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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, 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.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 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. 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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. 12 R. Saadati, M. Dehghan, S. M. Vaezpour, and M. Saravi, “The convergence of He’s variational iteration method for solving integral equations,” Computers and Mathematics with Applications. In press. . 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. 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. 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
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