Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces
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In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1. Introduction Throughout this paper, we denote by N the set of positive integers and by R the set of real numbers. For an arbitrary set A, we also denote by A the cardinal number of A. Let C be a closed convex subset of a Banach space E. Let T be a nonexpansive mapping on C, that is, Tx − T y ≤ x − y
(1.1)
for all x, y ∈ C. We denote by F(T) the set of fixed points of T. We know F(T) = ∅ in the case that E is uniformly convex and C is bounded; see Browder [2], G¨ohde [9], and Kirk [13]. Common fixed point theorems for families of nonexpansive mappings are proved in [2, 4, 5], and other references. Many convergence theorems for nonexpansive mappings and families of nonexpansive mappings have been studied; see [1, 3, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21] and others. For example, in 1979, Ishikawa proved the following theorem. Theorem 1.1 [12]. Let C be a compact convex subset of a Banach space E. Let {T1 ,T2 ,...,Tk } be a finite family of commuting nonexpansive mappings on C. Let {βi }ki=1 be a finite sequence in (0,1) and put Si x = βi Ti x + (1 − βi )x for x ∈ C and i = 1,2,...,k. Let x1 ∈ C and define a sequence {xn } in C by
xn+1 =
n nk−1 =1
Sk
n k −1
Sk−1 · · · S3
nk−2 =1
n2 n1 =1
S2
n1
S1
···
x1
n0 =1
for n ∈ N. Then {xn } converges strongly to a common fixed point of {T1 ,T2 ,...,Tk }. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 103–123 DOI: 10.1155/FPTA.2005.103
(1.2)
104
Convergence to common fixed point
The author thinks this theorem is one of the most interesting convergence theorems for families of nonexpansive mappings. In the case of k = 4, this iteration scheme is as follows: x2 = S4 S3 S2 S1 x1 , x3 = S4 S3 S2 S1 S1 S2 S1 S3 S2 S1 x2 , x4 = S4 S3 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S2 S1 S3 S2 S1 x3 , x5 = S4 S3 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S2 S1 S3 S2 S1 x4 , x6 = S4 S3 S2 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1
(1.3)
S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S2 S1 S3 S2 S1 x5 , x7 = S4 S3 S2 S1 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S1 S1 S1 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S1 S1 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S1 S2 S1 S1 S2 S1 S3 S2 S1 S1 S2 S1 S3 S2 S1 x6 . We remark that Si S j = S j Si does not hold in general. Very recently, in 2002, the following theorem was proved in [19]. Theorem 1.2 [19]. Let C be a compact convex subset of a Banach space E and let S and T be nonexpansive mappings on C with ST = TS. Let x1 ∈ C and define a sequence {xn } in C by αn i j S T xn + 1 − αn xn n2 i=1 j =1 n
xn+1 =
n
(1.4)
for n ∈ N, where {αn } is
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