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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 393245, 9 pages
doi:10.1155/2009/393245
Review Article
T-Stability Approach to Variational Iteration
Method for Solving Integral Equations
R. Saadati,
1
S. M. Vaezpour,
1
and B. E. Rhoades
2
1
Department of Mathematics and Computer Science, Amirkabir University of Technology,
424 Hafez Avenue, Tehran 15914, Iran
2
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Correspondence should be addressed to B. E. Rhoades,
Received 16 February 2009; Accepted 26 August 2009
Recommended by Nan-jing Huang
We consider T-stability definition according to Y. Qing and B. E. Rhoades 2008 and we show that
the variational iteration method for solving integral equations is T-stable. Finally, we present some
text examples to illustrate our result.
Copyright q 2009 R. Saadati et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and Preliminaries
Let X, · be a Banach space and T a self-map of X.Letx
n1
fT, x
n
be some iteration
procedure. Suppose that FT, the fixed point set of T, is nonempty and that x
n
converges to
apointq ∈ FT.Let{y
n
}⊆X and define
n
y
n1
− fT, y
n
. If lim
n
0 implies that
lim y
n
q, then the iteration procedure x
n1
fT, x
n
is said to be T-stable. Without loss of
generality, we may assume that {y
n
} is bounded, for if {y
n
} is not bounded, then it cannot
possibly converge. If these conditions hold for x
n1
Tx
n
, that is, Picard’s iteration, then we
will say that Picard’s iteration is T-stable.
Theorem 1.1 see 1. Let X, · be a Banach s pace and T a self-map of X satisfying
Tx − Ty
≤ L
x − Tx
α
x − y
1.1
for all x,y ∈ X,whereL ≥ 0, 0 ≤ α<1. Suppose that T has a fixed point p. Then, T is Picard
T-stable.
Various kinds of analytical methods and numerical methods 2–10 were used to
solve integral equations. To illustrate the basic idea of the method, we consider the general
2 Fixed Point Theory and Applications
nonlinear system:
L
u
t
N
u
t
g
t
, 1.2
where L is a linear operator, N is a nonlinear operator, and gt is a given continuous function.
The basic character of the method is to construct a functional for the system, which reads
u
n1
x
u
n
x
t
0
λ
s
Lu
n
s
N u
n
s
− g
s
ds,
1.3
where λ is a Lagrange multiplier which can be identified optimally via variational theory, u
n
is the nth approximate solution, and u
n
denotes a restricted variation; that is, δu
n
0.
Now, we consider the Fredholm integral equation of second kind in the general case,
which reads
u
x
f
x
λ
b
a
K
x, t
u
t
dt,
1.4
where Kx, t is the kernel of the integral equation. There is a simple iteration formula for
1.4 in the form
u
n1
x
f
x
λ
b
a
K
x, t
u
n
t
dt.
1.5
Now, we show that the nonlinear mapping T, defined by
u
n1
x
T
u
n
x
f
x
λ
b
a
K
x, t
u
n
t
dt,
1.6
is T-stable in L
2
a, b.
First, we show that the nonlinear mapping T has a fixed point. For m, n ∈ N we have
T
u
m
x
− T
u
n
x
u
m1
x
− u
n1
x
λ
b
a
K
x, t
u
m
t
− u
n
t
dt
≤
|
λ
|
b
a
K
2
x, tdxdt
1/2
u
m
x
− u
n
x
.
1.7
Fixed Point Theory and Applications 3
Therefore, if
|
λ
|
<
b
a
K
2
x, tdxdt
−1/2
,
1.8
then, the nonlinear mapping T has a fixed point.
Second, we show that the nonlinear mapping T satisfies 1.1.Let1.6 hold. Putting
L 0andα |λ|
b
a
K
2
x, tdxdt
1/2
shows that 1.1 holds for the nonlinear mapping T.
All of the conditions of Theorem 1.1 hold for the nonlinear mapping T and hence it is
T-stable. As a result, we can state the following theorem.
Theorem 1.2. Use the iteration scheme
u
0
x
f
x
,
u
n1
x
T
u
n
x
f
x
λ
b
a
K
x, t
u
n
t
dt,
1.9
for n 0, 1, 2, ,to construct a sequence of successive iterations {u
n
x} to the solution of 1.4.In
addition, if
|
λ
|
<
b
a
K
2
x, tdxdt
−1/2
,
1.10
L 0 and α |λ|
b
a
b
a
K
2
x, tdxdt
1/2
. Then the nonlinear mapping T, in the norm of L
2
a, b,is
T-stable.
Theorem 1.3 see 11. Use the iteration scheme
u
0
x
f
x
,
u
n1
x
f
x
λ
b
a
K
x, t
u
n
t
dt,
1.11
for n 0, 1, 2, ,to construct a sequence of successive iteration {u
n
x} to the solution of 1.4.In
addition, let
b
a
K
2
x, t
dxdt B
2
< ∞,
1.12
and assume that fx ∈ L
2
a, b. Then, if |λ| < 1/B, the above iteration converges, in the norm of
L
2
a, b to the solution of 1.4.
4 Fixed Point Theory and Applications
Corollary 1.4. Consider the iteration scheme
u
0
x
f
x
,
u
n1
x
T
u
n
x
f
x
λ
b
a
K
x, t
u
n
t
dt,
1.13
for n 0, 1, 2, If
|
λ
|
<
b
a
K
2
x, tdxdt
−1/2
,
1.14
L 0 and α |λ|
b
a
b
a
K
2
x, tdxdt
1/2
, then stability of the nonlinear mapping T in the norm of
L
2
a, b is a coefficient condition for the above iteration to converge in the norm of L
2
a, b, and to the
solution of 1.4.
2. Test Examples
In this section we present some test examples to show that the stability of the iteration method
is a coefficient condition for the convergence in the norm of L
2
a, b to the solution of 1.4.
In fact the stability interval is a subset of converges interval.
Example 2.1 see 12. Consider the integral equation
u
x
√
x λ
1
0
xtu
t
dt.
2.1
The iteration formula reads
u
n1
x
√
x λ
1
0
xtu
n
t
dt,
2.2
u
0
x
√
x. 2.3
Substituting 2.3 into 2.2, we have the following results:
u
1
x
√
x λ
1
0
xt
√
tdt
√
x
2λx
5
,
u
2
x
√
x λ
1
0
xt
√
t
2λt
5
dt
√
x
2λ
5
2λ
2
15
x,
u
3
x
√
x λ
1
0
xt
√
t
2λ
5
2λ
2
15
t
dt
√
x
2λ
5
2λ
2
15
2λ
3
45
x.
2.4
Fixed Point Theory and Applications 5
Continuing this way ad infinitum, we obtain
u
n
x
√
x
2
5.3
0
λ
2
5.3
1
λ
2
2
5.3
2
λ
3
···
x, 2.5
then
u
n
x
√
x
2
5
n
i1
λ
i
3
i−1
x.
2.6
The above sequence is convergent if |λ| < 3, and the exact solution is
lim
n →∞
u
n
x
√
x
6λ
5
3 − λ
x u
x
.
2.7
On the other hand we have
b
a
K
2
x, tdxdt
1/2
1
0
xt
2
dxdt
1/2
1
3
.
2.8
Then if |λ| < 3 for mapping
u
n1
x
T
u
n
x
√
x λ
1
0
xtu
n
t
dt,
2.9
we have
T
u
m
x
− T
u
n
x
u
m1
x
− u
n1
x
λ
1
0
xt
u
m
t
− u
n
t
dt
≤
|
λ
|
1
0
xt
2
dxdt
1/2
u
m
x
− u
n
x
≤
|
λ
|
3
u
m
x
− u
n
x
,
2.10
which implies that T has a fixed point. Also, putting L 0andα |λ|/3 shows that 1.1
holds for the nonlinear mapping T. All of the conditions of Theorem 1.1 hold for the nonlinear
mapping T and hence it is T-stable.
6 Fixed Point Theory and Applications
Example 2.2 see 12. Consider the integral equation
u
x
x λ
1
0
1 − 3xt
u
t
dt,
2.11
its iteration formula reads
u
n1
x
x λ
1
0
1 − 3xt
u
n
t
dt,
u
0
x
x.
2.12
Then we have
u
n
x
x
n
j1
λ
j
1
0
···
1
0
1 − 3xt
1
1 − 3t
1
t
2
···
1 − 3t
j−1
t
j
t
j
dt
j
···dt
1
.
2.13
By 2.13, we have the following results:
u
1
x
x λ
1
0
1 − 3xt
tdt
1 − λ
x
1
2
λ,
u
2
x
x λ
1
0
1 − 3xt
1 − λ
t
1
2
λ
dt
1 − λ
x
1
2
λ
λ
2
4
x,
u
3
x
x λ
1
0
1 − 3xt
1 − λ
t
1
2
λ
λ
2
4
t
dt
1 − λ
x
λ
2
4
1 − λ
x
1
2
λ
λ
3
8
.
2.14
Continuing this way ad infinitum, we obtain
u
n
x
n
j0
3
−1
j
− 1
2
λ
2
j
x
1
−1
i1
2
λ
2
j
.
2.15
The above sequence is convergent if |λ/2| < 1, that is, −2 <λ<2 and the exact solution
is
lim
n →∞
u
n
x
2λ
4 − λ
2
4
1 − λ
4 − λ
2
x u
x
.
2.16
Fixed Point Theory and Applications 7
On the other hand we have
b
a
K
2
x, tdxdt
1/2
1
0
1 − 3xt
2
dxdt
1/2
1
√
2
.
2.17
Then if |λ| <
√
2, for mapping
u
n1
x
T
u
n
x
x λ
1
0
1 − 3xt
u
n
t
dt,
2.18
we have
T
u
m
x
− T
u
n
x
u
m1
x
− u
n1
x
λ
1
0
xt
u
m
t
− u
n
t
dt
≤
|
λ
|
1
0
1 − 3xt
2
dxdt
1/2
u
m
x
− u
n
x
≤
|
λ
|
√
2
u
m
x
− u
n
x
,
2.19
which implies that T has a fixed point. Also, putting L 0andα |λ|/
√
2 shows that 1.1
holds for the nonlinear mapping T. All of conditions of Theorem 1.1 hold for the nonlinear
mapping T and hence it is T-stable.
Example 2.3. Consider the integral equation
u
x
sin ax λ
a
2
π/2a
0
cos
ax
u
t
dt,
2.20
its iteration formula reads
u
n1
x
sin ax λ
a
2
π/2a
0
cos
ax
u
n
t
dt,
2.21
u
0
x
sin ax. 2.22
8 Fixed Point Theory and Applications
Substituting 2.22 into 2.21, we have the following results:
u
1
x
sin ax λ
a
2
π/2a
0
cos
ax
sin
at
dt sin
ax
λ
2
cos
ax
,
u
2
x
sin
ax
λ
a
2
π/2a
0
cos
ax
sin
at
λ
2
cos
at
dt
sin
ax
cos
ax
λ
2
λ
2
4
,
u
3
x
sin
ax
λ
a
2
π/2a
0
cos
ax
sin
at
λ
2
λ
2
4
cos
at
dt
sin
ax
cos
ax
λ
2
λ
2
4
λ
3
8
.
2.23
Continuing this way ad infinitum, we obtain
u
n
x
sin
ax
cos
ax
∞
i1
λ
2
i
.
2.24
The above sequence is convergent if |λ/2| < 1; that is, −2 <λ<2, and the exact solution is
lim
n →∞
u
n
x
sin
ax
λ
2 − λ
cos
ax
u
x
.
2.25
On the other hand we have
b
a
K
2
x, tdxdt
1/2
π/2a
0
a
2
cosax
2
dxdt
1/2
π
2
32
.
2.26
Then if |λ| < 1/
π
2
/32
∼
1.800, for mapping
u
n1
x
T
u
n
x
x λ
a
2
π/2a
0
cos
ax
u
n
t
dt,
2.27
Fixed Point Theory and Applications 9
we have
T
u
m
x
− T
u
n
x
u
m1
x
− u
n1
x
λ
1
0
xt
u
m
t
− u
n
t
dt
≤
|
λ
|
π/2a
0
a
2
cosax
2
dxdt
1/2
u
m
x
− u
n
x
≤
|
λ
|
π
2
32
u
m
x
− u
n
x
,
2.28
which implies that T has a fixed point. Also, putting L 0andα |λ|
π
2
/32 shows that
1.1 holds for the nonlinear mapping T. All of the conditions of Theorem 1.1 hold for the
nonlinear mapping T and hence it is T-stable.
Acknowledgments
The authors would like to thank referees and area editor Professor Nan-jing Huang for giving
useful comments and suggestions for the improvement of this paper. This paper is dedicated
to Professor Mehdi Dehghan
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