T -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

1Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran

2Department 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. Letx_n1 fT, xnbe some iteration procedure. Suppose thatFT, the fixed point set ofT, is nonempty and thatxnconverges to a pointq ∈FT. Let{yn} ⊆ X and define_n y_n1−fT, yn. If lim_n 0 implies that limyn q, then the iteration procedurex_n1 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 forx_n1 Tx_n, 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 that T has a fixed pointp. 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

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

u_n1x u_nx _t

0

λs

Lu_ns Nu_ns−gs

ds, 1.3

whereλis a Lagrange multiplier which can be identified optimally via variational theory,u_n 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

u_n1x fx λ _b

a

Kx, tuntdt. 1.5

Now, we show that the nonlinear mappingT, defined by

u_n1x Tunx fx λ _b

a

Kx, tuntdt, 1.6

isT-stable inL²a, b.

First, we show that the nonlinear mappingThas a fixed point. Form, n∈Nwe have Tumx−Tunxu_m1x−u_n1x

λ

_b

a

Kx, tumt−u_ntdt

≤ |λ|

b a

K²x, tdxdt 1/2

umx−unx.

1.7

Therefore, if

|λ|<

_b

a

K²x, 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

aK²x, tdxdt^1/2shows 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, u_n1x 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

L0 andα|λ| ^b_a ^b_aK²x, tdxdt^1/2. Then the nonlinear mappingT, in the norm ofL²a, b, is T-stable.

Theorem 1.3see11. Use the iteration scheme

u0x fx, u_n1x 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∈ L²a, b. Then, if|λ|< 1/B, the above iteration converges, in the norm of L²a, bto the solution of1.4.

Corollary 1.4. Consider the iteration scheme

u₀x fx, u_n1x Tunx fx λ

_b

a

Kx, tuntdt, 1.13

forn0,1,2, . . . .If

|λ|<

_b

a

K²x, tdxdt _−1/2

, 1.14

L 0 andα |λ| ^b_a ^b_aK²x, tdxdt^1/2, then stability of the nonlinear mappingT in the norm of L²a, bis a coefficient condition for the above iteration to converge in the norm ofL²a, 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 ofL²a, 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

u_n1x √ xλ

₁

0

xtu_ntdt, 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.

2.4

Continuing this way ad infinitum, we obtain

u_nx √ x

2

5.3⁰λ 2

5.3¹λ² 2

5.3²λ³· · ·

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

nlim→ ∞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

u_n1x Tunx √ xλ

₁

0

xtuntdt, 2.9

we have

Tumx−Tunxum1x−u_n1x

λ ₁

0

xtumt−u_ntdt

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

Example 2.2see12. Consider the integral equation

ux xλ ₁

0

1−3xtutdt, 2.11

its iteration formula reads

u_n1x xλ ₁

0

1−3xtuntdt,

u0x x.

2.12

Then we have

u_nx xⁿ

j1

λ^j ₁

0

· · · ₁

0

1−3xt₁1−3t₁t₂· · ·

1−3t_j−1t_j

t_jdt_j· · ·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 2λλ²

4t

dt

1−λxλ²

4 1−λx 1 2λλ³

8 .

2.14

Continuing this way ad infinitum, we obtain

unx ⁿ

j0

3−1^j−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

nlim→ ∞u_nx 2λ

4−λ² 41−λ

4−λ² xux. 2.16

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

u_n1x Tunx xλ ₁

0

1−3xtuntdt, 2.18

we have

Tumx−Tunxum1x−u_n1x

λ ₁

0

xtumt−untdt

≤ |λ|

₁

0

1−3xt²dxdt _1/2

umx−u_nx

≤ √|λ|

2umx−u_nx,

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

u_n1x sinaxλa 2

_π/2a

0

cosaxuntdt, 2.21

u0x sinax. 2.22

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

u₁x sinaxλa 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 λa 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

u_nx 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

nlim→ ∞unx sinax λ

2−λcosax ux. 2.25

On the other hand we have _b

a

K²x, tdxdt 1/2

_π/2a

0

a

2cosax2

dxdt 1/2

π²

32. 2.26

Then if|λ|<1/

π²/32∼1.800, for mapping

u_n1x Tunx xλa 2

_π/2a

0

cosaxuntdt, 2.27

we have

Tumx−Tunxum1x−u_n1x

λ ₁

0

xtumt−untdt

≤ |λ|

_π/2a

0

a

2cosax₂ dxdt

_1/2

umx−u_nx

≤ |λ|

π²

32umx−u_nx,

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

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