Journal of Inequalities in Pure and
Applied Mathematics
APPLICATION OF A TRAPEZOID INEQUALITY TO NEUTRAL
FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS IN
BANACH SPACES
volume 7, issue 5, article 173,
2006.
ALEXANDRU MIAHI BICA, VASILE AUREL CĂUŞ AND
SORIN MUREŞAN
Received 27 January, 2006;
accepted 19 November, 2006.
Communicated by: S.S. Dragomir
Department of Mathematics
University of Oradea,
Str. Universitatii no.1
410087, Oradea, Romania
Abstract
EMail: abica@uoradea.ro
EMail: vcaus@uoradea.ro
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2000
Victoria University
ISSN (electronic): 1443-5756
028-06
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Abstract
A new approach for neutral Fredholm integro-differential equations in Banach
spaces, using the Perov’s fixed point theorem of existence, uniqueness and approximation is presented. The approximation of the solution and of its derivative
is realized using the method of successive approximations and a trapezoidal
quadrature rule in Banach spaces for Lipschitzian functions. The interest is
focused on the error estimation.
2000 Mathematics Subject Classification: Primary 45J05, Secondary 45N05, 45B05,
65L05.
Key words: Nonlinear neutral Fredholm integro-differential equations in Banach
spaces, Perov’s fixed point theorem, Method of successive approximations, Trapezoidal quadrature rule.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Existence, Uniqueness and Approximation . . . . . . . . . . . . . . . 5
3
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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Page 2 of 24
1.
Introduction
Consider the neutral Fredholm integro-differential equation
Z b
(1.1)
x (t) =
f (t, s, x (s) , x0 (s)) ds + g (t) , t ∈ [a, b]
a
where f : [a, b] × [a, b] × X × X → X is continuous, X is a Banach space and
g ∈ C 1 ([a, b] , X) .
To obtain the existence, uniqueness and global approximation of the solution of (1.1) we will use Perov’s fixed point theorem. To this purpose, we differentiate the equation (1.1) with respect to t and assume that f (·, s, u, v) ∈
C 1 ([a, b] , X) , ∀s ∈ [a, b] , ∀u, v ∈ X, where x0 = y. Hence (1.1) reduces to
the following system of Fredholm integral equations:

Rb
 x (t) = a f (t, s, x (s) , y (s)) ds + g (t)
, t ∈ [a, b] .
(1.2)
R b ∂f

0
y (t) = a ∂t (t, s, x (s) , y (s)) ds + g (t)
The Perov fixed point theorem will be applied to the system (1.2) obtaining
also the approximation of the solution of (1.1) and its derivative.
The Perov fixed point theorem appeared for the first time in [16] and was
later used for two point boundary value problems of second order differential
equations in [5]. The Perov fixed point theorem was also used in [1], [10],
[18] and [19]. Bica and Muresan have used the Perov fixed point theorem for
delay neutral integro-differential equations in [6] and [7]. In this paper the
authors have constructed a method of approximating the solution of (1.1) and
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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Page 3 of 24
its derivative using a sequence of successive approximations and a trapezoidal
quadrature rule from [8]. Some of the existing numerical methods applied to
Fredholm integro-differential equations can be found in the papers [2], [3], [4],
[9], [11], [12], [13], [14], [15], [17]. The tools utilised in these papers are:
the tau method, direct methods, collocation methods, Runge-Kutta methods,
wavelet methods and spline approximation.
In this paper, our interest will be focused on the error estimation of the
method presented in the Section 3.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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2.
Existence, Uniqueness and Approximation
Consider the following conditions:
(i) (continuity): f ∈ C ([a, b] × [a, b] × X × X, X) , g ∈ C 1 ([a, b] , X) and
f (·, s, u, v) ∈ C 1 ([a, b] , X) ,for any s ∈ [a, b] , u, v ∈ X
(ii) (Lipschitz conditions): there exist α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 , η ∈ R∗+ such
that for any t, t0 , s, s1 , s2 ∈ [a, b] and u, v, u1 , u2 , v1 , v2 ∈ X, we have:
(2.1)
(2.2)
kf (t, s, u1 , v1 ) − f (t, s, u2 , v2 )kX
≤ α1 ku1 − u2 kX + β1 kv1 − v2 kX ,
∂f
∂f
∂t (t, s, u1 , v1 ) − ∂t (t, s, u2 , v2 )
X
≤ α2 ku1 − u2 kX + β2 kv1 − v2 kX ,
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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(2.3)
kf (t, s1 , u, v) − f (t, s2 , u, v)kX ≤ γ1 |s1 − s2 | ,
(2.4)
∂f
∂f
≤ γ2 |s1 − s2 | ,
(t,
s
,
u,
v)
−
(t,
s
,
u,
v)
1
2
∂t
∂t
X
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(2.5)
kf (t, s, u, v) − f (t0 , s, u, v)kX ≤ δ1 |t − t0 | ,
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(2.6)
∂f
∂f
0
0
∂t (t, s, u, v) − ∂t (t , s, u, v) ≤ δ2 |t − t | ,
X
(2.7)
kg 0 (t) − g 0 (t0 )kX ≤ η · |t − t0 | .
We use Perov’s fixed point theorem (see [16], [5] and [10]):
Theorem 2.1. Let (X, d) be a complete generalized metric space such that
d (x, y) ∈ Rn for x, y ∈ X. Suppose that there exists a function A : X → X
such that:
d (A (x) , A (y)) ≤ Q · d (x, y)
for any x, y ∈ X, where Q ∈ Mn (R+ ). If all eigenvalues of Q lie in the open
unit disc from R2 then Qm → 0 for m → ∞ and the operator A has a unique
fixed point x∗ ∈ X. Moreover, the sequence of successive approximations xm =
A (xm−1 ) converges to x∗ in X for any x0 ∈ X and the following estimation
holds:
(2.8)
∗
m
−1
d (xm , x ) ≤ Q (In − Q)
· d (x0 , x1 ) ,
for each m ∈ N
∗
where In is the unity matrix in Mn (R) .
We recall the notion of generalized metric, which is a function d : Y × Y →
R on a nonempty set Y with the properties:
n
a) d (x, y) ≥ 0, for any x, y ∈ Y ;
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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b) d (x, y) = 0 ⇔ x = y, for x, y ∈ Y ;
c) d (y, x) = d (x, y) , for any x, y ∈ Y ;
d) d (x, z) ≤ d (x, y) + d (y, z) , for any x, y, z ∈ Y.
Here, the order relation on Rn is defined by
x ≤ y ⇔ xi ≤ yi ,
R, for each i = 1, n,
in
x i , yi ∈ R
for x = (x1 , . . . , xn ) , y = (y1 , . . . , yn ) ∈ Rn . The pair (Y, d) denote a generalized metric space.
In the following, we will use the notation
kukC = max {ku (t)kX : t ∈ [a, b]} ,
for u ∈ C ([a, b] , X) .
We consider the product space Y = C ([a, b] , X) × C ([a, b] , X) and define
the generalized metric d : Y × Y → R2 by,
d ((u1 , v1 ) , (u2 , v2 )) = (ku1 − u2 kC , kv1 − v2 kC ) ,
for (u1 , v1 ) , (u2 , v2 ) ∈ Y.
It is easy to prove that (Y, d) is a complete generalized metric space. We
define the operator A : Y → Y, A = (A1 A2 ) by,
Z
A1 (x, y) (t) =
b
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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f (t, s, x (s) , y (s)) ds + g (t)
a
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Page 7 of 24
and
b
Z
(2.9)
A2 (x, y) (t) =
a
∂f
(t, s, x (s) , y (s)) ds + gt (t) ,
∂t
t ∈ [a, b] .
The following result concerning the existence and uniqueness of the solution
for the equation (1.1) holds.
Theorem 2.2. In the conditions (i), (2.1), (2.2), if (α1 + β2 ) (b − a) < 2 and
(2.10) 1 + (b − a)4 (α1 β2 − α2 β1 )2 + 2α2 β1 (b − a)2 > (b − a)2 α12 + β22
then the operator A has a unique fixed point (x∗ , y ∗ ) such that
x∗ ∈ C 1 ([a, b] , X) ,
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
y ∗ = (x∗ )0
and x∗ is the unique solution of the equation (1.1). Moreover, the sequence of
the successive approximations given by,
(2.11)
(x0 (t) , y0 (t)) = (g (t) , g 0 (t)) ,
(2.12)
xm+1 (t) =
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J
f (t, s, xm (s) , ym (s)) ds + g (t) ,
t ∈ [a, b]
a
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and
Z
(2.13)
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t ∈ [a, b]
b
Z
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
ym+1 (t) =
a
b
∂f
(t, s, xm (s) , ym (s)) ds + g 0 (t) ,
∂t
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t ∈ [a, b] ,
Page 8 of 24
converges in Y to (x∗ , y ∗ ) and the following error estimation holds:
(2.14) d ((xm , ym ) , (x∗ , y ∗ )) ≤ Qm (In − Q)−1 · d ((x0 , y0 ) , (x1 , y1 )) ,
for any m ∈ N∗ ,
where
Q = (b − a)
α1 β1
α2 β2
.
Proof. From condition (i) we infer that A (Y ) ⊂ Y. For (u1 , v1 ) , (u2 , v2 ) ∈ Y,
t ∈ [a, b] we have
kA1 (u1 , v1 ) (t) − A1 (u2 , v2 ) (t)kX
Z b
≤
[α1 ku1 (s) − u2 (s)kX + β1 kv1 (s) − v2 (s)kX ]
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
a
≤ (b − a) · [α1 ku1 − u2 kC + β1 kv1 − v2 kC ] ,
for any t ∈ [a, b]
Contents
and
kA2 (u1 , v1 ) (t) − A2 (u2 , v2 ) (t)kX
≤ (b − a) · [α2 ku1 − u2 kC + β2 kv1 − v2 kC ] ,
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for any t ∈ [a, b] .
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We infer that
d (A (u1 , v1 ) , A (u2 , v2 )) ≤ Q · d ((u1 , v1 ) , (u2 , v2 )) ,
for any (u1 , v1 ) , (u2 , v2 ) ∈ Y,
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Page 9 of 24
where
(b − a) α1 (b − a) β1
!
Q=
.
(b − a) α2 (b − a) β2
It is easy to see that the eigenvalues of Q are real. The inequalities (2.10) and
(α1 + β2 ) (b − a) < 2 lead to µ1 , µ2 ∈ (−1, 1) , where µ1 and µ2 are these
eigenvalues. From the Perov fixed point theorem we infer that Qm → 0 for
m → ∞ and the operator A has a unique fixed point (x∗ , y ∗ ) ∈ Y. Then,
Z b
∗
x (t) =
f (t, s, x∗ (s) , y ∗ (s)) ds + g (t) , ∀t ∈ [a, b]
a
and
Z
b
∂f
(t, s, x∗ (s) , y ∗ (s)) ds + g 0 (t) , ∀t ∈ [a, b] .
∂t
a
Since f (·, s, u, v) ∈ C 1 ([a, b] , X) , for any s ∈ [a, b] , u, v ∈ X and g ∈
C 1 ([a, b] , X) we infer that x∗ ∈ C 1 ([a, b] , X) . If we differentiate the first
equality with respect to t we obtain y ∗ = (x∗ )0 . Then x∗ is the unique solution
of (1.1). From the relations (2.12) and (2.13) and from (2.9) we infer that the
sequences given in (2.12), (2.13) fulfil the recurrence relation
∗
y (t) =
(xm+1 , ym+1 ) = A ((xm , ym )) ,
∀m ∈ N.
Now, the inequality (2.14) follows from the estimation (2.8). Since Qm → 0 for
m → ∞ in M2 (R) we infer that
lim d ((xm , ym ) , (x∗ , y ∗ )) = (0, 0) .
m→∞
This proves the theorem.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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3.
The Main Result
To compute the terms of the sequence of successive approximations we use
in the calculus of integrals in (2.12), (2.13) the trapezoidal quadrature rule for
Lipschitzian functions from [8]:
Z
b
F (x) dx
(3.1)
a
"
=
(b − a)
F (a) + 2
2n
n−1
X
i=1
F
a+
i (b − a)
n
#
+ F (b) + Rn (F )
with
(3.2)
L (b − a)2
kRn (F )kX ≤
,
4n
where L is the Lipschitz constant of F : [a, b] → X.
In this respect consider the uniform partition of [a, b] ,
(3.3)
∆ : a = t0 < t1 < · · · < tn−1 < tn = b
with ti = a + i(b−a)
, i = 0, n and compute xm (ti ) , ym (ti ) , i = 0, n, m ∈
n
∗
N.
From (2.12) and (2.13) we have
Z b
(3.4)
xm+1 (ti ) =
f (ti , s, xm (s) , ym (s)) ds + g (ti )
a
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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and
Z
(3.5)
ym+1 (ti ) =
a
b
∂f
(ti , s, xm (s) , ym (s)) ds + g 0 (ti ) ,
∂t
for any i = 0, n,
m ∈ N.
We define the functions
Fm,i , Gm,i : [a, b] → X,
m ∈ N,
i = 0, n
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
by
(3.6)

 Fm,i (s) = f (ti , s, xm (s) , ym (s))

Gm,i (s) =
∂f
∂t
, for any s ∈ [a, b] .
(ti , s, xm (s) , ym (s))
Definition 3.1. A set Z ⊂ C ([a, b] , X) is equally Lipschitz if there exists L ≥ 0
such that for any h ∈ Z,
kh (t) − h (t0 )kX ≤ L · |t − t0 | ,
for each t, t0 ∈ [a, b] .
Theorem 3.1. The subsets
{{Fm,i }m∈N ,
i = 0, n} ⊂ C ([a, b] , X)
{{Gm,i }m∈N ,
i = 0, n} ⊂ C ([a, b] , X)
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and
defined in (3.6), are equally Lipschitz, if the conditions (i) and (2.1) – (2.7) are
true.
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Proof. Let
µ = kg 0 kC = max {kg 0 (t)kX : t ∈ [a, b]} .
For m ∈ N∗ and i = 0, n we have
kFm,i (s1 ) − Fm,i (s2 )kX
≤ γ1 |s1 − s2 | + α1 kxm (s1 ) − xm (s2 )kX + β1 kym (s1 ) − ym (s2 )kX
≤ γ1 |s1 − s2 | + α1 [δ1 (b − a) + µ] · |s1 − s2 |
+ β1 [δ2 (b − a) · |s1 − s2 | + kg 0 (s1 ) − g 0 (s2 )kX ]
≤ [γ1 + α1 µ + β1 η + (b − a) (α1 δ1 + β1 δ2 )] · |s1 − s2 | ,
for any s1 , s2 ∈ [a, b] and
kGm,i (s1 ) − Gm,i (s2 )kX
≤ γ2 |s1 − s2 | + α2 kxm (s1 ) − xm (s2 )kX + β2 kym (s1 ) − ym (s2 )kX
≤ [γ2 + α2 µ + β2 η + (b − a) (α2 δ1 + β2 δ2 )] · |s1 − s2 | ,
Alexandru Miahi Bica,
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Sorin Mureşan
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for any s1 , s2 ∈ [a, b] . Moreover, for any i = 0, n we have,
kF0,i (s1 ) − F0,i (s2 )kX ≤ (γ1 + α1 µ + β1 η)·|s1 − s2 | ,
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
for each s1 , s2 ∈ [a, b]
and
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kG0,i (s1 ) − G0,i (s2 )kX ≤ (γ2 + α2 µ + β2 η)·|s1 − s2 | ,
for each s1 , s2 ∈ [a, b] .
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Let
L1 = γ1 + α1 µ + β1 η + (b − a) (α1 δ1 + β1 δ2 ) ,
Page 13 of 24
L2 = γ2 + α2 µ + β2 η + (b − a) (α2 δ1 + β2 δ2 ) .
From the above, we infer that for any i = 0, n, we have,
kFm,i (s1 ) − Fm,i (s2 )kX ≤ L1 · |s1 − s2 | ,
for each s1 , s2 ∈ [a, b]
and
kGm,i (s1 ) − Gm,i (s2 )kX ≤ L2 ·|s1 − s2 | ,
for each s1 , s2 ∈ [a, b] and m ∈ N.
This concludes the proof of the theorem.
Applying in (3.4), (3.5) the quadrature rule (3.1) – (3.2) we obtain the numerical method:
x0 (ti ) = g (ti ) , y0 (ti ) = g 0 (ti ) ,
(3.7)
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
for i = 0, n
"
(b − a)
· f (ti , a, xm−1 (a) , ym−1 (a))
(3.8) xm (ti ) = g (ti ) +
2n
+2
n−1
X
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Contents
#
f (ti ; tj , xm−1 (tj ) , ym−1 (tj )) + f (ti , b, xm−1 (b) , ym−1 (b))
j=1
+ Rm,i ,
for i = 0, n and m ∈ N∗
and
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(3.9) ym (ti ) = g 0 (ti ) +
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
(b − a)
∂f
·
(ti , a, xm−1 (a) , ym−1 (a))
2n
∂t
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+2
n−1
X
∂f
j=1
∂f
(ti ; tj , xm−1 (tj ) , ym−1 (tj )) +
(ti , b, xm−1 (b) , ym−1 (b))
∂t
∂t
+ Rm,i ,
#
for i = 0, n and m ∈ N∗ .
with the remainder estimations
(3.10)
(3.11)
kRm,i kX ≤
kωm,i kX ≤
L1 (b − a)2
,
4n
L2 (b − a)2
,
4n
for any m ∈ N∗ and i = 0, n,
or any m ∈ N∗ and i = 0, n.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
These lead to the following algorithm:
x0 (ti ) = g (ti ) , y0 (ti ) = g 0 (ti ) ,
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(3.12)
x1 (ti )
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"
= g (ti ) +
+2
JJ
J
(b − a)
· f (ti , a, g (a) , g 0 (a))
2n
n−1
X
#
f (ti ; tj , g (tj ) , g 0 (tj )) + f (ti , b, g (b) , g 0 (b)) + R1,i
j=1
= x1 (ti ) + R1,i ,
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Page 15 of 24
(3.13)
(3.14)
(b − a) ∂f
y1 (ti ) = g (ti ) +
·
(ti , a, g (a) , g 0 (a))
2n
∂t
n−1
X
∂f
+2
(ti ; tj , g (tj ) , g 0 (tj ))
∂t
j=1
∂f
0
+
(ti , b, g (b) , g (b)) + ω1,i
∂t
= y1 (ti ) + ω1,i ,
0
(b − a)
· f (ti , t0 , x1 (t0 ) + R1,0 , y1 (t0 ) + ω1,0 )
x2 (ti ) = g (ti ) +
2n
n−1
X
+2
f (ti ; tj , x1 (tj ) + R1,j , y1 (tj ) + ω1,j )
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
j=1
+ f (ti , tn , x1 (tn ) + R1,n , y1 (tn ) + ω1,n ) + R2,i
(b − a)
= g (ti ) +
· f (ti , t0 , x1 (t0 ) , y1 (t0 ))
2n
n−1
X
+2
f (ti ; tj , x1 (tj ) , y1 (tj ))
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j=1
+ f (ti , tn , x1 (tn ) , y1 (tn )) + R2,i
= x2 (ti ) + R2,i ,
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Page 16 of 24
"
(b
−
a)
∂f
·
(ti , t0 , x1 (t0 ) + R1,0 , y1 (t0 ) + ω1,0 )
(3.15) y2 (ti ) = g 0 (ti ) +
2n
∂t
+2
n−1
X
∂f
j=1
∂t
(ti ; tj , x1 (tj ) + R1,j , y1 (tj ) + ω1,j )
#
∂f
+
(ti , tn , x1 (tn ) + R1,n , y1 (tn ) + ω1,n ) + ω2,i
∂t
"
(b − a)
∂f
0
·
(ti , t0 , x1 (t0 ) , y1 (t0 ))
= g (ti ) +
2n
∂t
+2
n−1
X
j=1
+
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
∂f
(ti ; tj , x1 (tj ) , y1 (tj ))
∂t
#
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∂f
(ti , tn , x1 (tn ) , y1 (tn )) + ω2,i
∂t
= y2 (ti ) + ω2,i ,
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when i = 0, n.
By induction, for m ≥ 3, we obtain for i = 0, n that
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(3.16) xm (ti )
Close
"
= g (ti ) +
(b − a)
· f ti , t0 , xm−1 (t0 ) + Rm−1,0 , ym−1 (t0 ) + ω m−1,0
2n
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Page 17 of 24
+2
n−1
X
f ti ; tj , xm−1 (tj ) + Rm−1,j , ym−1 (tj ) + ω m−1,j
j=1
#
+ f ti , tn , xm−1 (tn ) + Rm−1,n , ym−1 (tn ) + ω m−1,n
+ Rm,i
"
(b − a)
· f (ti , t0 , xm−1 (t0 ) , ym−1 (t0 ))
= g (ti ) +
2n
+2
n−1
X
f (ti ; tj , xm−1 (tj ) , ym−1 (tj ))
j=1
#
+ f (ti , tn , xm−1 (tn ) , ym−1 (tn )) + Rm,i
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
= xm (ti ) + Rm,i
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and
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(3.17) ym (ti )
(b − a) ∂f
ti , t0 , xm−1 (t0 ) + Rm−1,0 , ym−1 (t0 )
2n
∂t
n−1
X
∂f
ti ; tj , xm−1 (tj )
+ ω m−1,0 + 2
∂t
j=1
+ Rm−1,j , ym−1 (tj ) + ω m−1,j
= g 0 (ti ) +
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∂f
+
ti , tn , xm−1 (tn ) + Rm−1,n , ym−1 (tn ) + ω m−1,n + ωm,i
∂t
"
(b
−
a)
∂f
= g 0 (ti ) +
·
(ti , t0 , xm−1 (t0 ) , ym−1 (t0 ))
2n
∂t
+2
n−1
X
∂f
j=1
∂t
(ti ; tj , xm−1 (tj ) , ym−1 (tj ))
#
∂f
+
(ti , tn , xm−1 (tn ) , ym−1 (tn )) + ωm,i
∂t
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
= ym (ti ) + ωm,i .
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
At the remainder estimation we have for any i = 0, n
L1 (b − a)2
kR1,i kX ≤
,
4n
L2 (b − a)2
kω1,i kX ≤
.
4n
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Using in (3.14) the Lipschitz property (2.1) we obtain:
2
R2,i ≤ [1 + (b − a) (α1 + β1 )] · L1 (b − a) ,
4n
for any i = 0, n.
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Using in (3.15) the Lipschitz property (2.2) we obtain:
L2 (b − a)2
,
kω 2,i k ≤ [1 + (b − a) (α2 + β2 )] ·
4n
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for each i = 0, n.
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By induction, we obtain for m ≥ 2 and i = 0, n,
(3.18) Rm,i ≤ 1 + (b − a) (α1 + β1 ) + · · · + (b − a)m−1 (α1 + β1 )m−1
L1 (b − a)2
·
4n
1 − (b − a)m (α1 + β1 )m L1 (b − a)2
=
·
1 − (b − a) (α1 + β1 )
4n
and
(3.19) kω m,i k ≤ 1 + (b − a) (α2 + β2 ) + · · · + (b − a)m−1 (α2 + β2 )m−1
L2 (b − a)2
4n
1 − (b − a)m (α2 + β2 )m L2 (b − a)2
=
·
.
1 − (b − a) (α2 + β2 )
4n
·
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
Title Page
Finally, we can state the following result:
Theorem 3.2. With the conditions (i), (2.1) – (2.7), (2.10), and if (b − a) (α1 + β1 )
< 1 and (b − a) (α2 + β2 ) < 1, then the solution of the system (1.2) is approximated on the knots of the uniform partition ∆ given in (3.3), by the sequence
{(xm (ti ) , ym (ti ))}m∈N ,
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
i = 0, n
obtained in (3.12) – (3.17) and the following error estimation holds:
!
kx∗ (ti ) − xm (ti )kX
(3.20)
≤ Qm (I2 − Q)−1 d (x0 , x1 )
∗
ky (ti ) − ym (ti )kX
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
+
L1 (b−a)2
4n[1−(b−a)(α1 +β1 )]
L2 (b−a)2
4n[1−(b−a)(α2 +β2 )]

,
for each m ∈ N∗ and i = 0, n.
Proof. Follows from (2.14), (3.18) and (3.19) since
kx∗ (ti ) − xm (ti )kX ≤ kx∗ (ti ) − xm (ti )kX + kxm (ti ) − xm (ti )kX
for each m ∈ N∗ and i = 0, n and,
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
ky ∗ (ti ) − ym (ti )kX ≤ ky ∗ (ti ) − ym (ti )kX + kym (ti ) − ym (ti )kX
for each m ∈ N∗ and ∀i = 0, n.
Remark 1. When f (t, s, u, v) = H (t, s) · f (s, u, v) with H 1 ∈ C [a, b]2 , X
we obtain the existence, uniqueness and approximation of the solution for
Hammerstein-Fredholm integro-differential equations in Banach spaces. Moreover, in the particular case H (t, s) = G (t, s) , the Green function, we obtain
a new approach for two point boundary value problems associated to second
order differential equations in Banach spaces.
Remark 2. For X = Rn we obtain a new method in analysing systems of Fredholm integro-differential equations and for X = R we obtain similar results
for the approximation of the solution of a scalar Fredholm integro-differential
equation.
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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References
[1] S. ANDRAS, A note on Perov’s fixed point theorem, Fixed Point Theory,
4(1) (2003), 105–108.
[2] A. AYAD, Spline approximation for first order Fredholm delay integrodifferential equations, Int. J. Comput. Math., 70(3) (1999), 467–476.
[3] A. AYAD, Spline approximation for first order Fredholm integrodifferential equations, Studia Univ. Babes-Bolyai Math., 41(3) (1996), 1–
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[4] S.H. BEHIRY AND H. HASHISH, Wavelet methods for the numerical solution of Fredholm integro-differential equations, Int. J. Appl. Math., 11(1)
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[7] A. BICA AND S. MUREŞAN, Applications of the Perov’s fixed point theorem to delay integro-differential equations, Chap. 3 in Fixed Point Theory
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Inc., New York, 2006.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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[8] C. BUŞE, S.S. DRAGOMIR, J. ROUMELIOTIS AND A. SOFO, Generalized trapezoid type inequalities for vector-valued functions and applications, Math. Inequal. Appl., 5(3) (2002), 435–450.
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Pure Math. Appl., 11(2) (2000), 183–186.
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(1990), 237–246.
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Math. Modelling, 27(2) (2003), 145–154.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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[13] HU QIYA, Interpolation correction for collocation solutions of Fredholm
integro-differential equations, Math. Comput., 67(223) (1998), 987–999.
[14] L.M. LIHTARNIKOV, Use of Runge-Kutta method for solving Fredholm
type integro-differential equations, Zh. Vychisl. Mat. Fiz., 7 (1967), 899–
905.
[15] G. MICULA AND G. FAIRWEATHER, Direct numerical spline methods
for first order Fredholm integro-differential equations, Rev. Anal. Numer.
Theor. Approx., 22(1) (1993), 59–66.
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[16] A.I. PEROV AND A.V. KIBENKO, On a general method to study the
boundary value problems, Iz. Acad. Nauk., 30 (1966), 249–264.
[17] J. POUR MAHMOUD, M.Y. RAHIMI-ARDABILI AND S. SHAHMORAD, Numerical solution of the system of Fredholm integrodifferential equations by the tau method, Appl. Math. Comput., 168(1)
(2005), 465–478.
[18] I.A. RUS, Fiber Picard operators on generalized metric spaces and an application, Scripta Sci. Math., 1(2) (1999), 355–363.
[19] I.A. RUS, A fiber generalized contraction theorem and applications, Mathematica, 41(1) (1999), 85–90.
Application of a Trapezoid
Inequality to Neutral Fredholm
Integro-Differential Equations in
Banach Spaces
Alexandru Miahi Bica,
Vasile Aurel Căuş and
Sorin Mureşan
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