scholarly journals Weak KKM set-valued mappings in hyperconvex metric spaces

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 157-167
Author(s):  
Ravi Agarwal ◽  
Mircea Balaj ◽  
Donal O’Regan
Keyword(s):  
Metric Spaces ◽  
Variational Inequality Problem ◽  
Strong Solutions ◽  
Vector Spaces ◽  
Minimax Inequalities ◽  
Topological Vector Spaces ◽  
Set Valued Mapping ◽  
General Variational Inequality ◽  
General Variational Inequality Problem ◽  
Existence Criteria

In this paper, the concept of weak KKM set-valued mapping is extended from topological vector spaces to hyperconvex metric spaces. For these mappings we obtain several intersection theorems that prove to be useful in establishing existence criteria for weak and strong solutions of the general variational inequality problem and minimax inequalities.

scholarly journals Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Optimization Problems ◽  
Time Function ◽  
Vector Spaces ◽  
Directional Derivatives ◽  
Minimal Time Function ◽  
Topological Vector Spaces ◽  
Minimal Time ◽  
Set Valued Mapping ◽  
Bounded Set ◽  
Normed Vector

For a set-valued mappingMdefined between two Hausdorff topological vector spacesEandFand with closed convex graph and for a given point(x,y)∈E×F, we study the minimal time function associated with the images ofMand a bounded setΩ⊂Fdefined by𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of𝒯M,Ωat those points of(x,y)∈E×F(both cases: points in the graphgph Mand points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph ofMat points insidegph Mand to the graph of the enlargement set-valued mapping at points outsidegph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function𝒯M,Ωto the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

scholarly journals The Hierarchical Minimax Inequalities for Set-Valued Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yen-Cherng Lin ◽  
Chin-Tzong Pang
Keyword(s):  
Equilibrium Problems ◽  
Vector Spaces ◽  
Minimax Inequalities ◽  
Topological Vector Spaces ◽  
Set Valued Mappings

We study the minimax inequalities for set-valued mappings with hierarchical process and propose two versions of minimax inequalities in topological vector spaces settings. As applications, we discuss the existent results of solutions for set equilibrium problems. Some examples are given to illustrate the established results.

scholarly journals The Fuzzification of Classical Structures: A General View

2015 ◽  
Vol 10 (6) ◽  
pp. 12 ◽  
Author(s):  
Ioan Dzitac
Keyword(s):  
Metric Spaces ◽  
50Th Anniversary ◽  
Vector Spaces ◽  
Fuzzy Mathematics ◽  
Future Research ◽  
Topological Spaces ◽  
General View ◽  
Linear Spaces ◽  
Topological Vector Spaces ◽  
Fuzzy Topological Spaces

The aim of this survey article, dedicated to the 50th anniversary of Zadeh’s pioneering paper "Fuzzy Sets" (1965), is to offer a unitary view to some important spaces in fuzzy mathematics: fuzzy real line, fuzzy topological spaces, fuzzy metric spaces, fuzzy topological vector spaces, fuzzy normed linear spaces. We believe that this paper will be a support for future research in this field.

scholarly journals Contractive mappings on a premetric space

1985 ◽  
Vol 8 (3) ◽  
pp. 469-475
Author(s):  
C. Y. Shen ◽  
A. Sound
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Property ◽  
Vector Spaces ◽  
Point Property ◽  
Topological Vector Spaces ◽  
Contractive Mappings

In this paper, we study the fixed point property of certain types of contractive mappings defined on a premetric space. The applications of these results to topological vector spaces and to metric spaces are also discussed.

scholarly journals Existence Results for Generalized Vector Equilibrium Problems with Multivalued Mappings via KKM Theory

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
A. P. Farajzadeh ◽  
A. Amini-Harandi ◽  
D. O'Regan
Keyword(s):  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Vector Spaces ◽  
Multivalued Mappings ◽  
Vector Equilibrium Problems ◽  
Solution Sets ◽  
Topological Vector Spaces ◽  
Set Valued Mapping ◽  
Generalized Vector Equilibrium Problems ◽  
Vector Equilibrium

We first define upper sign continuity for a set-valued mapping and then we consider two types of generalized vector equilibrium problems in topological vector spaces and provide sufficient conditions under which the solution sets are nonempty and compact. Finally, we give an application of our main results. The paper generalizes and improves results obtained by Fang and Huang in (2005).

scholarly journals ON PSEUDOMONOTONE SET-VALUED MAPPINGS IN TOPOLOGICAL VECTOR SPACES

2008 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
A. P. FARAJZADEH
Keyword(s):  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Vector Spaces ◽  
Inequality Problem ◽  
Topological Vector Spaces ◽  
Set Valued Mappings

AbstractIn this paper we extend results of Inoan and Kolumban on pseudomonotone set-valued mappings to topological vector spaces. An application is made to a variational inequality problem.

scholarly journals w-b-Cone Distance and Its Related Results: A Survey

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 171 ◽  
Author(s):  
Reza Babaei ◽  
Hamidreza Rahimi ◽  
Manuel De la Sen ◽  
Ghasem Soleimani Rad
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Vector Spaces ◽  
Minkowski Functionals ◽  
Topological Vector Spaces

In this work, we define the concept of a w-b-cone distance in t v s -cone b-metric spaces which differs from generalized c-distance in cone b-metric spaces, and we discuss its properties. Our results are significant, since all of the results in fixed point theory with respect to a generalized c-distance can be introduced in the version of w-b-cone distance. Moreover, using Minkowski functionals in topological vector spaces, we prove the equivalence between some fixed point results with respect to a w t -distance in general b-metric spaces and a w-b-cone distance in t v s -cone b-metric spaces.

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