J. Borwein, S. Reich, and I. Shafrir. Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin, 35:21--28, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... mappings in the following setting: Let (X, X convex and f : C nonexpansive, i.e. #x, C(#f(x) f(y)# # #x y#) Let (# n ) n#IN be a sequence of real numbers in [0, 1) Then Mann iteration starting from x 0 : x C is defined as x n 1 : 1 # n )x n # n f(x n ) In [2], the following important result is proved: If (# n ) n#IN is divergent in sum and is bounded away from 1 then C(#x n r C (f) where r C (f) inf #x f(x)# C . In many cases, e.g. for bounded C, r C (f) can be shown to be 0, i.e 0 which (for bounded C) was first proved ....

.... of explicit bounds from large classes of prima facie ine#ective existence proofs together with procedures to transform such proofs into new ones from which these bounds can be read o# (see [9] 10] 11] and, for a general survey, 15] The proof given by Borwein, Reich and Shafrir in [2] of the result just cited happens to be of the required form. In [13] as a result of the logical transformation of the proof, a new quantitative version of the Borwein Reich Shafrir theorem was obtained. From this version, explicit uniform bounds for the case of bounded C could simply be read o#. ....

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Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992).

....for nding a xed point of an operator T : H H is governed by the recursion (8n 2 N) xn 1 = T xn ; 6) where xn denotes a convex combination of the points (x j ) 0 j n , say xn = j=0 n;j x j . Further work on this type of iterative process for certain types of operators was carried out in [4, 6, 11, 17, 18, 26, 31]. Most existing convergence results for the Mann iterates (6) require explicitly, e.g. 11, 15, 17, 31] or implicitly, e.g. 4, 6, 18, 26] that the averaging matrix A = n;j ] be segmenting, i.e. 8n 2 N) 8j 2 f0; ng) n 1;j = 1 n 1;n 1 ) n;j : 7) This property implies that ....

....points (x j ) 0 j n , say xn = j=0 n;j x j . Further work on this type of iterative process for certain types of operators was carried out in [4, 6, 11, 17, 18, 26, 31] Most existing convergence results for the Mann iterates (6) require explicitly, e.g. 11, 15, 17, 31] or implicitly, e.g. [4, 6, 18, 26], that the averaging matrix A = n;j ] be segmenting, i.e. 8n 2 N) 8j 2 f0; ng) n 1;j = 1 n 1;n 1 ) n;j : 7) This property implies that the points (x n ) n 0 generated in (6) satisfy xn 1 = n 1;n 1 xn 1 n 1;j x j = n 1;n 1 xn 1 (1 n 1;n 1 ) n;j x j = ....

J. Borwein, S. Reich, and I. Shafrir, Krasnoselski-Mann iterations in normed spaces, Canad. Math. Bull. 35 (1992), 1-28. 15

....n and if, 8i; r (y) r(y) F (y) Gamma y then r n = Deltax n and the choice (5) leads to r n Gamma r n = Gamma1) n = Gamma1) n . Such an iterative method falls into the framework of the method of Mann [50, 51] for which an extensive literature exists; see, for example, [2, 38]. We shall now examine the cases k = 1 and k = 2. 3.1 The LM and MW schemes When k = 1, that is for z n = Gamma Deltax n , and when r(y) r(y) F (y) Gamma y, the choice (4) or (5) leads to a method due to Lemar echal [48] denoted by LM in the sequel) F (x n ) Gamma x n ; F (F (x n ....

J. Borwein, S. Reich, I. Shafrir, Krasnoselski--Mann iterations in normed spaces, Canad. Math. Bull., 35 (1992) 21--28.

....we assume that (# k ) k#IN satisfies the following conditions: # k ) is divergent in sum, i.e. A) #n, i # IN#k # IN 0 i k X j=i # j # n 1 A . lim sup k## # k 1, i.e. B) #K, k 0 # IN#k # k 0 (# k # 1 1 K ) Theorem 10.11 (Borwein Reich Shafrir 1992,[16]) Let (X, ##) be a normed space, C # X convex, f : C # C nonexpansive and (# k ) k#IN be a sequence in [0, 1] which satisfies (A) B) above. Then for the Krasnoselski Mann iteration (x n ) starting from x # C one has #x n f(x n )# n## # r C (f) where r C (f) inf x#C #x ....

....and therefore do not ask for realizing functionals. By contrast (B) asks for a witnessing K 4 and (A) even for a witness function # which provides a k for given n, i. As we will see now these two additional inputs K and # play a most crucial role: Logical analysis of the proof given in [16] (which for the case of bounded C is basically identical with the proof in [42] yields the following quantitative version of theorem 10.11: Theorem 10.13 ( 85] X, # #) normed space, C # X convex and f : C # C a nonexpansive. Let (# k ) k#IN be a sequence in [0, 1] which is divergent in ....

Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992).

.... of rational numbers with fixed rate of convergence) is purely existential which is crucial for the possibility to extract an algorithm for n in (#) uniformly in x 0 and f (if X,C have a computable representation) We consider strong generalizations of Krasnoselski s result due to [11] 5] 7] and [3]. In [11] it is shown that Krasnoselski s fixed point theorem even holds without the assumption of X being uniformly convex. Even much more general so called Krasnoselski Mann iterations x k 1 : 1 # k )x k # k f(x k ) are allowed, where # k is a sequence in [0, 1] which is divergent in ....

....in sum and satisfies lim sup k## # k 1. In particular, it is proved in [11] that for such iterations (I) lim k## #x k f(x k )# = 0, where X is an arbitrary normed linear space, C a bounded convex subset of X and f : C # C is nonexpansive. This result is further generalized in [3] to the case where C no longer is required to be bounded. Then one has (II) lim k## #x k f(x k )# = r C (f) where r C (f) inf x#C #x f(x)# will in general be strictly positive. We give a complete quantitative analysis of (II) see theorem 2.7) as an instance of our general results ....

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Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992).

....T of C is called firmly nonexpansive, if kTx Gamma Tyk 2 hx Gamma y; Tx Gamma Tyi, for all x; y 2 C; see [9, 14, 16] since Genel and Lindenstrauss [8] supplied an example where (xn ) converges only weakly. Iteration (1) is then of type Krasnoselski Mann; see Borwein et al. s [4] for more. If each nonexpansive mapping is a projection onto a closed convex nonempty set, then algorithm (1) becomes the well known method of cyclic projections. However, even for N = 2 it is still not known whether or not convergence of the sequence (xn ) produced by the method of cyclic ....

J.M. BORWEIN, S. REICH, and I. SHAFRIR. Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin, 35(1):21--28, 1992. 8

....iteration was studied in Hilbert space. Some authors then implicitly used properties of Fej er monotone sequences; see [72, Proof of Theorem 1] and [87, Proof of Theorem 2] However, tremendous progress has been made and today the iteration is studied in normed or even more general spaces (see [15, 50] for further information) Example 2.20 (Example 2.14 continued) The sequence (T n x 0 ) converges weakly to some fixed point of T for every x 0 . Proof. T n x 0 ) is asymptotically regular (Example 2.14) and Fej er monotone w.r.t. Fix T (Example 2.18) By the Demiclosedness Principle, ....

J.M. BORWEIN, S. REICH, and I. SHAFRIR. Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin, 35(1):21--28, 1992.

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J. Borwein, S. Reich, and I. Shafrir. Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin, 35:21--28, 1992.

No context found.

Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992).

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J. Borwein, S. Reich, and I. Shafrir, Krasnoselski-Mann iterations in normed spaces, Canadian Mathematical Bulletin, vol. 35 (1992), pp. 21--28.

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Borwein, J., S. Reich, I. Shafrir (1992). Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35 pp 21-28.

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