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Fixed points and best approximation in Menger convex metric spaces

Ismat Beg, Mujahid Abbas (2005)

Archivum Mathematicum

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.

Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces

Daya Ram Sahu (2005)

Commentationes Mathematicae Universitatis Carolinae

We introduce the classes of nearly contraction mappings and nearly asymptotically nonexpansive mappings. The class of nearly contraction mappings includes the class of contraction mappings, but the class of nearly asymptotically nonexpansive mappings contains the class of asymptotically nonexpansive mappings and is contained in the class of mappings of asymptotically nonexpansive type. We study the existence of fixed points and the structure of fixed point sets of mappings of these classes in Banach...

Fixed points of Lipschitzian semigroups in Banach spaces

Jarosław Górnicki (1997)

Studia Mathematica

We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If T = T s : C → C : s ∈ G = [ 0 , ∞ ) is a Lipschitzian semigroup such that g = l i m i n f G ∋ α → ∞ i n f G ∋ δ ≥ 0 1 / α ʃ 0 α ∥ T β + δ ∥ p d β < 1 + c , where c > 0 is some constant, then there exists x ∈ C such that T s x = x for all s ∈ G.

Fixed points of periodic and firmly lipschitzian mappings in Banach spaces

Krzysztof Pupka (2012)

Commentationes Mathematicae Universitatis Carolinae

W.A. Kirk in 1971 showed that if T : C → C , where C is a closed and convex subset of a Banach space, is n -periodic and uniformly k -lipschitzian mapping with k < k 0 ( n ) , then T has a fixed point. This result implies estimates of k 0 ( n ) for natural n ≥ 2 for the general class of k -lipschitzian mappings. In these cases, k 0 ( n ) are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of k -lipschitzian mappings. In the paper we show that k 0 ( 3 ) > 2 in any Banach space. We also...

Fixed points of periodic mappings in Hilbert spaces

Víctor García, Helga Nathansky (2010)

Annales UMCS, Mathematica

In this paper we give new estimates for the Lipschitz constants of n-periodic mappings in Hilbert spaces, in order to assure the existence of fixed points and retractions on the fixed point set.

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