Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062, pptx

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062, 12 pages doi:10.1155/2010/169062 Research Article Comparison of the Rate of Convergence among Picard, Mann, Ishikawa, and Noor Iterations Applied to Quasicontractive Maps B. E. Rhoades 1 and Zhiqun Xue 2 1 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA 2 Department of Mathematics and Physics, Shijiazhuang Railway University, Shijiazhuang 050043, China Correspondence should be addressed to Zhiqun Xue, xuezhiqun@126.com Received 12 October 2010; Accepted 14 December 2010 Academic Editor: Juan J. Nieto Copyright q 2010 B. E. Rhoades and Z. Xue. 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. We provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij, Mann, Ishikawa, or Noor iteration for quasicontractive operators. We also compare the rates of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators. 1. Introduction Let X, d be a complete metric space, and let T be a self-map of X.IfT has a unique fixed point, which can be obtained as the limit of the sequence {p n },wherep n  T n p 0 ,p 0 any point of X,thenT is called a Picard operator see, e.g., 1, and the iteration defined by {p n } is called Picard iteration. One of the most general contractive conditions for which a map T is a Picard operator is that of ´ Ciri ´ c 2see also 3. A self-map T is called quasicontractive if it satisfies d  Tx,Ty  ≤ δ max  d  x, y  ,d  x, Tx  ,d  y, Ty  ,d  x, Ty  ,d  y, Tx  , 1.1 for each x, y ∈ X,whereδ is a real number satisfying 0 ≤ δ<1. Not every map which has a unique fixed point enjoys the Picard property. For example, let X 0, 1 with the absolute value metric, T : X → X defined by Tx  1 − x. Then, T has a unique fixed point at x  1/2, but if one chooses as a starting point x 0  a for any a /  1/2, then successive function iterations generate the bounded divergent sequence {a, 1 − a, a, 1 − a, }. 2 Fixed Point Theory and Applications To obtain fixed points for some maps for which Picard iteration fails, a number of fixed point iteration procedures have been developed. Let X be a Banach space, the corresponding quasicontractive mapping T : X → X is defined by   Tx − Ty   ≤ δ max    x − y   ,  x − Tx  ,   y − Ty   ,   x − Ty   ,   y − Tx    . 1.2 In this paper, we will consider the following four iterations. Krasnoselskij: ∀v 0 ∈ X, v n1   1 − λ  v n  λTv n ,n≥ 0, 1.3 where 0 <λ<1. Mann: ∀u 0 ∈ X, u n1   1 − a n  u n  a n Tu n ,n≥ 0, 1.4 where 0 <a n ≤ 1forn ≥ 0, and  ∞ n0 a n  ∞. Ishikawa: ∀x 0 ∈ X, x n1   1 − a n  x n  a n Ty n ,n≥ 0, y n   1 − b n  x n  b n Tx n ,n≥ 0, 1.5 where {a n }⊂0, 1, {b n }⊂0, 1. Noor: ∀w 0 ∈ X, z n   1 − c n  w n  c n Tw n ,n≥ 0, y n   1 − b n  w n  b n Tz n ,n≥ 0, w n1   1 − a n  w n  a n Ty n ,n≥ 0, 1.6 where {a n }⊂0, 1, {b n }, {c n }⊂0, 1. Three of these iteration schemes have also been used to obtain fixed points for some Picard maps. Consequently, it is reasonable to try to determine which process converges the fastest. In this paper, we will discuss this question for the above quasicontractions and for Zamfirescu operators. For this, we will need the following result, which is a special case of the Theorem in 4. Theorem 1.1. Let C be any nonempty closed convex subset of a Banach space X,andletT be a quasicontractive self-map of C.Let{x n } be the Ishikawa iteration process defined by 1.5,whereeach a n > 0 and  ∞ n0 a n  ∞.then{x n } converges strongly to the fixed point of T. Fixed Point Theory and Applications 3 2. Results for Quasicontractive Operators To avoid trivialities, we shall always assume that p 0 /  q,whereq denotes the fixed point of the map T. Let {f n }, {g n } be two convergent sequences with the same limit q,then{f n } is said to converge faster than {g n } see, e.g., 5 if lim n →∞   f n − q     g n − q    0. 2.1 Theorem 2.1. Let E be a Banach space, D a closed convex subset of E,andT a quasicontractive self- map of D,then,for0 <λ<1 − δ 2 , Picard iteration converges faster than Krasnoselskij iteration. Proof. From Theorem 1 of 2 and 1.2,   p n1 − q       T n1 p 0 − q    ≤ δ n1 1 − δ   Tp 0 − p 0   ≤ δ n1 1 − δ    Tp 0 − Tq      p 0 − q    ≤ δ n1 1 − δ  δ max    p 0 − q   ,   p 0 − q      Tp 0 − Tq       p 0 − q    ≤ δ n1 1 − δ  δ    p 0 − q    δ 1 − δ   p 0 − q       p 0 − q    ≤ δ n1  1 − δ  2   p 0 − q   , 2.2 where q is the fixed point of T. From 1.3,withv 0 /  q,   v n1 − q   ≥  1 − λ    v n − q   − λ   Tv n − Tq   ≥  1 − λ 1 − δ    v n − q   ≥··· ≥  1 − λ 1 − δ  n1   v 0 − q   . 2.3 By setting each β n  0andeachα n  λ, it follows from Theorem 1.1 that {v n } converges to q. 4 Fixed Point Theory and Applications Therefore,   p n1 − q     v n1 − q   ≤  δ 1 − δ − λ  n1  1 − δ  n−1   p 0 − q     v 0 − q   −→ 0 , 2.4 as n →∞,sinceλ<1 − δ 2 . Theorem 2.2. Let E, D,andT be as in Theorem 2.1.Andlet0 <a n <θ1 − δ,b n ,c n ∈ 0, 1 for all n>0. A If the constant 0 <θ<1 − δ, then Picard iteration converges faster than Mann iteration. B If the constant 0 <θ<1 − δ 2 /1 − δ  δ 2 , then Picard iteration converges faster than Ishikawa iteration. C If the constant 0 <θ<1 − δ 3 /1 − 2δ  2δ 2 , then Picard iteration converges faster than Noor iteration. Proof. We have the following cases Case A Mann Iteration.UsingTheorem 1.1 with each β n  0, {u n } converges to q.Using 1.4,   u n1 − q   ≥  1 − a n    u n − q   − a n   Tu n − Tq   ≥  1 − a n 1 − δ    u n − q   ≥···≥ n  i0  1 − a i 1 − δ    u 0 − q   . 2.5 Therefore,   p n1 − q     u n1 − q   ≤ δ n1   p 0 − q    1 − δ  2  n i0  1 − a i /  1 − δ    u 0 − q   −→ 0 , 2.6 as n →∞,sincea n <θ1 − δ for each n>0. Case B Ishikawa Iteration.FromTheorem 1.1, {x n } converges to q.Using1.5,   x n1 − q   ≥  1 − a n    x n − q   − a n   Ty n − Tq   ≥  1 − a n    x n − q   − a n δ 1 − δ   y n − q   ≥  1 − a n    x n − q   − a n δ 1 − δ    x n − q    b n   Tx n − Tq    Fixed Point Theory and Applications 5 ≥  1 − a n − a n δ 1 − δ    x n − q   − a n b n δ 2  1 − δ  2   x n − q   ≥  1 − a n − a n δ 1 − δ − a n δ 2  1 − δ  2    x n − q   ≥··· ≥ n  i0  1 − a i − a i δ 1 − δ − a i δ 2  1 − δ  2    x 0 − q   . 2.7 Hence,   p n1 − q     x n1 − q   ≤ δ n1   p 0 − q    1 − δ  2  n i0  1 − a i − a i δ/  1 − δ  − a i δ 2 /  1 − δ  2    x 0 − q   −→ 0, 2.8 as n →∞,sincea n <θ1 − δ for each n>0. Case C Noor Iteration. First we must show that {w n } converges to q. The proof will follow along the lines of that of Theorem 1.1. Lemma 2.3. Define A n  { z i } n i0 ∪  y i  n i0 ∪ { w i } n i0 ∪ { Tz i } n i0 ∪  Ty i  n i0 ∪ { Tw i } n i0 , α n  diam  A n  , β n  max  max { w 0 − Tw i  :0≤ i ≤ n } , max    w 0 − Ty i   :0≤ i ≤ n  , max { w 0 − Tz i  :0≤ i ≤ n }} , 2.9 then {A n } is bounded. Proof. Case 1. Suppose that α n  Tz i − Tz j  for some 0 ≤ i, j ≤ n, then, from 1.2 and the definition of α n , α n    Tz i − Tz j   ≤ δ max    z i − z j   ,  z i − Tz i  ,   z j − Tz j   ,   z i − Tz j   ,   z j − Tz i    ≤ δα n , 2.10 a contradiction, since δ<1. Similarly, α n /  Ty i − Ty j , α n /  Tw i − Tw j , α n /  Tz i − Ty j , α n /  Tz i − Tw j ,and α n /  Ty i − Tw j  for any 0 ≤ i, j ≤ n. 6 Fixed Point Theory and Applications Case 2. Suppose that α n  w i − w j , without loss of generality we let 0 ≤ i<j≤ n. Then, from 1.6, α n    w i − w j   ≤  1 − a j−1    w i − w j−1    a j−1   w i − Ty j−1   ≤  1 − a j−1    w i − w j−1    a j−1 α n . 2.11 Hence, α n ≤w i − w j−1 ≤α n ,thatis,α n  w i − w j−1 . By induction on j,weobtainα n  w i − w i   0, a contradiction. Case 3. Suppose that α n  w i − Tw j  for some 0 ≤ i, j ≤ n.Ifi>0, then we have, using 1.6, α n    w i − Tw j   ≤  1 − a i−1    w i−1 − Tw j    a i−1   Ty i−1 − Tw j   ≤  1 − a i−1    w i−1 − Tw j    a i−1 α n , 2.12 which implies that α n ≤w i−1 − Tw j , and by induction on i,wegetα n  w 0 − Tw j . Case 4. Suppose that α n  w i − z j  or α n  z i − z j , y i − z j , z i − Ty j , y i − y j  for some 0 ≤ i, j ≤ n,then α n    w i − z j   ≤  1 − c j    w i − w j    c j   w i − Tw j   ≤ max    w i − w j   ,   w i − Tw j    . 2.13 From Cases 2 and 3, w i − w j  <α n ,andw i − Tw j ≤w 0 − Tw m  for some m ≤ j,thatis, α n  w 0 − Tw m .Ifα n  z i − z j ,weobtainthatα n ≤w i − z j . Therefore; α n  w 0 − Tw m , other cases, omitting. Case 5. Suppose that α n  w i −Tz j  or α n  z i −Tz j , w i −y j , y i −Tz j  for some 0 ≤ i, j ≤ n, then if i>0, α n    w i − Tz j   ≤  1 − a i−1    w i−1 − Tz j    a i−1   Ty i−1 − Tz j   ≤  1 − a i−1    w i−1 − Tz j    a i−1 α n , 2.14 it leads to α n ≤w i−1 − Tz j . Again by induction on i,wehaveα n  w 0 − Tz j . Similarly, if α n  z i − Tz j  or, α n  w i − y j ,wealsogetα n  w 0 − Tz j ; other cases, omitting. Fixed Point Theory and Applications 7 Case 6. Suppose that α n  z i − Tw j  or α n  y i − Tw j  for some 0 ≤ i, j ≤ n, then, using Case 1, α n    z i − Tw j   ≤  1 − c i    w i − Tw j    c i   Tw i − Tw j   ≤  1 − c i    w i − Tw j    c i α n , 2.15 or α n    y i − Tw j   ≤  1 − b i    w i − Tw j    b i   Tz i − Tw j   ≤  1 − b i    w i − Tw j    b i α n , 2.16 these imply that α n ≤w i − Tw j .ByCase3,weobtainthatα n  w 0 − Tw j . Case 7. Suppose that α n  w i − Ty j  or α n  y i − Ty j  for some 0 ≤ i, j ≤ n,thenifi>0, using Case 2, α n    w i − Ty j   ≤  1 − a i−1    w i−1 − Ty j    a i−1   Ty i−1 − Ty j   ≤  1 − a i−1    w i−1 − Ty j    a i−1 α n , 2.17 which implies that α n ≤w i−1 − Ty j . Using induction on i,wehaveα n  w 0 − Ty j . In view of the a bove cases, so we have shown that α n  β n . It remains to show that {α n } is bounded. Indeed, suppose that α n  w 0 − Tw j  for some 0 ≤ j ≤ n, then, using Case 1, α n    w 0 − Tw j   ≤  w 0 − Tw 0     Tw 0 − Tw j   ≤ B  δα n , 2.18 where B : w 0 − Tw 0 ,thenα n ≤ B/1 − δ. Similarly, if α n  w 0 − Ty j ,orα n  w 0 − Tz j  we again get α n ≤ B/1 − δ.Hence, {α n } is bounded, that is, {A n } is bounded. Lemma 2.4. Let E, D,andT be as in Theorem 2.1,andthat  a n  ∞,then{w n },asdefinedby 1.6, converges strongly to the unique fixed point q of T. Proof. From ´ Ciri ´ c 2, T has a unique fixed point q.Foreachn ∈ ,define B n  { w i } i≥n ∪  y i  i≥n ∪ { z i } i≥n ∪ { Tw i } i≥n ∪  Ty i  i≥n ∪ { Tz i } i≥n . 2.19 8 Fixed Point Theory and Applications Then, using the same proof as that of Lemma 2.3,itcanbeshownthat r n : diam  B n   max  sup    w n − Tw j   : j ≥ n  , sup    w n − Ty j   : j ≥ n  , sup    w n − Tz j   : j ≥ n  . 2.20 Using 1.2 and 1.6, r n    w n − Tw j   ≤  1 − a n−1    w n−1 − Tw j    a n−1   Tw n−1 − Tw j   ≤  1 − a n−1  r n−1  a n−1 δr n−1   1 − a n−1  1 − δ  r n−1 ≤··· ≤ r 0 n−1  i0  1 −  1 − δ  a i  , 2.21 lim r n  0, since  a n  ∞. For any m, n > 0withj ≥ 0,  w n − w m  ≤   w n − Tw j      Tw j − w m    r n  r m , 2.22 and {w n } is Cauchy sequence. Since D is closed, there exists w ∞ ∈ D such that lim w n  w ∞ . Also, lim w n − Tw n   0. Using 1.2,  Tw ∞ − w ∞    Tw ∞ − Tw n  Tw n − w n  w n − w ∞   lim  Tw ∞ − Tw n  ≤ lim sup δ max { w ∞ − w n  ,  w ∞ − Tw ∞  ,  w n − Tw n  ,  w ∞ − Tw n  ,  w n − Tw ∞ }  δ  w ∞ − Tw ∞  . 2.23 Since δ<1, it follows that w ∞  Tw ∞ ,andw ∞ is a fixed point of T.Butthefixedpoint is unique. Therefore, w ∞  q. Fixed Point Theory and Applications 9 Returning to the proof of Case C, from 1.6,   w n1 − q   ≥  1 − a n    w n − q   − a n   Ty n − Tq   ≥  1 − a n    w n − q   − a n δ 1 − δ   y n − q   ≥  1 − a n    w n − q   − a n δ 1 − δ    w n − q    b n   Tz n − Tq    ≥  1 − a n − a n δ 1 − δ    w n − q   − a n δ 2  1 − δ  2   z n − q   ≥  1 − a n − a n δ 1 − δ − a n δ 2  1 − δ  2 − a n δ 3  1 − δ  3    w n − q   ≥··· ≥ n  i0  1 − a i − a i δ 1 − δ − a i δ 2  1 − δ  2 − a i δ 3  1 − δ  3    w 0 − q   . 2.24 So,   p n1 − q     w n1 − q   ≤ δ n1   p 0 − q    1 − δ  2  n i0  1 − a i − a i δ/1 − δ − a i δ 2 /  1 − δ  2 − a i δ 3 /  1 − δ  3    w 0 − q   −→ 0, 2.25 as n →∞,sincea n <θ1 − δ for n>0. It is not possible to compare the rates of convergence between the Krasnoselskij, Mann, and Noor iterations for quasicontractive maps. However, if one considers Zamfirescu maps, then some comparisons can be made. 3. Zamfirescu Maps AselfmapT is called a Zamfirescu operator if there exist real numbers a, b, c sa tisfying 0 < a<1, 0 <b,c < 1/2suchthat,foreachx, y ∈ X at least one of the following cond itions is true: 1 dTx,Ty ≤ adx, y, 2 dTx,Ty ≤ bdx, Txdy, Ty, 3 dTx,Ty ≤ cdx, Tydy, Tx. 10 Fixed Point Theory and Applications In 6 it was shown that the above set of conditions is equivalent to d  Tx,Ty  ≤ δ max  d  x, y  ,  d  x, Tx   d  y, Ty  2 ,  d  x, Ty   d  y, Tx  2  , 3.1 for some 0 <δ<1. In the following results, we shall use the representation 3.1. Theorem 3.1. Let E,andD be as in Theorem 2.1, T a Zamfirescu selfmap of D,thenifa n <λ1 − δθ/1  δ with the constant 0 <θ<1  δ for each n>0, Krasnoselskij iteration converges faster than Mann, Ishikawa, or Noor iteration. Proof. Since Zamfirescu maps are special cases of quasicontractive maps, from Theorem 1.1 {v n }, {x n },and{w n } converge to the unique fixed point of T, which we will call q. Using 1.2,   v n1 − q   ≤  1 − λ    v n − q    λ   Tv n − q   . 3.2 Using 3.1,   Tv n − q   ≤ δ max    v n − q   ,  v n − Tv n   0  2 ,    v n − q      q − Tv n    2   δ   v n − q   . 3.3 Therefore,   v n1 − q   ≤  1 − λ  1 − δ    v n − q   ≤··· ≤  1 − λ  1 − δ  n1   v 0 − q   , 3.4 and   u n1 − q   ≥  1 − a n  1  δ    u n − q   ≥··· ≥ n  i0  1 − a i  1  δ    u 0 − q   . 3.5 Thus,   v n1 − q     u n1 − q   ≤  1 − λ  1 − δ  n1   v 0 − q    n i0  1 − a i  1  δ    u 0 − q   −→ 0 , 3.6 as n →∞,sincea n <λ1 − δ. The proofs for Ishikawa and Noor iterations are similar. [...]... comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators,” Fixed Point Theory and Applications, Article ID 387504, 3 pages, 2008 8 G V R Babu and K N V V Vara Prasad, “Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators,” Fixed Point Theory and Applications, Article ID 49615, 6 pages, 2006 ... Mathematical Analysis and Applications, vol 167, no 2, pp 582–587, 1992 5 V Berinde, “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,” Fixed Point Theory and Applications, no 2, pp 97–105, 2004 6 A Rafiq, Fixed points of Ciri´ quasi-contractive operators in normed spaces,” Mathematical c Communications, vol 11, no 2, pp 115–120, 2006 7 Y Qing and B E Rhoades,.. .Fixed Point Theory and Applications 11 Theorem 3.2 Let E, D, and T be as in Theorem 3.1, then if λ 1 δ θ/ 1 − δ < an < 1 with the constant 0 < θ < 1 − δ for any n, Mann iteration converges faster than Krasnoselskij iteration Proof Using 1.4 and 3.1 , un 1 − q ≤ 1 − an un − q an Tun − q ≤ 1 − an 1 − δ un − q 3.7 ≤ ··· ≤ n 1 − ai 1 − δ u0 − q i 0 And again using 1.3 , 3.1 ,... converges, but for which neither Mann nor Ishikawa converges Acknowledgments The authors would like to thank the reviewers for valuable suggestions, and the National Natural Science Foundation of China Grant 10872136 for the financial support 12 Fixed Point Theory and Applications References 1 I A Rus, “Weakly Picard mappings,” Commentationes Mathematicae Universitatis Carolinae, vol 34, no 4, pp 769–773, 1993... Mann, Ishikawa, and Noor iterations, even for Zamfirescu maps Remark 3.3 It has been noted in 7 that the principal result in 8 is incorrect Remark 3.4 Krasnoselskij and Mann iterations were developed to obtain fixed point iteration methods which converge for some operators, such as nonexpansive ones, for which Picard iteration fails Ishikawa iteration was invented to obtain a convergent fixed point iteration . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062, 12 pages doi:10.1155/2010/169062 Research Article Comparison of the. Tw ∞  . 2.23 Since δ<1, it follows that w ∞  Tw ∞ ,andw ∞ is a fixed point of T.Butthefixedpoint is unique. Therefore, w ∞  q. Fixed Point Theory and Applications 9 Returning to the proof of Case C,. Qing and B. E. Rhoades, “Letter to the editor: comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators,” Fixed Point Theory and Applications, Article ID