Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces
- PDF / 168,292 Bytes
- 12 Pages / 468 x 680 pts Page_size
- 109 Downloads / 251 Views
We first introduce an iterative sequence for finding fixed points of relatively nonexpansive mappings in Banach spaces, and then prove weak and strong convergence theorems by using the notion of generalized projection. We apply these results to the convex feasibility problem and a proximal-type algorithm for monotone operators in Banach spaces. 1. Introduction Let E be a real Banach space and let A be a maximal monotone operator from E to E∗ , where E∗ is the dual space of E. It is well known that many problems in nonlinear analysis and optimization can be formulated as follows: find u∈E
such that 0 ∈ Au.
(1.1)
A well-known method for solving (1.1) in a Hilbert space H is the proximal point algorithm: x0 ∈ H and xn+1 = Jrn xn ,
n = 0,1,2,...,
(1.2)
where {rn } ⊂ (0, ∞) and Jr = (I + rA)−1 for all r > 0. This algorithm was first introduced by Martinet [9]. In [16], Rockafellar proved that if liminf n→∞ rn > 0 and A−1 0 = ∅, then the sequence {xn } defined by (1.2) converges weakly to an element of solutions of (1.1). On the other hand, Kamimura and Takahashi [4] considered an algorithm to generate a strong convergent sequence in a Hilbert space. Further, Kamimura and Takahashi’s result was extended to more general Banach spaces by Kohsaka and Takahashi [7]. They introduced and studied the following iteration sequence: x = x0 ∈ E and
xn+1 = J −1 αn Jx + 1 − αn JJrn xn ,
n = 0,1,2,...,
(1.3)
−1 J for all r > 0. Kohsaka and Takawhere J is the duality mapping on E and Jr = (J + rA) ∞ −1 hashi [7] proved that if A 0 = ∅, limn→∞ αn = 0, n=0 αn = ∞, and limn→∞ rn = ∞, then the sequence generated by (1.3) converges strongly to an element of A−1 0.
Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:1 (2004) 37–47 2000 Mathematics Subject Classification: 47H09, 47H05, 47J25 URL: http://dx.doi.org/10.1155/S1687182004310089
38
Weak and strong convergence theorems
On the other hand, Reich [13] studied an iteration sequence of nonexpansive mappings in a Banach space which was first introduced by Mann [8]: x0 ∈ C and
xn+1 = αn xn + 1 − αn Sxn ,
n = 0,1,2,...,
(1.4)
where S is a nonexpansive mapping from a closed convex subset C of E into itself and ∞ {αn } ⊂ [0,1]. He proved that if F(T) is nonempty and n=0 αn (1 − αn ) = ∞, then the sequence generated by (1.4) converges weakly to some fixed point of S. Motivated by Kohsaka and Takahashi [7], and Reich [13], our purpose in this paper is to prove weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces which were first introduced by Butnariu et al. [3] and further studied by the authors [10]. For this purpose, we consider the following iterative sequence: x0 ∈ C and
xn+1 = ΠC J −1 αn Jxn + 1 − αn JTxn ,
n = 0,1,2,...,
(1.5)
where T is a relatively nonexpansive mapping from C into itself and ΠC is the generalized projection onto C. Notice that if E is a Hilbert space and S = T, then the sequences (1.4) and (1.5) are equivalent. We prove that if F(T) is nonemp
Data Loading...
