We give a correction to the main result from [13]

Ş. M. Şolutuz

Şolutuz, Ş. M.

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Mathematical Communications 7(2002), 61-64 61A correction for a result on convergence ofIshikawa iteration for strongly pseudocontractivemapsS¸tefan M. S¸oltuz∗Abstract. We give a correction to the main result from [13].Key words: Ishikawa iteration, strongly pseudocontractive mapsAMS subject classifications: Primary 47H10, Secondary 47H06Received December 11, 2001 Accepted June 27, 20021. IntroductionLet X be a real Banach space. Let B be a nonempty, convex subset of X . LetT : B → B be a map. Let x1 ∈ B. We consider the following iteration, see [4]:xn+1 = (1− αn)xn + αnTyn, (I)yn = (1− βn)xn + βnTxn, n = 1, 2, ... .We suppose that (αn)n, (βn)n ⊂ (0, 1), and the sequence (αn)n satisfies0 < w ≤ αn ≤ 1. (1)For βn = 0, ∀n ∈ N we get Mann iteration, see [5]. Ishikawa iteration with condition(1) is studied in [13]. In [7] it was proven that two assumptions of the main theoremfrom [13] are contradictory. In this note we will prove that renouncing to oneassumption from [13] and supposing true an assumption a` la [3], the above theoremfrom [13] is true.The map J : X → 2X∗ given byJx := {f ∈ X∗ : 〈x, f〉 = ‖x‖2 , ‖f‖ = ‖x‖}, ∀x ∈ X,is called the normalized duality mapping. The Hahn-Banach theorem assures thatJx = ∅, ∀x ∈ X. It is easy to see that we have〈j(x), y〉 ≤ ‖x‖ ‖y‖ , ∀x, y ∈ X, ∀j(x) ∈ J(x). (2)∗S¸tefan M. S¸oltuz, Kurt Schumacher Str. 48, Ap. 38, 67663 Kaiserslautern, Germany, e-mail:ssoltuz@yahoo.com, soltuz@itwm.fhg.de62 S¸.M. S¸oltuzDefinition 1. Let X be a real Banach space, let B be a nonempty subset. Amap T : B → B is called strongly pseudocontractive if for all x, y ∈ B, there existsj(x− y) ∈ J(x − y) such that∃γ ∈ (0, 1) : 〈Tx− Ty, j(x− y)〉 ≤ γ ‖x− y‖2 . (3)The following Lemma could be found in [6], [12], with different proofs. A par-ticular form of this lemma is in [11].Lemma 1. [6], [11], [12] If X is a real Banach space, then the following relationis true‖x+ y‖2 ≤ ‖x‖2 + 2 〈y, j(x+ y)〉 , ∀x, y ∈ X, ∀j(x+ y) ∈ J(x+ y). (4)The following result is from [9]. Three other proofs could be found in [10].Proposition 1. [9], [10].Let (an)n be a nonnegative sequence which satisfiesan+1 ≤ (1− w)an + σnS, (5)where w ∈ (0, 1) , S > 0 are fixed numbers, σn ≥ 0, ∀n ∈ N, limn→∞ σn = 0. Thenlimn→∞ an = 0.2. Main resultWe are able now to give the following result.Theorem 1. Let X be a real Banach space, and let T : X → X be a continuous,strongly pseudocontractive with bounded range map. If limn→∞ ‖Tyn − Txn+1‖ = 0,and γ ∈ (0, 1/2), then the iteration (I) strongly converges to the unique fixed pointof T.Proof. . The existence follows from [2] and the uniqueness from strongly pseudo-contractivity. Let x∗ = Tx∗. Using (4) for the first inequality, (2) and (3) for thethird one, we can see:‖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∗)〉≤ (1 − αn)2 ‖xn − x∗‖2 + 2αn ‖Tyn − Txn+1‖ ‖xn+1 − x∗‖+ 2αnγ ‖xn+1 − x∗‖2 , ∀j(xn+1 − x) ∈ J(xn+1 − x).There results(1− 2αnγ) ‖xn+1 − x∗‖2 ≤ (1 − αn)2 ‖xn − x∗‖2 + 2αn ‖Tyn − Txn+1‖ ‖xn+1 − x∗‖ ,‖xn+1 − x∗‖2 ≤ (1− αn)2(1− 2αnγ) ‖xn − x∗‖2 + 2αn(1− 2αnγ) ‖Tyn − Txn+1‖ ‖xn+1 − x∗‖ .A correction for a result on Ishikawa iteration 63Because γ ∈ (0, 1/2), αn ∈ (0, 1)⇒ 2(1−γ)−αn1−2αnγ ≥ 1 i.e. −(2(1−γ)−αn1−2αnγ)≤ −1, wehave(1 − αn)2(1 − 2αnγ) =1− 2αn + αn2(1 − 2αnγ) =((1− 2αnγ) + 2αnγ − 2αn + α2n)(1 − 2αnγ) == 1−(2(1− γ)− αn1− 2αnγ)αn ≤ 1− αn. (6)Also, the sequence (xn) is bounded. We will prove that by induction. Let us denoteby d := sup{‖Tx‖ : x ∈ B} + ‖x∗‖ . Because the range of T is bounded we haved <∞. We denote by M := d+ ‖x0 − x∗‖+ 1. Observe that‖x1 − x∗‖ ≤ (1 − α0) ‖x0 − x∗‖+ α0 ‖Ty0 − x∗‖≤ (1 − α0)M + α0(‖Ty0‖+ ‖x∗‖) ≤ (1− α0)M + α0M = M.Supposing ‖xn − x∗‖ ≤M, we will prove that ‖xn+1 − x∗‖ ≤M. Indeed we have‖xn+1 − x∗‖ ≤ (1− αn) ‖xn − x∗‖+ αn ‖Tyn − x∗‖≤ (1− αn)M + αn(‖Tyn‖+ ‖x∗‖) ≤ (1− αn)M + αnM = M.Thus we have∃M > 0 : ‖xn+1 − x∗‖ ≤M, ∀n ≥ 0. (7)Conditions (6), (7) and (8) lead us to‖xn+1 − x∗‖2 ≤ (1− αn) ‖xn − x∗‖2 + ‖Tyn − Txn+1‖ 2αn(1 − 2αnγ)M.But (1− αn) ≤ (1− w) , and 2αn(1−2αnγ) ≤ 21−2γ . So, we have‖xn+1 − x∗‖2 ≤ (1− w) ‖xn − x∗‖2 + ‖Tyn − Txn+1‖ 21− 2γM.Let us denote be an := ‖xn − x∗‖2 , σn := ‖Tyn − Txn+1‖ , and S := 21−2γM. Thenwe have limn→∞ an = 0. Thus limn→∞ xn = x∗. ✷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] K.Deimling, Zeroes of Accretive Operators, Manuscripta Math. 13(1974),365-374.[3] GuFeng, Iteration processes for approximating fixed points of operators ofmonotone type, Proc. Amer. Math. Soc. 129(2001), 2293-2300.64 S¸.M. S¸oltuz[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc.44(1974), 147-150.[5] W.R.Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4(1953), 506-510.[6] C.Morales, J. S. Jung, Convergence of paths for pseudocontractive map-pings in Banach spaces, Proc. Amer. Math. Soc. 128:11(2000), 3411-3419.[7] M.O.Osilike, A note on the stability of iteration procedures for stronglypseudocontractions and strongly accretive type equations, J. Math. Anal. Appl.250(2000), 726-730.[8] J.A. Park, Mann-iteration for strictly pseudocontractive maps, J. KoreanMath. Soc. 31(1994), 333-337.[9] S¸.M. S¸oltuz, Some sequences supplied by inequalities and their applications,Revue d’analyse nume´rique et de the´orie de l’approximation, Tome 29(2000),207-212.[10] S¸.M. S¸oltuz, Three proofs for the convergence of a sequence, OCTOGONMath. Mag. 9(2001), 503-505.[11] H.Y.Zhou, Y. Jia, Approximation of fixed points of strongly pseudocontrac-tive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997),1705-1709.[12] H.Y.Zhou, Y. J. Cho, Ishikawa and Mann iterative process with errors fornonlinear φ− strongly quasi-accretive mappings in normed spaces, J. KoreanMath. Soc. 36(1999), 1061-1073.[13] H.Y.Zhou, Stable iteration procedures for strong pseudocontractions and non-linear equations involving accretive operators without Lipschitz assumption, J.Math. Anal. Appl. 230(1999), 1-30.

A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps

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