In this note we give a correction to the main result of Zhou in [14] on the convergence of the Ishikawa iteration process to a unique fixed point of a strongly pseudocontractive operator in arbitrary real Banach spaces. Our results extend the recent result of Soltuz [11] to arbitrary strongly pseudocontractive operators

J. S. Ume

Lj. Ćirić

Ćirić, Lj.

Ume, J. S.

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Mathematical Communications 8(2003), 43-48 43Ishikawa iterative process for stronglypseudocontractive operators in arbitrary BanachspacesLjubomir C´iric´∗ and Jeong S.Ume†Abstract. In this note we give a correction to the main result ofZhou in [14] on the convergence of the Ishikawa iteration process to aunique fixed point of a strongly pseudocontractive operator in arbitraryreal Banach spaces. Our results extend the recent result of Soltuz [11]to arbitrary strongly pseudocontractive operators.Key words: Ishikawa iteration process, strongly pseudocontractiveoperatorAMS subject classifications: Primary 47H10; Secondary 47H06,54H25Received January 10, 2003 Accepted March 13, 20031. Introduction and preliminariesLet X be a real Banach space and D a nonempty, convex subset of X . Let X∗ bethe duality space of X and 〈. , .〉 be the pairing between X and X∗. The mappingJ : X → 2X∗ defined byJ(x) = {f ∈ X∗ : 〈x, f〉 = ‖x‖2, ‖f‖ = ‖x‖}, x ∈ Xis said to be a normalized duality mapping. The Hahn-Banach theorem assuresthat J(x) = Ø for each x ∈ X . It is easy to see (c.f. [11]) that〈x, j(y)〉 ≤ ‖x‖‖y‖ (1)for all x, y ∈ X and each j(y) ∈ J(y).An operator T : D ⊂ X → X is called strongly pseudocontractive if for allx, y ∈ D there exist j(x− y) ∈ J(x − y) and a constant k ∈ (0, 1) such that〈Tx− Ty, j(x− y)〉 ≤ k‖x− y‖2. (2)∗University of Belgrade, Aleksinacˇkih rudara 12/35, 11080 Belgrade, Serbia, e-mail:ciric@alfa.mas.bg.ac.yu. The author was sponsored by Fuji-film.†Department of Applied Mathematics, Changwon National University, Changwon 641-773, Ko-rea, e-mail: jsume@sarim.changwon.ac.kr44 Lj. C´iric´ and Jeong S.UmeOne of the effective methods for approximating fixed points of an operator T :D(T ) ⊂ X → X is the Ishikawa iteration process [5], starting with arbitrary x0 ∈D(T ) and for n ≥ 0 defined byxn+1 = (1− αn)xn + αnTyn,yn = (1− βn)xn + βnTyn,where αn, βn ∈ [0, 1] satisfy suitable conditions (see e.g. [1]-[4], [6]-[9], [11]-[14]). Ifβn = 0 for each n ≥ 0, then Ishikawa iterations reduce to the Krasnoselski-Manniterations [6]. In the literature which considers the convergence of the Ishikawaiteration sequence associated with accretive or pseudocontractive operators, one ofhypotheses for parameters αn is, in general, that αn → 0 as n → ∞. RecentlyZhou [14] considered the Ishikawa iteration process with parameters αn ≥ a > 0.Osilike in [8] have proved that two assumptions of the main theorem in [14] arecontradictory. Recently Soltuz [11] presented a correction for the result of Zhou[14] for a subclass of strongly pseudocontractive operators, namely for operators Twhich satisfy (2) with k < 12 .The purpose of this note is to extend the result of Soltuz [11] to all stronglypseudocontractive operators which satisfy (2) with k < 1 and the parameters αnin the Ishikawa iteration process satisfy the condition 0 < a ≤ αn ≤ b < 2(1 − k),where a, b ∈ (0, 1] are some constants.For our result we need the following two lemmas:Lemma 1 [[7], [12], [13]]. Let X be a real Banach space and let J : X → 2X∗be a normalized duality mapping. Then‖x+ y‖2 ≤ ‖x‖2 + 2〈y, j(x+ y)〉 (3)for all x, y ∈ X and each j(x+ y) ∈ J(x + y).Lemma 2 [[9], [10]]. Let {ρn} be a sequence of non-negative real numberswhich satisfyρn+1 ≤ (1− ω)ρn + σn, (4)where ω ∈ (0, 1) is a fixed number and σn ≥ 0 is such that σn → 0 as n→∞. Thenρn → 0 as n→∞.2. Main resultsNow we prove the following theorems on approximation.Theorem 1. Let X be a real Banach space, D a non-empty, convex subsetof X and T : D → D a continuous and strongly pseudocontractive mapping with apseudocontractive parameter k ∈ (0, 1). Let x0 ∈ D be arbitrary and let the Ishikawaiteration sequence {xn} be defined byxn+1 = (1− αn)xn + αnTyn,yn = (1− βn)xn + βnTxn, n = 0, 1, 2, . . . ,(5)where αn, βn ∈ [0, 1] and constants a, b ∈ (0, 1] are such that0 < a ≤ αn ≤ b < 2(1− k). (6)Ishikawa iterative process 45If sequences {Txn} and {Tyn} are bounded and‖Txn+1 − Tyn‖ → 0 as n→∞, (7)then the sequence {xn} converges strongly to a unique fixed point of T in D.Proof. The existence of a fixed point follows from the result of Deimling [3],and the uniqueness from the strongly pseudocontractivity of T . Let x∗ be such thatTx∗ = x∗.PutM = 1 + ‖x0 − x∗‖+ sup {‖Txn − x∗‖ : xn ∈ D}+ sup {‖Tyn − x∗‖ : yn ∈ D} .(8)Since {Txn} and {Tyn} are bounded, we have that M < +∞.We show that the sequence {xn} is bounded. We shall use the mathematical in-duction to prove that‖xn − x∗‖ ≤M for all n ≥ 0. (9)For n = 0, (9) follows from the definition of M . Suppose now that (9) holds forsome n ≥ 0. From (5), (8) and (9) we get‖xn+1 − x∗‖ = ‖(1− αn)(xn − x∗) + αn(Tyn − x∗)‖≤ (1− αn)‖xn − x∗‖+ αnM.Now, by the induction hypothesis we obtain ‖xn+1−x∗‖ ≤ (1−αn)M+αnM =M .Thus, by induction, we conclude that (9) holds for all n ≥ 0.From Lemma 1 and (5) we have‖xn+1 − x∗‖2 = ‖(1− αn)(xn − x∗) + αn(Tyn − x∗)‖2≤ (1− αn)2‖xn − x∗‖2 + 2αn〈Tyn − x∗, j(xn+1 − x∗)〉≤ (1− αn)2‖xn − x∗‖2 + 2αn〈Tyn − Txn+1, j(xn+1 − x∗)〉+2αn〈Txn+1 − x∗, j(xn+1 − x∗)〉.Hence, by strongly pseudocontractivity of T , we get‖xn+1 − x∗‖2 ≤ (1 − αn)2‖xn − x∗‖2 + 2αnk‖xn+1 − x∗‖2+ 2αn〈Tyn − Txn+1, j(xn+1 − x∗)〉(10)for each j(xn+1 − x∗) ∈ J(xn+1 − x∗). From (10), (1) and (9) we obtain(1− 2αnk)‖xn+1 − x∗‖2 ≤ (1− αn)2‖xn+1 − x∗‖2 + 2αn‖Tyn − Txn+1‖M. (11)From (6) it follows that1− 2αnk ≥ 1− 2kb > (2k − 1)2 ≥ 0.Thus, from (11) we have‖xn+1 − x∗‖2 ≤ (1− αn)21− 2αnk ‖xn − x∗‖2 + 2αnM1− 2αnk ‖Txn+1 − Tyn‖. (12)46 Lj. C´iric´ and Jeong S.UmeSince (1− αn)2 < 1− 2αnk for 0 < a ≤ αn ≤ b < 2(1− k), we get(1− αn)21− 2αnk < (1− αn)2 + 2αnk≤ 1− 2αn + αnb+ 2αnk= 1− [2(1− k)− b]αn≤ 1− [2(1− k)− b]a;2αn1− 2kαn ≤21− 2kb.Thus, from (12) we have‖xn+1 − x∗‖2 ≤ (1− ω)‖xn − x∗‖2 + σn, (13)whereω = [2(1− k)− b] · a,σn =2M1− 2kb‖Txn+1 − Tyn‖.From (7) we have thatlimn→∞σn = 0.Taking ρn = ‖xn − x∗‖2, from (13) and Lemma 2 we getlimn→∞ ‖xn − x∗‖ = 0.Thus we proved that the sequence {xn} converges strongly to a unique fixed pointof T in D. ✷Remark 1. For k < 12 condition (6) in Theorem1 becomes 0 < a ≤ αn, sincein this case 2(1 − k) > 1. Thus, Theorem1 contains Theorem 1 of Soltuz [11] as acorollary.Let X be a real Banach space and S : X → X a mapping on X . If for anyx, y ∈ X there exist j(x− y) ∈ J(x− y) and a constant k ∈ (0, 1) such that〈Sx− Sy, j(x− y)〉 ≤ k‖x− y‖2,then S is called a strongly accretive operator.Lemma 3 [[1]]. If T : X → X is a strongly accretive operator, then, forany f ∈ X, mapping S : X → X, defined by Sx = f − Tx + x is a stronglypseudocontractive operator, i.e. for any x, y ∈ X:〈Sx− Sy, j(x− y)〉 ≤ (1− k)‖x− y‖2,where k ∈ (0, 1) is the strongly accretive constant of T .Theorem 2. Let X be a real Banach space and S : X → X a continuousstrongly accretive operator with a strongly accretive constant k ∈ (0, 1). For anygiven f ∈ X, define a mapping T : X → X byTx = f − Sx+ xIshikawa iterative process 47for all x ∈ X. Let {αn}, {βn} be two real sequences in [0, 1] and a, b ∈ (0, 1] besuch that0 < a ≤ αn ≤ b < 2k.If the range of (I −S) is bounded, then for arbitrary x0 ∈ X the sequence {xn}, de-fined by (5) and satisfying (7) in Theorem1, converges strongly to a unique solutionof the equation Sx = f .Proof. Obviously, if x∗ ∈ X is a solution of the equation Sx = f , then x∗is a fixed point of T . Also it is easy to prove that T is continuous and stronglypseudocontractive with the strongly pseudocontractivity constant (1− k). Clearly,since the range of (I−S) is bounded, it follows that {Txn} and {Tyn} are bounded.Thus, Theorem 2 follows from Theorem 1. ✷References[1] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, S.M.Kang, Iterative ap-proximations of fixed points and solutions for strongly accretive and stronglypseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl.224(1998), 149–165.[2] Lj. B. C´iric´, Convergence theorems for a sequence of Ishikawa iterations fornonlinear quasi-contractive mappings, Indian J. Pure Appl. Math. 30(1999),425–433.[3] K.Deimling, Zeroes of accretive operators, Manuscripta Math. 13(1974), 365–374.[4] GuFeng, Iteration processes for approximating fixed points of operators ofmonotone type, Proc. Amer. Math. Soc. 129(2001), 2293–2300.[5] S. 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Ishikawa iterative process for strongly pseudocontractive operators in arbitrary Banach spaces

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