-Stability Approach to Variational Iteration Method for Solving Integral Equations

Volume 2009, Article ID 393245,9pages doi:10.1155/2009/393245

Review Article

T

-Stability Approach to Variational Iteration

Method for Solving Integral Equations

R. Saadati,

_{S. M. Vaezpour,}

_{and B. E. Rhoades}

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,[email protected]

Received 16 February 2009; Accepted 26 August 2009

Recommended by Nan-jing Huang

We considerT-stability definition according to Y. Qing and B. E. Rhoades2008and we show that the variational iteration method for solving integral equations isT-stable. Finally, we present some text examples to illustrate our result.

Copyrightq2009 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

LetX, · be a Banach space andT a self-map ofX. Letxn1 fT, xnbe some iteration procedure. Suppose thatFT, the fixed point set ofT, is nonempty and thatxnconverges to a pointq ∈FT. Let{yn} ⊆ X and definen yn1−fT, yn. If limn 0 implies that limyn q, then the iteration procedurexn1 fT, xnis said to beT-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 forxn1 Txn, that is, Picard’s iteration, then we will say that Picard’s iteration isT-stable.

Theorem 1.1see1. LetX, · be a Banach space andT a self-map ofXsatisfying

Tx−Ty≤Lx−Txαx−y 1.1

for allx, y ∈ X, whereL ≥ 0, 0 ≤ α < 1. Suppose thatT has a fixed pointp. Then, T is Picard

T-stable.

nonlinear system:

Lut Nut gt, 1.2

whereLis a linear operator,Nis a nonlinear operator, andgtis a given continuous function. The basic character of the method is to construct a functional for the system, which reads

un1x unx _t

0

λsLuns Nuns−gs

ds, 1.3

whereλis a Lagrange multiplier which can be identified optimally via variational theory,un is thenth approximate solution, andundenotes a restricted variation; that is,δun0.

Now, we consider the Fredholm integral equation of second kind in the general case, which reads

ux fx λ

b

a

Kx, tutdt, 1.4

whereKx, tis the kernel of the integral equation. There is a simple iteration formula for

1.4in the form

un1x fx λ

b

a

Kx, tuntdt. 1.5

Now, we show that the nonlinear mappingT, defined by

un1x Tunx fx λ _b

a

Kx, tuntdt, 1.6

isT-stable inL2_{a, b.}

First, we show that the nonlinear mappingThas a fixed point. Form, n∈Nwe have

Tumx−Tunxum1x−un1x

λ

_b

a

Kx, tumt−untdt

≤ |λ|

b

a

K2x, tdxdt

1/2

umx−unx.

Therefore, if

|λ|<

b

a

K2_{x, tdxdt}

−1/2

, 1.8

then, the nonlinear mappingT has a fixed point.

Second, we show that the nonlinear mappingT satisfies1.1. Let1.6hold. Putting

L0 andα|λ| b_aK2_{x, tdxdt}1/2_{shows that1.1holds for the nonlinear mappingT.}

All of the conditions ofTheorem 1.1hold for the nonlinear mappingTand hence it is

T-stable. As a result, we can state the following theorem.

Theorem 1.2. Use the iteration scheme

u0x fx,

un1x Tunx fx λ _b

a

Kx, tuntdt,

1.9

forn0,1,2, . . . ,to construct a sequence of successive iterations{unx}to the solution of1.4. In addition, if

|λ|<

b

a

K2x, tdxdt

−1/2

, 1.10

L0andα|λ| b_a b_aK2_{x, tdxdt}1/2_{. Then the nonlinear mappingT, in the norm ofL}2_{a, b, is}

T-stable.

Theorem 1.3see11. Use the iteration scheme

u0x fx,

un1x fx λ

b

a

Kx, tuntdt,

1.11

forn0,1,2, . . . ,to construct a sequence of successive iteration{unx}to the solution of 1.4. In addition, let

_b

a

K2x, tdxdtB2 <∞, 1.12

and assume thatfx∈ L2_{a, b. Then, if|λ|< 1/B, the above iteration converges, in the norm of}

Corollary 1.4. Consider the iteration scheme

u0x fx,

un1x Tunx fx λ b

a

Kx, tuntdt,

1.13

forn0,1,2, . . . .If

|λ|<

b

a

K2x, tdxdt

₋1/2

, 1.14

L 0and α |λ| b_{a a}bK2_{x, tdxdt}1/2_{, then stability of the nonlinear mappingT in the norm of}

L2_{a, bis a coefficient condition for the above iteration to converge in the norm ofL}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 ofL2_{a, bto the solution of1.4.}

In fact the stability interval is a subset of converges interval.

Example 2.1see12. Consider the integral equation

ux √xλ

1

0

xtutdt. 2.1

The iteration formula reads

un1x

√

xλ

₁

0

xtuntdt, 2.2

u0x √x. 2.3

Substituting2.3into2.2, we have the following results:

u1x

√

xλ

₁

0

xt√tdt√x2λx

5 ,

u2x

√

xλ

1

0

xt

_√

t2λt

5

dt√x

2λ

5 2λ2

15

x,

u3x

√

xλ

1

0

xt

√

t

2λ

5 2λ2

15

t

dt√x

2λ

5 2λ2

15

2λ3

45

x.

Continuing this way ad infinitum, we obtain

unx √

x

2 5.30λ

2 5.31λ

2 2

5.32λ 3_{· · ·}

x, 2.5

then

unx √

x

2 5

n

i1

λi

3i−1

x. 2.6

The above sequence is convergent if|λ|<3, and the exact solution is

lim n→ ∞unx

√

x 6λ

53−λxux. 2.7

On the other hand we have

b

a

K2x, tdxdt

1/2

1

0

xt2dxdt

1/2 1

3. 2.8

Then if|λ|<3 for mapping

un1x Tunx √

xλ

1

0

xtuntdt, 2.9

we have

Tumx−Tunxum1x−un1x

λ

₁

0

xtumt−untdt

≤ |λ|

1

0

xt2dxdt

1/2

umx−unx

≤ |λ|

3 umx−unx,

2.10

which implies thatT has a fixed point. Also, puttingL 0 andα |λ|/3 shows that1.1

Example 2.2see12. Consider the integral equation

ux xλ

₁

0

1−3xtutdt, 2.11

its iteration formula reads

un1x xλ

₁

0

1−3xtuntdt,

u0x x.

2.12

Then we have

unx x n

j1

λj ₁

0

· · · ₁

0

1−3xt11−3t1t2· · ·

1−3tj−1tj

tjdtj· · ·dt1. 2.13

By2.13, we have the following results:

u1x xλ

1

0

1−3xttdt 1−λx1

2λ,

u2x xλ

₁

0

1−3xt

1−λt1

2λ

dt

1−λx1

2λ

λ2

4 x,

u3x xλ

₁

0

1−3xt

1−λt1

2λ

λ2

4t

dt

1−λxλ

2

4 1−λx 1 2λ

λ3

8 .

2.14

Continuing this way ad infinitum, we obtain

unx n

j0

3−1j−1 2

λ

2 j

x

1 −1i1 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→ ∞unx

2λ

4−λ2

41−λ

On the other hand we have

b

a

K2x, tdxdt

1/2

1

0

1−3xt2dxdt

1/2 √1

2. 2.17

Then if|λ|<√2, for mapping

un1x Tunx xλ ₁

0

1−3xtuntdt, 2.18

we have

Tumx−Tunxum1x−un1x

λ

1

0

xtumt−untdt

≤ |λ|

1

0

1−3xt2dxdt

1/2

umx−unx

≤ √|λ|

2umx−unx,

2.19

which implies thatT has a fixed point. Also, puttingL 0 andα|λ|/√2 shows that1.1

holds for the nonlinear mappingT. All of conditions ofTheorem 1.1hold for the nonlinear mappingTand hence it isT-stable.

Example 2.3. Consider the integral equation

ux sinaxλa

2 π/2a

0

cosaxutdt, 2.20

its iteration formula reads

un1x sinaxλ

a

2 _π/2a

0

cosaxuntdt, 2.21

Substituting2.22into2.21, we have the following results:

u1x sinaxλ

a

2 _π/2a

0

cosaxsinatdtsinax λ

2cosax,

u2x sinax λa

2 π/2a

0

cosax

sinat λ

2cosat

dt

sinax cosax

λ

2

λ2

4

,

u3x sinax λ

a

2 _π/2a

0

cosax

sinat

λ

2

λ2

4

cosat

dt

sinax cosax

λ

2

λ2

4

λ3

8

.

2.23

Continuing this way ad infinitum, we obtain

unx sinax cosax ∞

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→ ∞unx sinax

λ

2−λcosax ux. 2.25

On the other hand we have

b

a

K2x, tdxdt

1/2

π/2a

0

_a

2cosax 2

dxdt

1/2

π2

32. 2.26

Then if|λ|<1/π2_/32∼_{1.800, for mapping}

un1x Tunx xλ

a

2 _π/2a

0

we have

Tumx−Tunxum1x−un1x

λ

1

0

xtumt−untdt

≤ |λ|

π/2a

0

_a

2cosax 2

dxdt

1/2

umx−unx

≤ |λ|

π2

32umx−unx,

2.28

which implies thatT has a fixed point. Also, puttingL 0 andα |λ|π2_{/32 shows that} 1.1holds for the nonlinear mappingT. All of the conditions of Theorem 1.1hold for the nonlinear mappingTand hence it isT-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

References

1 Y. Qing and B. E. Rhoades, “T-stability of Picard iteration in metric spaces,”Fixed Point Theory and Applications, vol. 2008, Article ID 418971, 4 pages, 2008.

2 J. Biazar and H. Ghazvini, “He’s variational iteration method for solving hyperbolic differential equations,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 311– 314, 2007.

3 J. H. He, “Variational iteration method—a kind of nonlinear analytical technique: some examples,”

International Journal of Non-Linear Mechanics, vol. 34, pp. 699–708, 1999.

4 J.-H. He, “A review on some new recently developed nonlinear analytical techniques,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000.

5 J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,”Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007.

6 J.-H. He, “Variational iteration method—some recent results and new interpretations,”Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.

7 Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006.

8 H. Ozer, “Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics,”International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 4, pp. 513–518, 2007.

9 A. M. Wazwaz and S. A. Khuri, “Two methods for solving integral equations,”Applied Mathematics and Computation, vol. 77, no. 1, pp. 79–89, 1996.

10 A. M. Wazwaz, “A reliable treatment for mixed Volterra-Fredholm integral equations,” Applied Mathematics and Computation, vol. 127, no. 2-3, pp. 405–414, 2002.

11 C.-E. Fr ¨oberg,Introduction to Numerical Analysis, Addison-Wesley, Reading, Mass, USA, 1969.