Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. (English) Zbl 0865.47040

Lin, Pei-Kee; Tan, Kok-Keong; Xu, Hong Kun

Nonlinear Anal., Theory Methods Appl. 24, No. 6, 929-946 (1995).

The demiclosedness principle of F. E. Browder [Bull. Am. Math. Soc. 74, 660-665 (1968; Zbl 0164.44801)] states that if \(X\) is a uniformly convex Banach space, if \(C\) is a nonempty closed convex subset of \(X\), and if \(T:C\to X\) is a nonexpansive map, then \(I-T\) is demiclosed at each \(y\) in \(X\). In this paper, authors prove this principle at zero for asymptotically nonexpansive maps either in a Banach space with the locally uniform Opial condition or in a nearly uniformly convex Banach space with Opial’s condition. They also study the asymptotic behavior of the iterates for an asymptotically nonexpansive map. Finally, they prove that the uniform Opial condition in a Banach space \(Y\) implies the fixed point property for asymptotically nonexpansive maps defined on weakly compact convex subsets of \(Y\).

Reviewer: S.L.Singh (Rishikesh)

MSC:

47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B20	Geometry and structure of normed linear spaces
47H10	Fixed-point theorems

Keywords:

Opial modulus; Bynum’s weakly convergent sequence coefficient; Lipschitzian semigroup; Banach spaces; demiclosedness principle; uniformly convex Banach space; asymptotically nonexpansive maps; locally uniform Opial condition; nearly uniformly convex Banach space; fixed point property; weakly compact convex subsets

Citations:

Zbl 0164.44801

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