scholarly journals Ekeland-type variational principle with applications to nonconvex minimization and equilibrium problems

2019 ◽  
Vol 24 (3) ◽  
pp. 407-432
Author(s):  
Iram Iqbal ◽  
Nawab Hussain
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Lower Semicontinuous ◽  
Minimax Theorem ◽  
Nonconvex Minimization ◽  
Equilibrium Theorem

The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

scholarly journals A minimization theorem in quasi-metric spaces and its applications

2002 ◽  
Vol 31 (7) ◽  
pp. 443-447 ◽  
Author(s):  
Jeong Sheok Ume
Keyword(s):  
Fixed Point ◽  
Variational Principle ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Caristi’S Fixed Point Theorem ◽  
Caristi's Fixed Point Theorem

We prove a new minimization theorem in quasi-metric spaces, which improves the results of Takahashi (1993). Further, this theorem is used to generalize Caristi's fixed point theorem and Ekeland'sϵ-variational principle.

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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