Comparison of the Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions in Banach Spaces

Transcription

1 Thai Journal of Mathematics Volume 11 (2013) Number 11 : ISSN Comparison of the Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions in Banach Spaces Withun Phuengrattana and Suthep Suantai,,1 Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand Materials Science Research Center, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand withun ph@yahoocom (W Phuengrattana) sutheps@cmuacth (S Suantai) Abstract : In this article, we propose a new iterative method for finding fixed points of a class of weak contractions in arbitrary Banach spaces Comparison of the rate of convergence between Mann, Ishikawa, Noor iterations, and the proposed iterative method are also discussed It is shown that the proposed iterative method converges faster than the others Moreover, we give a numerical example for comparing the rate of convergence of those methods Keywords : fixed point; rate of convergence; weak contractions; Banach spaces 2010 Mathematics Subject Classification : 47H09; 47H10 1 Introduction Let C be a nonempty convex subset of a Banach space X, and T : C C be a mapping A point x C is a fixed point of T if Tx = x The set of all fixed points of T is denoted by F(T) The Mann iteration is defined by u 1 C and 1 Corresponding author u n+1 = (1 α n )u n + α n Tu n, for all n N, (11) Copyright c 2013 by the Mathematical Association of Thailand All rights reserved

2 218 Thai J Math 11 (2013)/ W Phuengrattana and S Suantai where {α n } is a sequence in [0, 1] For α n = λ (constant), the iteration (11) reduces to the so-called Krasnoselskij iteration The Ishikawa iteration is defined by s 1 C and { rn = (1 β n )s n + β n Ts n, (12) s n+1 = (1 α n )s n + α n Tr n, for all n N, where {α n } and {β n } are sequences in [0, 1] The Noor iteration is defined by w 1 C and h n = (1 γ n )w n + γ n Tw n, q n = (1 β n )w n + β n Th n, (13) w n+1 = (1 α n )w n + α n Tq n, for all n N, where {α n }, {β n }, and {γ n } are sequences in [0, 1] Clearly, Mann and Ishikawa iterations are special cases of Noor iteration There are many papers have been concentrated on the iteration methods for approximation of fixed points of nonlinear mappings, see for instance [1, 2, 3, 4] However, there are only a few paper concerning comparison of those iteration methods in order to establish which one converges faster In 2004, Berinde [5] provided the following concept to compare the rate of convergence of the iteration methods Definition 11 Let {a n } and {b n } be two sequences of positive numbers that converge to a, b, respectively Assume there exists a n a l = lim n b n b If l = 0, then it is said that the sequence {a n } converges to a faster than the sequence {b n } to b If 0 < l <, then we say that the sequence {a n } and {b n } have that same rate of convergence Suppose that for two fixed point iterations {x n } and {y n } converging to the same fixed point z of a mapping T, the error estimates x n z a n and y n z b n, n N, are available, where {a n } and {b n } are two sequences of positive real numbers (converging to zero) Then, in view of Definition 11, the following concept appears to be very natural Definition 12 If {a n } converges faster than {b n }, then we say that the fixed point iteration {x n } converges faster than the fixed point iteration {y n } to z We observe that the comparison of the rate of convergence in above definition depends on the choice of sequences {a n } and {b n } which are error bounds of {x n } and {y n }, respectively It seem not to be clear if we use above definition for comparing the rate of convergence So, we modify above definition for comparing the rate of convergence as follows:

3 Comparison of the Rate of Convergence of Various Iterative Methods 219 Definition 13 Suppose that {x n } and {y n } are two iteration methods converging to the same fixed point z of a mapping T We say that {x n } converges faster than {y n } to z if x n z lim n y n z = 0 Berinde [5] compared the rate of convergence of the Picard and Mann iterations for a class of Zamfirescu operators in arbitrary Banach spaces In 2006, Babu and Prasad [6] showed that the Mann iteration converges faster than the Ishikawa iteration for this class of operators Two year later, Qing and Rhoades [7] provided an example to show that the claim of Babu and Prasad [6] is false Later, Xue [8] compared convergence speed of the Picard, Krasnoselskij, Mann and Ishikawa iterations for the class of Zamfirescu operators He proved that the Picard iteration converges faster than the Mann iteration and the Krasnoselskij iteration converges faster than the Ishikawa iteration under some appropriate conditions In 2010, Rhoades and Xue [9] provided sufficient conditions for Picard iteration to converge faster than Krasnoselskij, Mann, Ishikawa, or Noor iteration for quasi-contractive operators, and also compared the rate of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators The following class of operators is more general than that of Zamfirescu operators Definition 14 Let C be a nonempty subset of a Banach space X A mapping T : C C is called weak contraction if there exist a constant δ (0, 1) and some L 0 such that Tx Ty δ x y + L y Tx for all x, y C This mapping is also partially cover the quasi-contraction operator Many researcher studied the existence and convergence theorems for finding fixed points of the class of weak contractions, Zamfirescu operators, quasi-contraction operators, see for instance [5, 6, 8, 9, 10, 11] The following weak contraction theorem obtained by Berinde in 2004 Proposition 15 ([11]) Let C be a nonempty closed convex subset of a Banach space X and T : C C be a weak contraction Then F(T) Moreover, the Picard iteration {x n } defined by x n+1 = Tx n for all n N, converges to a fixed point of T In this article, we propose a new iterative method as follows: z n = (1 γ n )x n + γ n Tx n, y n = (1 β n )z n + β n Tz n, (14) x n+1 = (1 α n λ n )y n + α n Ty n + λ n Tz n, for all n N, where x 1 C, {α n }, {β n }, {γ n }, {λ n }, and {α n + λ n } are sequences in [0, 1] Then, we prove the strong convergence of the proposed iterative method to a fixed point of a weak contraction, and also compare the rate of convergence of this iterative method with Mann, Ishikawa and Noor iterations

4 220 Thai J Math 11 (2013)/ W Phuengrattana and S Suantai 2 Main Results Firstly, we prove the following convergence theorems to a fixed point of a weak contraction in Banach spaces Theorem 21 Let C be a nonempty closed convex subset of a Banach space X and T : C C be a weak contraction with the following condition: there exist a constant δ (0, 1) and some L 0 such that Tx Ty δ x y + L y Ty for all x, y C (21) Suppose the sequence {x n } generated by (14) and {α n }, {β n }, {γ n }, {λ n }, {α n + λ n } are sequences in [0, 1] which satisfy one of the following conditions: (i) n=1 α n = ; (ii) n=1 β n = ; (iii) n=1 γ n = Then {x n } converges strongly to a unique fixed point of T Proof Proposition 15 guarantees that T has a fixed point Obviously, by condition (21), the fixed point of T is unique, say p By condition (21) again, we have Tx n p δ x n p, Ty n p δ y n p and Tz n p δ z n p Thus, it follows by (14) that x n+1 p (1 α n λ n ) y n p + α n Ty n p + λ n Tz n p (1 α n (1 δ ) λ n ) y n p + λ n δ z n p ((1 α n (1 δ ) λ n )(1 β n (1 δ )) + λ n δ ) z n p ((1 α n (1 δ ) λ n )(1 β n (1 δ )) + λ n δ )(1 γ n (1 δ ))) x n p = (((1 α n (1 δ ))(1 β n (1 δ )) λ n (1 β n )(1 δ )))(1 γ n (1 δ )) x n p ((1 α n (1 δ ))(1 β n (1 δ ))(1 γ n (1 δ )) x n p n ((1 α k (1 δ ))(1 β k (1 δ ))(1 γ k (1 δ )) x 1 p (22) By the assumption, we can conclude that {x n } converges to p Theorem 22 Assume X, C, T are as in Theorem 21 and {α n }, {β n }, {γ n } are sequences in [0, 1] such that n=1 α n = Suppose {u n }, {s n }, {w n } are sequences generated by Mann iteration (11), Ishikawa iteration (12), and Noor iteration (13), respectively Then {u n }, {s n }, {w n } converge strongly to a unique fixed point of T

5 Comparison of the Rate of Convergence of Various Iterative Methods 221 Proof By Proposition 15 and condition (21), T has a unique fixed point, say p Using (13) and condition (21), we have w n+1 p (1 α n ) w n p + α n δ q n p (1 α n + α n δ (1 β n )) w n p + α n β n δ 2 h n p (1 α n (1 δ (1 β n ) β n δ 2 (1 γ n (1 δ )))) w n p = (1 α n (1 δ + β n δ (1 δ (1 γ n (1 δ )))) w n p (23) (1 α n (1 δ )) w n p n (1 α k (1 δ )) w 1 p This implies by n=1 α n = that {w n } converges to p Since Mann and Ishikawa iterations are special cases of Noor iteration, we can conclude that {s n } and {u n } converge to a unique fixed point of T Remark 23 It is known that only weak contraction does not guarantee the uniqueness of fixed point of T But if T also satisfies the condition (21), its fixed point must be unique The following results, we consider the rate of convergence of the proposed iterative method (14) and the well-known iterative methods Theorem 24 Let C be a nonempty closed convex subset of a Banach space X and T : C C be a weak contraction with condition (21) Suppose {u n }, {s n }, {w n } are sequences generated by Mann iteration (11), Ishikawa iteration (12) and Noor iteration (13), respectively, and {α n }, {β n }, {γ n }, {λ n }, {α n +λ n } are sequences in [0, 1] which satisfy the conditions: (C1) 0 < η β n < 1 or 0 < η γ n < 1; (C2) 0 < α n < 1 1+δ, n=1 α n = and lim n α n = 0 Then the sequence {x n } generated by (14) converges faster than Mann, Ishikawa and Noor iterations to a unique fixed point of T, provided that x 1 = u 1 = s 1 = w 1 C Proof By Theorems 21 and 22, the sequence {x n } generated by (14), Mann iteration {u n }, Ishikawa iteartion {s n } and Noor iteration {w n } converge to a

6 222 Thai J Math 11 (2013)/ W Phuengrattana and S Suantai unique fixed point of T, say p Firstly, by Noor iteration (13), w n+1 p (1 α n ) w n p α n Tq n p (1 α n ) w n p α n δ q n p (1 α n ) w n p α n δ ((1 β n ) w n p + β n δ h n p ) (1 α n α n δ (1 β n )) w n p α n β n δ 2 (1 γ n (1 δ )) w n p = (1 α n (1 + δ (1 β n (1 δ (1 γ n (1 δ )))))) w n p (1 α n (1 + δ )) w n p n (1 α k (1 + δ )) w 1 p (24) It follows by (C1), (22) and (24) that x n+1 p w n+1 p (1 η(1 δ )) n n (1 α k(1 + δ )) Put θ n = É (1 η(1 δ )) n n (1 α k (1+δ )) By the assumption 0 < α n < 1 1+δ and lim n α n = 0, we obtain θ n+1 (1 η(1 δ )) n+1 n lim = lim (1 α k(1 + δ )) n θ n n n+1 (1 α k(1 + δ )) (1 η(1 δ )) n 1 η(1 δ ) = lim n 1 α n+1 (1 + δ ) = 1 η(1 δ ) < 1 By the ratio test, we conclude that n=1 θ n < So, lim n θ n = 0 This implies that the sequence generated by (14) converges faster than Noor iteration Secondly, by Ishikawa iteration (12), s n+1 p (1 α n ) s n p α n Tr n p (1 α n ) s n p α n δ r n p (1 α n ) s n p α n δ (1 β n + β n δ ) s n p = (1 α n (1 + δ (1 β n (1 δ )))) s n p (1 α n (1 + δ )) s n p n (1 α k (1 + δ )) s 1 p (25)

7 Comparison of the Rate of Convergence of Various Iterative Methods 223 It follows by (C1), (22) and (25) that x n+1 p s n+1 p (1 η(1 δ )) n n (1 α k(1 + δ )) By the assumption and same argument as above, we can show that the sequence generated by (14) converges faster than Ishikawa iteration Finally, by Mann iteration (11), u n+1 p (1 α n (1 + δ )) u n p n (1 α k (1 + δ )) u 1 p (26) It follows by (C1), (22) and (26) that x n+1 p u n+1 p (1 η(1 δ )) n n (1 α k(1 + δ )) By the assumption and same argument as above, we can show that the sequence generated by (14) converges faster than Mann iteration Set γ n = 0 for all n N in (14), we get the following two-step iterative method: { yn = (1 β n )x n + β n Tx n, (27) x n+1 = (1 α n λ n )y n + α n Ty n + λ n Tx n, for all n N, where x 1 C, {α n }, {β n }, {λ n }, {α n + λ n } are sequences in [0, 1] By Theorem 21, we obtain the following convergence theorem of the iterative scheme (27) and by Theorem 24, we obtain the following result for comparing the rate of convergence of two-step iteration (27) with Mann, Ishikawa and Noor iterations Theorem 25 Assume X, C, T are as in Theorem 21 and {α n }, {β n }, {λ n }, {α n +λ n } are sequences in [0, 1] such that n=1 α n = or n=1 β n = Then the sequence {x n } generated by two-step iteration (27) converges strongly to a unique fixed point of T Theorem 26 Assume X, C, T, {u n }, {s n }, {w n } are as in Theorem 24 and {α n }, {β n }, {λ n }, {α n + λ n } are sequences in [0, 1] which satisfy the conditions: (C1) 0 < η β n < 1; (C2) 0 < α n < 1 1+δ, n=1 α n = and lim n α n = 0 Then the sequence {x n } generated by two-step iteration (27) converges faster than Mann, Ishikawa and Noor iterations to a unique fixed point of T

8 224 Thai J Math 11 (2013)/ W Phuengrattana and S Suantai It is worth to observe that above theorem shows that the two-step iteration (27) convergence faster than three-step iteration (13) under some suitable control conditions Remark 27 (i) If a mapping T satisfies only the condition (21) with F(T), under the same control conditions, Theorems 21, 22, 24, 25 and 26 hold true for this class of mappings (ii) Any Zamfirescu operators, quasi-contraction operators with h (0, 1/2) are weak contractions and also satisfy condition (21) Therefore, Theorems 21, 22, 24, 25 and 26 are obtained with these mappings The following example shows that our iterative method converges faster than Mann, Ishikawa and Noor iterations Example 28 Let T : [0, 1] [0, 1] be defined by x 3, if x [ ) 0, 2 5 Tx = 2x 5, if x [ 2 5, 1] Obviously, if x, y [0, 2 5 ) or x, y [2 5, 1], then T is satisfies the weak contraction condition and condition (21) For the case x [0, 2 5 ) and y [2 5, 1], or x [2 5, 1] and y [0, 2 5 ), it follows that and Tx Ty 1 3 x y + 1 y Ty, 9 Tx Ty 1 x y + y Tx 3 Then T is a weak contraction and satisfies condition (21) The comparison of the convergence of the sequence {x n } generated by (14) and (27), Mann, Ishikawa and Noor iterations to the exact fixed point p = 0 are given in the following table, with the initial point x 1 = u 1 = s 1 = w 1 = 1, and the control conditions α n = λ n = γ n = 1 n and β n = n 5n+1 It is clear that these control conditions satisfy those in Theorems 24 and 26

9 Comparison of the Rate of Convergence of Various Iterative Methods 225 Mann Ishikawa Noor iteration iteration iteration iteration iteration (27) (14) n u n s n w n x n x n Tx n x n E E E E-09 Table 21: Comparison of the rate of convergence of the iterative methods (14), (27), Mann, Ishikawa and Noor iterations for the mapping given in Example 28 From Table 21, we see that the iterative methods (14) and (27) converge faster than Mann, Ishikawa and Noor iterations Acknowledgements : The authors would like to thank the National Research University Project under Thailand s Office of the Higher Education Commission for financial support The first author is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand References [1] WR Mann, Mean value methods in iteration, Proc Amer Math Soc 4 (1953) [2] MA Krasnoselskij, Two remarks on the method of successive approximations (Russian), Uspehi Mat Nauk 10 (1955) [3] S Ishikawa, Fixed points by a new iteration method, Proc Amer Math Soc 44 (1974) [4] MA Noor, New approximation schemes for general variational inequalities, J Math Anal Appl 251 (2000) [5] V Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory and Applications 2 (2004) [6] GV Babu, KN Prasad, Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators, Fixed Point Theory and Appl, Volume 2006 (2006), Article ID49615, 6 pages [7] Y Qing, BE Rhoades, Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to Zamfirescu Operators, Fixed Point Theory and Appl, Volume 2008 (2008), Article ID , 3 pages

10 226 Thai J Math 11 (2013)/ W Phuengrattana and S Suantai [8] Z Xue, The comparison of the convergence speed between Picard, Mann, Krasnoselskij and Ishikawa iterations in Banach spaces, Fixed Point Theory and Appl, Volume 2008 (2008), Article ID , 5 pages [9] BE Rhoades, Z Xue, Comparison of the rate of convergence among Picard, Mann, Ishikawa, and Noor iterations applied to quasicontractive maps, Fixed Point Theory and Appl, Volume 2010 (2010), Article ID , 12 pages [10] V Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J Math 19 (2003) 7 22 [11] V Berinde, Approximation fixed points of weak contractions using the Picard iteration, Nonlinear Anal Forum 9 (2004) (Accepted 4 December 2012) Thai J Math