The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators

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1 Mathematical Communications 10(2005), The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators Ştefan M. Şoltuz Abstract. We show that the Ishikawa iteration, the corresponding Mann iteration and the Picard iteration are equivalent when applied to quasi-contractive operators. Key words: Picard iteration, Mann iteration, Ishikawa iteration, quasi-contractive operators AMS subject classifications: 47H10 Received March 22, 2005 Accepted June 9, Introduction Let X be a real Banach space, D a nonempty, convex subset of X, and T aselfmap of D, let x 0 = u 0 = p 0 D. The Mann iteration (see [3]) is defined by The Ishikawa iteration is defined (see [2]) by where {α n } (0, 1), {β n } [0, 1). The Picard iteration is given by u n+1 =(1 α n )u n + α n Tu n. (1) x n+1 =(1 α n )x n + α n Ty n, (2) y n =(1 β n )x n + β n Tx n, p n+1 = Tp n. (3) Definition 1. [4] The operator T : X X satisfies condition Z if and only if there exist real numbers a, b, c satisfying 0 <a<1, 0 <b,c<1/2 such that for each pair x, y in X, at least one condition is true (z 1 ) Tx Ty a x y, (z 2 ) Tx Ty b ( x Tx + y Ty ), T. Popoviciu Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, s: smsoltuz@gmail.com, soltuzul@yahoo.com, stefanmsoltuz@yahoo.com

2 82 Ş. M. Şoltuz (z 3 ) Tx Ty c ( x Ty + y Tx ). It is known, see Rhoades [5], that (z 1 ), (z 2 )and(z 3 ) are independent conditions. In [6] the following conjecture was given: if the Mann iteration converges, then so does the Ishikawa iteration. In a series of papers [6], [7], [8], [9], [10], Professor B. E. Rhoades and I have given a positive answer to this Conjecture, showing the equivalence between Mann and Ishikawa iterations for strongly and uniformly pseudocontractive maps. In this paper we show that the convergence of the Mann iteration is equivalent to the convergence of the Ishikawa iteration and both are equivalent to the Picard iteration, when applied to a map which satisfies condition Z. A map satisfying condition Z is independent, see Rhoades [4], of the class of strongly pseudocontractive maps. Lemma 1. [12] Let (a n ) n be a nonnegative sequence which satisfies the following inequality a n+1 (1 λ n )a n + σ n, (4) where λ n (0, 1), n n 0, n=1 λ n =, and σ n = o(λ n ). Then lim n a n =0. 2. Main result Let F (T ) denote the fixed point set with respect to D for the map T. Suppose that x F (T ). Theorem 1. Let X be a normed space, D a nonempty, convex, closed subset of X and T : D D an operator satisfying condition Z. If u 0 = x 0 D, then the following are equivalent: (i) the Mann iteration (1) converges to x, (ii) the Ishikawa iteration (2) converges to x. Proof. Consider x, y D. Since T satisfies condition Z, at least one of the conditions from (z 1 ), (z 2 )and(z 3 ) is satisfied. If (z 2 ) holds, then Tx Ty b ( x Tx + y Ty ) b ( x Tx +( y x + x Tx + Tx Ty )), thus (1 b) Tx Ty b x y +2b x Tx. From 0 b<1weget Tx Ty If (z 3 ) holds, then we obtain b 2b x y + x Tx. 1 b 1 b Tx Ty c ( x Ty + y Tx ) c ( x Tx + Tx Ty + x y + x Tx ),

3 Picard, Mann and Ishikawa iterations 83 hence Denote to obtain Finally, we get (1 c) Tx Ty c x y +2c x Tx i.e. Tx Ty c 2c x y + x Tx. 1 c 1 c { } b δ := max a, 1 b, c, 1 c 0 δ<1. Tx Ty δ x y +2δ x Tx, x, y D. (5) Formula (5) was obtained as in [1]. We will prove the implication (i) (ii). Suppose that lim n u n = x. Using lim n x n u n =0, (6) and 0 x x n u n x + x n u n we get lim x n = x. n The proof is complete if we prove relation (6). Using now (1), (2) and (5) with we have x := u n, y := y n, u n+1 x n+1 (1 α n )(u n x n )+α n (Tu n Ty n ) (7) (1 α n ) u n x n + α n Tu n Ty n (1 α n ) u n x n + α n δ u n y n +2α n δ u n Tu n. Using (5) with x := u n,y:= y n, we have u n y n (1 β n )(u n x n )+β n (u n Tx n ) (8) (1 β n ) u n x n + β n u n Tx n (1 β n ) u n x n + β n u n Tu n +β n Tu n Tx n (1 β n ) u n x n + β n u n Tu n +β n δ u n x n +2δβ n u n Tu n =(1 β n (1 δ)) u n x n +β n u n Tu n (1 + 2δ).

4 84 Ş. M. Şoltuz Relations (7) and (8) lead to Denote by u n+1 x n+1 (1 α n ) u n x n (9) +α n δ (1 β n (1 δ)) u n x n +α n β n δ u n Tu n (1 + 2δ) +α n δ u n y n =(1 α n (1 δ (1 β n (1 δ)))) u n x n +α n δ u n Tu n (β n (1 + 2δ)+2δ). a n := u n x n, λ n := α n (1 δ (1 β n (1 δ))) (0, 1), σ n := α n δ u n Tu n (β n (1 + 2δ)+2δ). Since lim n u n x =0,Tsatisfies condition Z, and x F (T ), from (5) we obtain 0 u n Tu n u n x + x Tu n (δ +1) u n x 0asn. Hence lim n u n Tu n =0;thatis,σ n = o (λ n ). Lemma 1 leads to lim n u n x n =0. We will prove now that if the Ishikawa iteration converges, then the Mann iteration does too. Using (5) with we obtain x := y n, y := u n, x n+1 u n+1 (1 α n )(x n u n )+α n (Ty n Tu n ) (10) (1 α n ) x n u n + α n Ty n Tu n (1 α n ) x n u n + α n δ y n u n +2α n δ y n Ty n. The following relation holds y n u n (1 β n )(x n u n )+β n (Tx n u n ) (11) (1 β n ) x n u n + β n Tx n u n (1 β n ) x n u n + β n Tx n x n +β n x n u n x n u n + β n Tx n x n.

5 Picard, Mann and Ishikawa iterations 85 Substituting (11) in (10), we obtain x n+1 u n+1 (1 α n ) x n u n + α n δ ( x n u n + β n Tx n x n ) (12) +2α n δ y n Ty n (1 (1 δ) α n ) x n u n + α n β n δ Tx n x n +2α n δ y n Ty n. Denote by a n := x n u n, λ n := α n (1 δ) (0, 1), σ n := α n β n δ Tx n x n +2α n δ y n Ty n. Since lim n x n x =0,Tsatisfies condition Z, and x F (T ), from (5) we obtain 0 x n Tx n x n x + x Tx n (δ +1) x n x 0asn, and 0 y n Ty n y n x + x Ty n (δ +1) y n x (δ +1) [(1 β n ) x n x + β n Tx n x ] (δ +1)[(1 β n ) x n x + β n δ x n x ] (δ +1)(1 β n (1 δ)) x n x 0asn, Hence lim n x n Tx n = 0 and lim n y n Ty n =0,thatis,σ n = o (λ n ). Lemma 1 and (12) lead to lim n x n u n = 0. Thus, we get x u n x n u n + x n x 0. Theorem 2. Let X be a normed space, D a nonempty, convex, closed subset of X and T : D D an operator satisfying condition Z. Let u 0 = p 0 D, then: (i) If the Mann iteration (1) converges to x and u n+1 u n lim =0, n α n then the Picard iteration (3) converges to x. (ii) If the Picard iteration (3) converges to x and p n+1 p n lim =0, n α n then the Mann iteration (1) converges to x. Proof. Suppose that the Mann iteration converges. We will prove that the Picard iteration converges, too. Relations (1) and (5) with x := u n, y := p n,

6 86 Ş. M. Şoltuz lead to u n+1 p n+1 (1 α n ) u n Tp n + α n Tu n Tp n (1 α n ) u n Tp n + α n δ u n p n +2α n δ Tu n u n =(1 α n ) u n p n+1 + α n δ u n p n +2α n δ Tu n u n (1 α n ) u n+1 p n+1 +(1 α n ) u n+1 u n +α n δ u n p n +2α n δ Tu n u n. Thus, we obtain α n u n+1 p n+1 (1 α n ) u n+1 u n +α n δ u n p n +2α n δ Tu n u n i.e. u n+1 p n+1 δ u n p n +2δ Tu n u n + (1 α n) u n+1 u n α n Set in Lemma 1: a n := u n p n, 1 λ := δ (0, 1), σ n := 2δ Tu n u n + (1 α n) u n+1 u n, α n to obtain lim n u n p n = 0. Hence, one get x p n u n p n + u n x 0. Suppose now that the Picard iteration converges. We prove that the Mann iteration converges as well. Using (5) with and the following we get x := p n, y := u n, u n Tp n u n p n + p n Tp n, u n+1 p n+1 (1 α n ) u n Tp n + α n Tp n Tu n (1 α n ) u n p n +(1 α n ) p n Tp n +α n δ p n u n +2α n δ Tp n p n =(1 (1 δ) α n ) u n p n +(1 α n ) p n Tp n +2α n δ Tp n p n (1 (1 δ) α n ) u n p n +(1+2α n ) p n p n+1 =(1 (1 δ) α n ) u n p n + o (α n ).

7 Picard, Mann and Ishikawa iterations 87 Set in Lemma 1: a n := u n p n, α n := (1 δ) α n (0, 1), n N, σ n := (1 + 2α n ) p n p n+1, to obtain lim n u n p n =0. Hence, one obtains x u n u n p n + p n x 0. Theorem 1 and Theorem 2 lead to the following Corollary: Corollary 1. Let X beanormedspace, D a nonempty, convex, closed subset of X and T : D D an operator satisfying condition Z. If u 0 = x 0 D, then the following are equivalent: (i) the Mann iteration (1) converges to x, (ii) the Ishikawa iteration (2) converges to x, (iii) the Picard iteration (3) converges to x. Theorem 1 generalizes Theorem 2 from [11]. In Theorem 2, the map T satisfies only condition (z 1 ): Theorem 3. [11] Let X beanormedspace,andb a nonempty convex subset of X. LetT : B B be a contraction with constant L (0, 1). Let x 0 = u 0 B. The following two assertions are equivalent: (i) the Mann iteration (u n ) n converges to x, (ii) the Ishikawa iteration (x n ) n converges to x. References [1] V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae LXXIII(2004), [2] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), [3] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4(1953), [4] B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196(1974), [5] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226(1977), [6] B. E. Rhoades, Ş. M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci. 7(2003), [7] B. E. Rhoades, Ş. M.Şoltuz, The equivalence of the Mann and Ishikawa iteration for non-lipschitzian operators, Int. J. Math. Math. Sci. 42(2003), [8] B. E. Rhoades, Ş. M. Şoltuz, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically pseudocontractive map, J. Math. Anal. Appl. 283(2003),

8 88 Ş. M. Şoltuz [9] B. E. Rhoades, Ş. M. Şoltuz, The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps, TamkangJ.Math.35(2004), [10] B. E. Rhoades, Ş. M. Şoltuz, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289(2004), [11] Ş. M. Şoltuz, An equivalence between the convergences of Ishikawa, Mann and Picard iterations, Math. Comm. 8(2003), [12] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113(1991),