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.

scholarly journals A note on Kakutani type fixed point theorems

2000 ◽  
Vol 24 (4) ◽  
pp. 231-235 ◽  
Author(s):  
A. R. Khan ◽  
N. Hussain ◽  
L. A. Khan
Keyword(s):  
Fixed Point ◽  
Convex Space ◽  
Fixed Point Theorems ◽  
Locally Convex Space ◽  
Space Setting ◽  
Locally Convex

We present Kakutani type fixed point theorems for certain semigroups of self maps by relaxing conditions on the underlying set, family of self maps, and the mappings themselves in a locally convex space setting.

FIXED POINT THEOREMS IN A LOCALLY CONVEX SPACE

1996 ◽  
Vol 19 (3-4) ◽  
pp. 505-515
Author(s):  
S. N. Mishra ◽  
S. L. Singh
Keyword(s):  
Fixed Point ◽  
Convex Space ◽  
Fixed Point Theorems ◽  
Locally Convex Space ◽  
Locally Convex

On Fixed Point Theorems for Mappings in a Separated Locally Convex Space

1972 ◽  
Vol 15 (4) ◽  
pp. 603-604 ◽  
Author(s):  
Cheng-Ming Lee
Keyword(s):  
Fixed Point ◽  
Convex Space ◽  
Fixed Point Theorems ◽  
Nonempty Subset ◽  
Locally Convex Space ◽  
Banach Contraction Principle ◽  
Locally Convex Spaces ◽  
Banach Contraction ◽  
Locally Convex ◽  
The Banach Contraction Principle

The Banach contraction principle has been generalized by Tan [6] to the mappings in separated locally convex spaces. We show that the result of Sehgal [5] and also of Holmes [3] can be generalized in the same way.Throughout this note, we let X be a separated locally convex space, U a base for the closed absolutely convex neighborhoods of the origin O in X, K a nonempty subset of X, and Ta mapping from K to K.

scholarly journals On some optimization problems in not necessarily locally convex space

2008 ◽  
Vol 18 (2) ◽  
pp. 167-172
Author(s):  
Ljiljana Gajic
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Convex Space ◽  
Optimization Problems ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Equilibrium Existence ◽  
Cooperative Equilibrium ◽  
Locally Convex

In this note, by using O. Hadzic's generalization of a fixed point theorem of Himmelberg, we prove a non - cooperative equilibrium existence theorem in non - compact settings and a generalization of an existence theorem for non - compact infinite optimization problems, all in not necessarily locally convex spaces.

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 J-nonexpansive mappings in uniform spaces and applications

1991 ◽  
Vol 43 (2) ◽  
pp. 331-339 ◽  
Author(s):  
Vasil G. Angelov
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Functional Differential Equations ◽  
Uniform Convexity ◽  
Existence Theory ◽  
Uniform Spaces ◽  
Geometric Properties ◽  
Functional Differential

The purpose of the paper is to introduce a class of “j-nonexpansive” mappings and to prove fixed point theorems for such mappings. They naturally arise in the existence theory of functional differential equations. These mappings act in spaces without specific geometric properties as, for instance, uniform convexity. Critical examples are given.

scholarly journals Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Kazimierz Włodarczyk ◽  
Robert Plebaniak
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Dynamic Systems ◽  
Metric Spaces ◽  
Lower Semicontinuous ◽  
Type Theorem ◽  
Uniform Spaces ◽  
Asymmetric Structures ◽  
Symmetric Structures ◽  
Locally Convex

In uniform spacesX, Dwith symmetric structures determined by theD-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by theJ-families of generalized pseudodistances onXare constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined byJ-families. Results are new also in locally convex and metric spaces. Examples are provided.

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