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Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

✍ Scribed by C.O. Chidume; G. De Souza

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
234 KB
Volume
69
Category
Article
ISSN
0362-546X

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✦ Synopsis

Let E be a real reflexive Banach space with uniformly GΓ’teaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T : K β†’ K be a strictly pseudo-contractive map and let L > 0 denote its Lipschitz constant. Assume F(T ) := {x ∈ K : T x = x} = βˆ… and let z ∈ F(T ). Fix Ξ΄ ∈ (0, 1) and let Ξ΄ * be such that Ξ΄ * := Ξ΄L ∈ (0, 1). Define S n x := (1 -Ξ΄ n )x + Ξ΄ n T x βˆ€x ∈ K , where Ξ΄ n ∈ (0, 1) and lim Ξ΄ n = 0. Let {Ξ± n } be a real sequence in (0, 1) which satisfies the following conditions: C1 : lim Ξ± n = 0; C2 :

Then, {x n } converges strongly to a fixed point of T.

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