Some Caristi type fixed point theorems

Abstract

In 1976, Caristi fixed point theorem was proved for ordinary metric spaces. After that, it was shown that this result is equivalent to Ekeland variational principle which has a great number of applications in many branches of mathematics. In this paper, we first introduce a new concept so called strong $$M_{b}$$ -metric to remedy lackness of continuity of $$M_{b}$$ -metric. Then, we investigate whether Caristi fixed point theorem can be extended to this space. Next, we obtain Caristi type fixed point theorem and some generalizations on strong $$M_{b}$$ -metric spaces. Also, we provide some illustrative and interesting examples showing that our theorems extend the results existing in the literature. Finally, we present some applications of our results to ordinary metric spaces.