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Zbl 1181.65098
Chang, S.S.; Lee, H.W.J.; Chan, Chi Kin; Kim, J.K.
Approximating solutions of variational inequalities for asymptotically nonexpansive mappings. (English)
[J] Appl. Math. Comput. 212, No. 1, 51-59 (2009). ISSN 0096-3003

Let $E$ be a real Banach space with a uniformly Gâteaux differentiable norm and possessing a uniform normal structure. Iterative sequences are constructed which involve a contractive and an asymptotically nonexpanding mappings $K\to K$, where $K$ is a bounded closed convex subset of $E$. Conditions are given for convergence of these sequences to a fixed point which is also the unique solution of some variational inequalities. Thus previous results on asymptotically nonexpanding mappings are generalized [see e.g. {\it C.~Chidume, J.~Li} and {\it A.~Udomene}, Proc. Am. Math. Soc. 133, No.~2, 473--480 (2005; Zbl 1073.47059)].
[Mihail M. Konstantinov (Sofia)]

MSC 2000:: *65K15
49J40 Variational methods including variational inequalities

Keywords: nonexpansive mapping; fixed point; uniform normal structure; iscosity approximation; fixed point; uniform normal structure; uniformly Gâteaux differentiable norm; normalized duality mapping; Banach space; convergence; variational inequalities

Citations: Zbl 1073.47059

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