Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces

Rabian Wangkeeree

International Journal of Mathematics and Mathematical Sciences, 2007, vol. 2007, 1-10

Abstract:

Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H , P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H . For a contraction f on C and { t n } ⊆ ( 0 , 1 ) , let x n be the unique fixed point of the contraction x ↦ t n f ( x ) + ( 1 − t n ) (1 / n) ∑ j = 1 n ( P T ) j x . Consider also the iterative processes { y n } and { z n } generated by y n + 1 = α n f ( y n ) + ( 1 − α n ) (1 / (n + 1)) ∑ j = 0 n ( P T ) j y n , n ≥ 0 , and z n + 1 = (1 / (n + 1)) ∑ j = 0 n P ( α n f ( z n ) + ( 1 − α n ) ( T P ) j z n ) , n ≥ 0 , where y 0 , z 0 ∈ C , { α n } is a real sequence in an interval [ 0 , 1 ] . Strong convergence of the sequences { x n } , { y n } , and { z n } to a fixed point of T which solves some variational inequalities is obtained under certain appropriate conditions on the real sequences { α n } and { t n } .

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:048648

DOI: 10.1155/2007/48648

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