Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

Jui-Chi Huang

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-10

Abstract:

Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E . In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by { T j : C → C } j = 1 r as follows: U n ( j ) = a n ( j ) I + b n ( j ) T j n U n ( j − 1 ) + c n ( j ) u n ( j ) , j = 1 , 2 , … , r , x 1 ∈ C , x n + 1 = a n ( r ) x n + b n ( r ) T r n U n ( r − 1 ) x n + c n ( r ) u n ( r ) , n ≥ 1 , where U n ( 0 ) : = I , I the identity map; and { u n ( j ) } are bounded sequences in C ; and { a n ( j ) } , { b n ( j ) } , and { c n ( j ) } are suitable sequences in [ 0 , 1 ] . We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:416271

DOI: 10.1155/S0161171201007244

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