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Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

✍ Scribed by H. Zegeye; E. Prempeh

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 665 KB
Volume: 44
Category: Article
ISSN: 0898-1221
DOI: 10.1016/s0898-1221(02)00152-9

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✦ Synopsis

Let E be a real q-uniformly smooth Banech space which is also uniformly convex (for example, L, or 1, spaces: 1 < p < 00) and K be a nonempty closed convex and bounded subset of E with 4 # int (K). Let T : K -+ K be a Lipschitzian pseudocontractive mapping such that for z E int(K), 1I.z -Trll < 112 -Tzll, f or all I E a(K). Then for zo E K arbitrary, the iteration process {z,,} defined by zn+l := (l-/~,,+l)z+~~+l~,,; vn := (1 -cu,)z, +cr,Tz, converges strongly to a fixed point of T, provided that {p,,} and {an} satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.

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