scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

scholarly journals A fixed point theorem for a family of nonexpansive mappings

1976 ◽  
Vol 15 (1) ◽  
pp. 111-116 ◽  
Author(s):  
V.M. Sehgal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Vector Space ◽  
Recent Result ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Bounded Subset ◽  
The Family ◽  
Commutative Family ◽  
Locally Convex

Let E be a separated, locally convex topological vector space and F a commutative family of nonexpansive mappings defined on a quasi-complete convex (not necessarily bounded) subset X of E. In this paper, it is proved that if one of the mappings in F is condensing with a bounded range then the family F has a common fixed point in X. This result improves several well-known results and supplements a recent result of E. Tarafdar (Bull. Austral. Math. Soc. 13 (1975), 241–254) for such mappings.

scholarly journals Fixed point theorems for condensing multivalued mappings on a locally convex topological space

1975 ◽  
Vol 12 (2) ◽  
pp. 161-170 ◽  
Author(s):  
E. Tarafdar ◽  
R. Výborný
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theorems ◽  
Linear Topological Space ◽  
General Definition ◽  
Multivalued Mappings ◽  
Locally Convex

A general definition for a measure of nonprecompactness for bounded subsets of a locally convex linear topological space is given. Fixed point theorems for condensing multivalued mappings have been proved. These fixed point theorems are further generalizations of Kakutani's fixed point theorems.

scholarly journals Fixed point theorems for generalized nonexpansive mappings

1974 ◽  
Vol 18 (3) ◽  
pp. 265-276 ◽  
Author(s):  
Chi Song Wong
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Nonexpansive Mappings ◽  
Real Numbers ◽  
Image Position

Let S, T be self-mappings on a (non-empty) complete metric space (X, d). Let ai, i = 1, 2, …, 5, be non-negative real numbers such that < 1 and for any x, y in X,

An ergodic theorem for asymptotically nonexpansive mappings

1994 ◽  
Vol 124 (1) ◽  
pp. 23-31
Author(s):  
Manfred Krüppel ◽  
Jaroslaw Górnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Ergodic Theorem ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Image Position

The purpose of this paper is to prove the following (nonlinear) mean ergodic theorem: Let E be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E and let T: C → C be an asymptotically nonexpansive mapping. Ifexists uniformly in r = 0, 1, 2,…, then the sequence {Tnx} is strongly almost-convergent to a fixed point y of T, that is,uniformly in i = 0, 1, 2, ….

scholarly journals Fixed-point theorems for multivalued non-expansive mappings without uniform convexity

2003 ◽  
Vol 2003 (6) ◽  
pp. 375-386 ◽  
Author(s):  
T. Domínguez Benavides ◽  
P. Lorenzo Ramírez
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Contractive Mapping ◽  
Uniform Convexity ◽  
Opial Condition ◽  
The Family

LetXbe a Banach space whose characteristic of noncompact convexity is less than1and satisfies the nonstrict Opial condition. LetCbe a bounded closed convex subset ofX,KC(C)the family of all compact convex subsets ofC, andTa nonexpansive mapping fromCintoKC(C). We prove thatThas a fixed point. The nonstrict Opial condition can be removed if, in addition,Tis a1-χ-contractive mapping.

scholarly journals Fixed point theorems for nonexpansive mappings in a locally convex space

1979 ◽  
Vol 20 (2) ◽  
pp. 179-186 ◽  
Author(s):  
P. Srivastava ◽  
S.C. Srivastava
Keyword(s):  
Fixed Point ◽  
Convex Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Uniform Space ◽  
Normal Structure ◽  
Locally Convex Space ◽  
Uniform Spaces ◽  
Locally Convex

Several fixed point theorems for nonexpansive self mappings in metric spaces and in uniform spaces are known. In this context the concept of orbital diameters in a metric space was introduced by Belluce and Kirk. The concept of normal structure was utilized earlier by Brodskiĭ and Mil'man. In the present paper, both these concepts have been extended to obtain definitions of β-orbital diameter and β-normal structure in a uniform space having β as base for the uniformity. The closed symmetric neighbourhoods of zero in a locally convex space determine a base β of a compatible uniformity. For 3-nonexpansive self mappings of a locally convex space, fixed point theorems have been obtained using the concepts of β-orbital diameter and β-normal structure. These theorems generalise certain theorems of Belluce and Kirk.

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