Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 173, 2006
APPLICATION OF A TRAPEZOID INEQUALITY TO NEUTRAL FREDHOLM
INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES
ALEXANDRU MIAHI BICA, VASILE AUREL CĂUŞ, AND SORIN MUREŞAN
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF O RADEA ,
S TR . U NIVERSITATII NO .1
410087, O RADEA , ROMANIA
abica@uoradea.ro
vcaus@uoradea.ro
Received 27 January, 2006; accepted 19 November, 2006
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Nonlinear neutral Fredholm integro-differential equations in Banach spaces, Perov’s fixed point theorem, Method of successive approximations, Trapezoidal quadrature rule.
2000 Mathematics Subject Classification. Primary 45J05, Secondary 45N05, 45B05, 65L05.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
028-06
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A LEXANDRU M IAHI B ICA , VASILE AUREL C ĂU Ş , AND S ORIN M URE ŞAN
(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)
(1.2)
, t ∈ [a, b] .
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 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.
2. E XISTENCE , U NIQUENESS AND A PPROXIMATION
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)
kf (t, s, u1 , v1 ) − f (t, s, u2 , v2 )kX ≤ α1 ku1 − u2 kX + β1 kv1 − v2 kX ,
(2.2)
∂f
∂f
≤ α2 ku1 − u2 k + β2 kv1 − v2 k ,
(t,
s,
u
,
v
)
−
(t,
s,
u
,
v
)
1
1
2
2
X
X
∂t
∂t
X
(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
(2.5)
kf (t, s, u, v) − f (t0 , s, u, v)kX ≤ δ1 |t − t0 | ,
(2.6)
∂f
∂f 0
≤ δ2 |t − t0 | ,
(t,
s,
u,
v)
−
(t
,
s,
u,
v)
∂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]):
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
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N EUTRAL F REDHOLM I NTEGRO -D IFFERENTIAL E QUATIONS
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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)
d (xm , x∗ ) ≤ Qm (In − Q)−1 · 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 → Rn on a
nonempty set Y with the properties:
a) d (x, y) ≥ 0, for any x, y ∈ Y ;
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 ,
in
R, for each i = 1, n,
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 b
A1 (x, y) (t) =
f (t, s, x (s) , y (s)) ds + g (t)
a
and
Z
(2.9)
A2 (x, y) (t) =
a
b
∂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) ,
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
y ∗ = (x∗ )0
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A LEXANDRU M IAHI B ICA , VASILE AUREL C ĂU Ş , AND S ORIN M URE ŞAN
and x∗ is the unique solution of the equation (1.1). Moreover, the sequence of the successive
approximations given by,
(x0 (t) , y0 (t)) = (g (t) , g 0 (t)) ,
(2.11)
b
Z
xm+1 (t) =
(2.12)
t ∈ [a, b]
f (t, s, xm (s) , ym (s)) ds + g (t) ,
t ∈ [a, b]
a
and
b
Z
ym+1 (t) =
(2.13)
a
∂f
(t, s, xm (s) , ym (s)) ds + g 0 (t) ,
∂t
t ∈ [a, b] ,
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 )) ,
where
Q = (b − a)
α1 β1
α2 β2
for any m ∈ N∗ ,
.
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 ]
a
≤ (b − a) · [α1 ku1 − u2 kC + β1 kv1 − v2 kC ] ,
for any t ∈ [a, b]
and
kA2 (u1 , v1 ) (t) − A2 (u2 , v2 ) (t)kX
≤ (b − a) · [α2 ku1 − u2 kC + β2 kv1 − v2 kC ] ,
for any t ∈ [a, b] .
We infer that
d (A (u1 , v1 ) , A (u2 , v2 )) ≤ Q · d ((u1 , v1 ) , (u2 , v2 )) ,
for any (u1 , v1 ) , (u2 , v2 ) ∈ Y,
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 (t) =
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
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N EUTRAL F REDHOLM I NTEGRO -D IFFERENTIAL E QUATIONS
5
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
∀m ∈ N.
(xm+1 , ym+1 ) = A ((xm , ym )) ,
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.
3. T HE M AIN R ESULT
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
n−1
X
i (b − a)
(b − a)
(3.1)
F (x) dx =
F (a) + 2
F a+
+ F (b) + Rn (F )
2n
n
a
i=1
with
L (b − a)2
,
(3.2)
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] ,
∆ : a = t0 < t1 < · · · < tn−1 < tn = b
(3.3)
with ti = a + i(b−a)
, i = 0, n and compute xm (ti ) , ym (ti ) , i = 0, 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 )
m ∈ N∗ .
a
and
b
Z
(3.5)
ym+1 (ti ) =
a
∂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
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)
and
{{Gm,i }m∈N , i = 0, n} ⊂ C ([a, b] , X)
defined in (3.6), are equally Lipschitz, if the conditions (i) and (2.1) – (2.7) are true.
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
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6
A LEXANDRU M IAHI B ICA , VASILE AUREL C ĂU Ş , AND S ORIN M URE ŞAN
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 | ,
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 | ,
for each s1 , s2 ∈ [a, b]
kG0,i (s1 ) − G0,i (s2 )kX ≤ (γ2 + α2 µ + β2 η) · |s1 − s2 | ,
for each s1 , s2 ∈ [a, b] .
and
Let
L1 = γ1 + α1 µ + β1 η + (b − a) (α1 δ1 + β1 δ2 ) ,
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)
for i = 0, n
"
(3.8) xm (ti ) = g (ti ) +
+2
n−1
X
(b − a)
· f (ti , a, xm−1 (a) , ym−1 (a))
2n
#
f (ti ; tj , xm−1 (tj ) , ym−1 (tj )) + f (ti , b, xm−1 (b) , ym−1 (b))
j=1
+ Rm,i ,
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
for i = 0, n and m ∈ N∗
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N EUTRAL F REDHOLM I NTEGRO -D IFFERENTIAL E QUATIONS
7
and
"
(3.9) ym (ti ) = g 0 (ti ) +
+2
(b − a)
∂f
·
(ti , a, xm−1 (a) , ym−1 (a))
2n
∂t
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)
kRm,i kX ≤
L1 (b − a)2
,
4n
for any m ∈ N∗ and i = 0, n,
(3.11)
L2 (b − a)2
,
kωm,i kX ≤
4n
or any m ∈ N∗ and i = 0, n.
These lead to the following algorithm:
x0 (ti ) = g (ti ) , y0 (ti ) = g 0 (ti ) ,
(3.12)
"
(b − a)
x1 (ti ) = g (ti ) +
· f (ti , a, g (a) , g 0 (a))
2n
+2
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 ,
(3.13)
"
(b
−
a)
∂f
y1 (ti ) = g 0 (ti ) +
·
(ti , a, g (a) , g 0 (a))
2n
∂t
+2
#
∂f
(ti ; tj , g (tj ) , g 0 (tj )) +
(ti , b, g (b) , g 0 (b)) + ω1,i
∂t
∂t
n−1
X
∂f
j=1
= y1 (ti ) + ω1,i ,
(3.14)
"
(b − a)
x2 (ti ) = g (ti ) +
· f (ti , t0 , x1 (t0 ) + R1,0 , y1 (t0 ) + ω1,0 )
2n
+2
n−1
X
f (ti ; tj , x1 (tj ) + R1,j , y1 (tj ) + ω1,j ) + f (ti , tn , x1 (tn )
j=1
#
+ R1,n , y1 (tn ) + ω1,n ) + R2,i
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A LEXANDRU M IAHI B ICA , VASILE AUREL C ĂU Ş , AND S ORIN M URE ŞAN
"
(b − a)
= g (ti ) +
· f (ti , t0 , x1 (t0 ) , y1 (t0 ))
2n
+2
n−1
X
#
f (ti ; tj , x1 (tj ) , y1 (tj )) + f (ti , tn , x1 (tn ) , y1 (tn )) + R2,i
j=1
= x2 (ti ) + R2,i ,
(3.15)
"
(b
−
a)
∂f
y2 (ti ) = g 0 (ti ) +
·
(ti , t0 , x1 (t0 ) + R1,0 , y1 (t0 ) + ω1,0 )
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 )
∂t
#
+ R1,n , y1 (tn ) + ω1,n ) + ω2,i
"
(b − a)
∂f
= g (ti ) +
·
(ti , t0 , x1 (t0 ) , y1 (t0 ))
2n
∂t
0
+2
#
∂f
(ti ; tj , x1 (tj ) , y1 (tj )) +
(ti , tn , x1 (tn ) , y1 (tn )) + ω2,i
∂t
∂t
n−1
X
∂f
j=1
= y2 (ti ) + ω2,i ,
when i = 0, n.
By induction, for m ≥ 3, we obtain for i = 0, n that
(3.16) xm (ti )
"
(b − a)
· f ti , t0 , xm−1 (t0 ) + Rm−1,0 , ym−1 (t0 ) + ω m−1,0
= g (ti ) +
2n
+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
"
= g (ti ) +
+2
(b − a)
· f (ti , t0 , xm−1 (t0 ) , ym−1 (t0 ))
2n
n−1
X
#
f (ti ; tj , xm−1 (tj ) , ym−1 (tj )) + f (ti , tn , xm−1 (tn ) , ym−1 (tn )) + Rm,i
j=1
= xm (ti ) + Rm,i
and
(3.17)
"
(b
−
a)
∂f
ym (ti ) = g 0 (ti ) +
·
ti , t0 , xm−1 (t0 ) + Rm−1,0 , ym−1 (t0 ) + ω m−1,0
2n
∂t
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N EUTRAL F REDHOLM I NTEGRO -D IFFERENTIAL E QUATIONS
+2
n−1
X
∂f
j=1
9
ti ; tj , xm−1 (tj ) + Rm−1,j , ym−1 (tj ) + ω m−1,j
∂t
#
∂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
= ym (ti ) + ωm,i .
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
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.
Using in (3.15) the Lipschitz property (2.2) we obtain:
kω 2,i k ≤ [1 + (b − a) (α2 + β2 )] ·
L2 (b − a)2
,
4n
for each i = 0, n.
By induction, we obtain for m ≥ 2 and i = 0, n,
(3.18)
2
Rm,i ≤ 1 + (b − a) (α1 + β1 ) + · · · + (b − a)m−1 (α1 + β1 )m−1 · L1 (b − a)
4n
m
m
2
1 − (b − a) (α1 + β1 )
L1 (b − a)
=
·
1 − (b − a) (α1 + β1 )
4n
and
(3.19)
2
m−1
m−1 L2 (b − a)
kω m,i k ≤ 1 + (b − a) (α2 + β2 ) + · · · + (b − a)
(α2 + β2 )
·
4n
1 − (b − a)m (α2 + β2 )m L2 (b − a)2
=
·
.
1 − (b − a) (α2 + β2 )
4n
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 ,
J. Inequal. Pure and Appl. Math., 7(5) Art. 173, 2006
i = 0, n
http://jipam.vu.edu.au/
10
A LEXANDRU M IAHI B ICA , VASILE AUREL C ĂU Ş ,
AND
S ORIN M URE ŞAN
obtained in (3.12) – (3.17) and the following error estimation holds:
∗
kx (ti ) − xm (ti )kX
≤ Qm (I2 − Q)−1 d (x0 , x1 )
(3.20)
ky ∗ (ti ) − ym (ti )kX
!
L (b−a)2
1
+
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,
ky ∗ (ti ) − ym (ti )kX ≤ ky ∗ (ti ) − ym (ti )kX + kym (ti ) − ym (ti )kX
for each m ∈ N∗ and ∀i = 0, n.
Remark 3.3. 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 integrodifferential 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 3.4. For X = Rn we obtain a new method in analysing systems of Fredholm integrodifferential equations and for X = R we obtain similar results for the approximation of the
solution of a scalar Fredholm integro-differential equation.
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