Abstract
Abstract Let K be a compact convex subset of a real Hilbert space H and T i : K → K , i = 1 , 2 , … , k , be a family of continuous hemicontractive mappings. Let α n , β n i ∈ [ 0 , 1 ] be such that α n + ∑ i = 1 k β n i = 1 and satisfying { α n } , β n i ⊂ [ δ , 1 − δ ] for some δ ∈ ( 0 , 1 ) , i = 1 , 2 , … , k . For arbitrary x 0 ∈ K , define the sequence { x n } by (1.9) see below, then { x n } converges strongly to a common fixed point in ⋂ i = 1 k F ( T i ) ≠ ∅ . MSC:05C38, 15A15, 05A15, 15A18.
Highlights
Tx – Ty ≤ x – y + (I – T)x – (I – T)y for all x, y ∈ H and T is said to be strongly pseudocontractive if there exists k ∈ (, ) such that
A mapping T : K → K is said to be hemicontractive if F(T) = ∅ and
It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings
Summary
If K is a compact convex subset of a Hilbert space H, T : K → K is a Lipschitzian pseudocontractive mapping and x is any point in K , the sequence {xn} converges strongly to a fixed point of T , where xn is defined iteratively by Another iteration process which has been studied extensively in connection with fixed points of pseudocontractive mappings is the following.
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