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A general iterative method for an infinite family of nonexpansive mappings

โœ Scribed by Yonghong Yao; Yeong-Cheng Liou; Rudong Chen

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

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โœฆ Synopsis

Let H be a real Hilbert space. Consider the iterative sequence

where ฮณ > 0 is some constant, f : H โ†’ H is a given contractive mapping, A is a strongly positive bounded linear operator on H and W n is the W -mapping generated by an infinite countable family of nonexpansive mappings T 1 , T 2 , . . . , T n , . . . and ฮป 1 , ฮป 2 , . . . , ฮป n , . . . such that the common fixed points set F := โˆž n=1 Fix(T n ) = โˆ…. Under very mild conditions on the parameters, we prove that {x n } converges strongly to p โˆˆ F where p is the unique solution in F of the following variational inequality:

which is the optimality condition for the minimization problem min xโˆˆF 1 2 Ax, xh(x).

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