Abstract
Let be a real -uniformly smooth Banach space which is also uniformly convex (e.g., or spaces , and a nonempty closed convex subset of . By constructing nonexpansive mappings, we elicit the weak convergence of Mann's algorithm for a -strictly pseudocontractive mapping of Browder-Petryshyn type on in condition thet the control sequence is chosen so that (i) (ii) , where . Moreover, we consider to find a common fixed point of a finite family of strictly pseudocontractive mappings and consider the parallel and cyclic algorithms for solving this problem. We will prove the weak convergence of these algorithms.
Highlights
Let E be a real Banach space and let Jq q > 1 denote the generalized duality mapping fromE into 2E∗ given by Jq x {f ∈ E∗ : x, f x q and f x q−1}, where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing
We show that xn − Sμxn is decreasing
We show that the real sequence { xn − p }∞n 0 is decreasing, limn→∞ xn − p exists
Summary
Let E be a real Banach space and let Jq q > 1 denote the generalized duality mapping from. The sequence {xn} is generated by the Mann’s algorithm: xn 1 1 − αn xn αnT xn, 1.7 converges weakly to a fixed point of T. N x ∈ F Ti , i1 where N ≥ 1 is a positive integer and {Ti}Ni 1 are N strictly pseudocontractive mappings defined on a closed convex subset K of a real Banach space E which is q-uniformly smooth and uniformly convex. Under appropriate assumptions on the sequences of the weights {λin }Ni 1 we will prove the weak convergence, to a solution of the problem 1.9 , of the algorithm 1.11. 1 for weak convergence; 2 ωW xn {x : ∃xnj x} denotes the weak ω-limit set of {xn}
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