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. Rhoades2
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, rhoades@indiana.edu
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. Let xn1 fT, xn be some iteration
procedure. Suppose that FT , the fixed point set of T , is nonempty and that xn converges to
a point q ∈ FT . Let {yn } ⊆ X and define n yn1 − fT, yn . If lim n 0 implies that
lim yn q, then the iteration procedure xn1 fT, xn is said to be T -stable. Without loss of
generality, we may assume that {yn } is bounded, for if {yn } is not bounded, then it cannot
possibly converge. If these conditions hold for xn1 T xn , 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 space and T a self-map of X satisfying
T x − T y ≤ Lx − T x αx − y
1.1
for all x, y ∈ X, where L ≥ 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:
Lut Nut gt,
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
un1 x un x t
λs Lun s N u
n s − gs ds,
1.3
0
where λ is a Lagrange multiplier which can be identified optimally via variational theory, un
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
ux fx λ
b
1.4
Kx, tutdt,
a
where Kx, t is the kernel of the integral equation. There is a simple iteration formula for
1.4 in the form
b
un1 x fx λ
Kx, tun tdt.
1.5
a
Now, we show that the nonlinear mapping T , defined by
un1 x T un x fx λ
b
Kx, tun tdt,
1.6
a
is T -stable in L2 a, b.
First, we show that the nonlinear mapping T has a fixed point. For m, n ∈ N we have
T um x − T un x um1 x − un1 x
b
λ Kx, tum t − un tdt
a
≤ |λ|
b
a
1/2
K x, tdxdt
2
um x − un x.
1.7
Fixed Point Theory and Applications
3
Therefore, if
|λ| <
b
−1/2
K x, tdxdt
2
,
1.8
a
then, the nonlinear mapping T has a fixed point.
Second, we show that the nonlinear mapping T satisfies 1.1. Let 1.6 hold. Putting
b
1/2
L 0 and α |λ| a K 2 x, tdxdt 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
u0 x fx,
un1 x T un x fx λ
b
Kx, tun tdt,
1.9
a
for n 0, 1, 2, . . . , to construct a sequence of successive iterations {un x} to the solution of 1.4. In
addition, if
|λ| <
b
−1/2
K x, tdxdt
2
,
1.10
a
L 0 and α |λ|
T -stable.
bb
a a
1/2
K 2 x, tdxdt
. Then the nonlinear mapping T , in the norm of L2 a, b, is
Theorem 1.3 see 11. Use the iteration scheme
u0 x fx,
un1 x fx λ
b
Kx, tun tdt,
1.11
a
for n 0, 1, 2, . . . , to construct a sequence of successive iteration {un x} to the solution of 1.4. In
addition, let
b
K 2 x, tdxdt B2 < ∞,
1.12
a
and assume that fx ∈ L2 a, b. Then, if |λ| < 1/B, the above iteration converges, in the norm of
L2 a, b to the solution of 1.4.
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Fixed Point Theory and Applications
Corollary 1.4. Consider the iteration scheme
u0 x fx,
un1 x T un x fx λ
b
Kx, tun tdt,
1.13
a
for n 0, 1, 2, . . . . If
|λ| <
b
−1/2
K x, tdxdt
2
,
1.14
a
bb
1/2
L 0 and α |λ| a a K 2 x, tdxdt , then stability of the nonlinear mapping T in the norm of
L2 a, b is a coefficient condition for the above iteration to converge in the norm of L2 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 L2 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
√
ux x λ
1
xtutdt.
2.1
0
The iteration formula reads
un1 x √
xλ
1
xtun tdt,
2.2
0
u0 x √
x.
2.3
Substituting 2.3 into 2.2, we have the following results:
1
√
√
2λx
,
xt tdt x 5
0
1 √
√
√
2λt
2λ 2λ2
x,
u2 x x λ xt t dt x 5
5
15
0
u1 x u3 x √
√
xλ
1 √
√
2λ 2λ2
2λ 2λ2 2λ3
t dt x x.
x λ xt t 5
15
5
15
45
0
2.4
Fixed Point Theory and Applications
5
Continuing this way ad infinitum, we obtain
un x √
x
2
2 2
2 3
λ
λ
λ
·
·
·
x,
5.30
5.32
5.31
2.5
then
√
un x x
n
λi
2
5 i1 3i−1
2.6
x.
The above sequence is convergent if |λ| < 3, and the exact solution is
lim un x √
n→∞
6λ
x ux.
53 − λ
x
2.7
On the other hand we have
b
1/2
K x, tdxdt
2
1
a
1/2
2
xt dxdt
0
1
.
3
2.8
Then if |λ| < 3 for mapping
un1 x T un x √
xλ
1
xtun tdt,
2.9
0
we have
T um x − T un x um1 x − un1 x
1
λ xtum t − un tdt
0
≤ |λ|
1
1/2
2
xt dxdt
2.10
um x − un x
0
≤
|λ|
um x − un x,
3
which implies that T has a fixed point. Also, putting L 0 and α |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.
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Fixed Point Theory and Applications
Example 2.2 see 12. Consider the integral equation
1
ux x λ
2.11
1 − 3xtutdt,
0
its iteration formula reads
1
un1 x x λ
1 − 3xtun tdt,
2.12
0
u0 x x.
Then we have
un x x 1 1
n
λj
· · · 1 − 3xt1 1 − 3t1 t2 · · · 1 − 3tj−1 tj tj dtj · · · dt1 .
j1
0
0
2.13
By 2.13, we have the following results:
1
1
1 − 3xttdt 1 − λx λ,
2
0
1
1
u2 x x λ 1 − 3xt 1 − λt λ dt
2
0
u1 x x λ
1
λ2
1 − λx λ x,
2
4
1
λ2
1
u3 x x λ 1 − 3xt 1 − λt λ t dt
2
4
0
1 − λx 2.14
λ2
λ3
1
1 − λx λ .
4
2
8
Continuing this way ad infinitum, we obtain
n
3−1j − 1 λ j
un x x
2
2
j0
1 −1i1
2
j
λ
.
2
2.15
The above sequence is convergent if |λ/2| < 1, that is, −2 < λ < 2 and the exact solution
is
lim un x n→∞
41 − λ
2λ
x ux.
2
4−λ
4 − λ2
2.16
Fixed Point Theory and Applications
7
On the other hand we have
b
1/2
K x, tdxdt
2
1
a
Then if |λ| <
1/2
2
1 − 3xt dxdt
0
1
√ .
2
2.17
√
2, for mapping
un1 x T un x x λ
1
1 − 3xtun tdt,
2.18
0
we have
T um x − T un x um1 x − un1 x
1
λ xtum t − un tdt
0
1
≤ |λ|
1/2
2
1 − 3xt dxdt
2.19
um x − un x
0
|λ|
≤ √ um x − un x,
2
√
which implies that T has a fixed point. Also, putting L 0 and α |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
ux sin ax λ
a
2
π/2a
cosaxutdt,
2.20
0
its iteration formula reads
un1 x sin ax λ
a
2
π/2a
cosaxun tdt,
2.21
0
u0 x sin ax.
2.22
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Fixed Point Theory and Applications
Substituting 2.22 into 2.21, we have the following results:
u1 x sin ax λ
a
2
u2 x sinax λ
π/2a
cosax sinatdt sinax 0
a
2
π/2a
0
λ
cosax,
2
λ
cosax sinat cosat dt
2
λ λ2
sinax cosax
,
2
4
a
u3 x sinax λ
2
π/2a
0
2.23
λ λ2
cosat dt
cosax sinat 2
4
λ λ2 λ 3
sinax cosax
.
2
4
8
Continuing this way ad infinitum, we obtain
un x sinax cosax
∞ i
λ
i1
2
.
2.24
The above sequence is convergent if |λ/2| < 1; that is, −2 < λ < 2, and the exact solution is
lim un x sinax n→∞
λ
cosax ux.
2−λ
2.25
On the other hand we have
b
a
1/2
K 2 x, tdxdt
π/2a
0
1/2 2
π2
a
cosax dxdt
.
2
32
2.26
Then if |λ| < 1/ π 2 /32 ∼
1.800, for mapping
a
un1 x T un x x λ
2
π/2a
0
cosaxun tdt,
2.27
Fixed Point Theory and Applications
9
we have
T um x − T un x um1 x − un1 x
1
λ xtum t − un tdt
0
≤ |λ|
π/2a
0
≤ |λ|
a
2
2
1/2
cosax dxdt
um x − un x
2.28
π2
um x − un x,
32
which implies that T has a fixed point. Also, putting L 0 and α |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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