−λ
1−λ a
1−λ a
Now an application of Lemma 2.1 to (4.21) yields (4.19), which shows the dependency of solutions to equations (4.6) and (4.7) on parameters.
Remark 3. We note that our approach to the study of the more general equation (1.1)
is different from those used in [1] and we believe that the results obtained here are
of independent interest.
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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References
[1] A.M. BICA, V.A. CǍUŞ AND S. MUREŞAN, Application of a trapezoid inequality to neutral Fredholm integro-differential equations in Banach spaces,
J. Inequal. Pure and Appl. Math., 7(5) (2006), Art. 173. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=791].
Volterra-Fredholm Type
Integral Equation
[2] A. BIELECKI, Un remarque sur l’application de la méthode de BanachCacciopoli-Tikhonov dans la théorie des équations differentilles ordinaires,
Bull. Acad. Polon. Sci. Sér. Sci. Math.Phys. Astr., 4 (1956), 261–264.
vol. 9, iss. 4, art. 116, 2008
[3] T.A. BURTON, Volterra Integral and Differential Equations, Academic Press,
New York, 1983.
Title Page
[4] C. CORDUNEANU, Bielecki’s method in the theory of integral equations,
Ann. Univ.Mariae Curie-Sklodowska, Section A, 38 (1984), 23–40.
[5] C. CORDUNEANU, Integral Equations and Applications, Cambridge University Press, 1991.
B.G. Pachpatte
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[6] M.A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1964.
Page 23 of 24
[7] M. KWAPISZ, Bieleck’s method, existence and uniqueness results for Volterra
integral equations in Lp spaces, J. Math. Anal. Appl., 154 (1991), 403–416.
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[8] R.K. MILLER, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo
Park CA, 1971.
[9] B.G. PACHPATTE, On a nonlinear Volterra Integral-Functional equation,
Funkcialaj Ekvacioj, 26 (1983), 1–9.
[10] B.G. PACHPATTE, Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies, Vol. 205 Elsevier Science, Amsterdam, 2006.
[11] B.G. PACHPATTE, On higher order Volterra-Fredholm integrodifferential
equation, Fasciculi Mathematici, 37 (2007), 35–48.
[12] W. WALTER, Differential and Integral Inequalities, Springer-Verlag, Berlin,
New York, 1970.
Volterra-Fredholm Type
Integral Equation
B.G. Pachpatte
vol. 9, iss. 4, art. 116, 2008
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