By using a Danes̆’ drop theorem in locally convex spaces we obtain a vectorial form of Ekelandtype variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk’s fixed-point theorem, Takahashi’s nonconvex minimization theorem, and Oettli-Théra’s theorem. Furthermore, we show that these results are equivalent to each other. Also, the existen...

In this paper, we generalize the renowned Ciric and Caristi type fixed point theorem some corollaries. Then give an example to illustrate our result is really better than theorem.

We prove two new fixed point theorems for generalized metric spaces and show that various fundamental fixed point principles, including: Banach Contraction Principle, Caristi fixed point theorem for metric spaces, Knaster-Tarski fixed point theorem for complete lattices, and the Bourbaki-Witt fixed point theorem for directed-complete orders, follow as corollaries of our results.

The well-known Caristi fixed point theorem has numerous generalizations and modifications. Recently there
 have appeared its equivalent dual forms based on new concept of lower semicontinuity
 from above by several authors. In the present article, we give proofs such versions their
 formulations applying our Metatheorem in ordered theory.

We introduce the new concepts of e-distance, e-type mapping with respect to some e-distance and S-complete quasimetric space, and prove minimization theorems, fixed point theorems, and variational principles on an S-complete quasimetric space. We also give some examples of quasimetrics, e-distances, and e-type mapping with respect to some e-distance. Our results extend, improve, and unify many ...

In 1996, Kada et al. 1 introduced the notion of w-distance. They elaborated, with the help of examples, that the concept of w-distance is general than that of metric on a nonempty set. They also proved a generalization of Caristi fixed point theorem employing the definition of w-distance on a complete metric space. Recently, Ilić and Rakočević 2 obtained fixed point and common fixed point theor...

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach Caristi Type Theorems b-Metric Spaces”. also characterize the completeness of b-metric spaces with help variant contractivity condition introduced authors aforementioned article.