Abstract
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.
Highlights
Introduction and preliminariesEkeland [17, 18] formulated a variational principle, which is considered the basis of modern calculus of variations
By getting motivation from above mentioned work, in present paper, we prove some generalizations of Ekeland’s variational principle in the setting of T -orbitally complete metric spaces for functions, which are not necessarily lower semicontinuous, by introducing T -orbitally lower semicontinuity
A variational principle is obtained in metric spaces, which are not necessarily complete, by introducing the notion of T -orbitally lower semicontinuous functions
Summary
Ekeland [17, 18] formulated a variational principle, which is considered the basis of modern calculus of variations. Ekeland’s variational principle has been widely used to prove the existence of approximate solutions of minimization problems for lower semicontinuous functions on c Vilnius University, 2019. There are many improvements and generalizations of Takahashi’s nonconvex minimization theorem, Caristi’s fixed point theorem and Ekeland’s variational principle in complete metric spaces by using generalized distances: for example, w-distances, τ -distances, τ -functions, weak τ -functions and Q-functions (see [2, 22, 30, 42, 44]). By getting motivation from above mentioned work, in present paper, we prove some generalizations of Ekeland’s variational principle in the setting of T -orbitally complete metric spaces for functions, which are not necessarily lower semicontinuous, by introducing T -orbitally lower semicontinuity. If φ : R+ → R+ is a comparison function, φ(t) < t for all t > 0
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