Approximating zero points of accretive operators with compact domains in general Banach spaces
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We prove strong convergence theorems of Mann’s type and Halpern’s type for resolvents of accretive operators with compact domains and apply these results to find fixed points of nonexpansive mappings in Banach spaces. 1. Introduction Let E be a real Banach space, let C be a closed convex subset of E, let T be a nonexpansive mapping of C into itself, that is, Tx − T y ≤ x − y for each x, y ∈ C, and let A ⊂ E × E be an accretive operator. For r > 0, we denote by Jr the resolvent of A, that is, Jr = (I + rA)−1 . The problem of finding a solution u ∈ E such that 0 ∈ Au has been investigated by many authors; for example, see [3, 4, 7, 16, 26]. We know the proximal point algorithm based on a notion of resolvents of accretive operators. This algorithm generates a sequence {xn } in E such that x1 = x ∈ E and xn+1 = Jrn xn
for n = 1,2,...,
(1.1)
where {rn } is a sequence in (0, ∞). Rockafellar [18] studied the weak convergence of the sequence generated by (1.1) in a Hilbert space; see also the original works of Martinet [12, 13]. On the other hand, Mann [11] introduced the following iterative scheme for finding a fixed point of a nonexpansive mapping T in a Banach space: x1 = x ∈ C and
xn+1 = αn xn + 1 − αn Txn
for n = 1,2,...,
(1.2)
where {αn } is a sequence in [0,1], and studied the weak convergence of the sequence generated by (1.2). Reich [17] also studied the following iterative scheme for finding a fixed point of a nonexpansive mapping T : x1 = x ∈ C and
xn+1 = αn x + 1 − αn Txn Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 93–102 DOI: 10.1155/FPTA.2005.93
for n = 1,2,... ,
(1.3)
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Approximating zero points of accretive operators
where {αn } is a sequence in [0,1]; see the original work of Halpern [6]. Wittmann [27] showed that the sequence generated by (1.3) in a Hilbert space converges strongly to the point of F(T), the set of fixed points of T, which is the nearest to x if {αn } satisfies ∞ limn→∞ αn = 0, ∞ α = ∞ , and n=1 n n=1 |αn+1 − αn | < ∞. Since then, many authors have studied the iterative schemes of Mann’s type and Halpern’s type for nonexpansive mappings and families of various mappings; for example, see [1, 2, 19, 20, 21, 22, 23, 24, 14, 15]. Motivated by two iterative schemes of Mann’s type and Halpern’s type, Kamimura and Takahashi [8, 9] introduced the following iterative schemes for finding zero points of m-accretive operators in a uniformly convex Banach space: x1 = x ∈ E and
xn+1 = αn x + 1 − αn Jrn xn
xn+1 = αn xn + 1 − αn Jrn xn
for n = 1,2,..., for n = 1,2,...,
(1.4)
where {αn } is a sequence in [0,1] and {rn } is a sequence in (0, ∞). They studied the strong and weak convergence of the sequences generated by (1.4). Such iterative schemes for accretive operators with compact domains in a strictly convex Banach space have also been studied by Kohsaka and Takahashi [10]. In this paper, we first deal with the strong convergence of resolvents of accretive operators defined in compact sets of smooth
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