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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.

Fixed points of set-valued maps with closed proximally ∞-connected values

Grzegorz Gabor (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Introduction Many authors have developed the topological degree theory and the fixed point theory for set-valued maps using homological techniques (see for example [19, 28, 27, 16]). Lately, an elementary technique of single-valued approximation (on the graph) (see [11, 1, 13, 5, 9, 2, 6, 7]) has been used in constructing the fixed point index for set-valued maps with compact values (see [21, 20, 4]). In [20, 4] authors consider set-valued upper semicontinuous...

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