In the present paper, considering the simulation function, we give a new class of Picard operators on complete metric spaces. We also provide a nontrivial example that shows the aforementioned class properly contains some earlier such classes.

In this paper, taking into account two metrics on a space, we present a new fixed point theorem for F-contractions. Our theorem includes both Agarwal and O?Regan?s andWardowski?s results as properly. Also we provide a nontrivial example showing this fact.

Recently, Wardowski [?] introduced the concept of F-contraction on complete metric space which is a proper generalization of Banach contraction principle. In the present paper, we proved a related fixed point theorem with F-contraction mappings on two complete metric spaces.

In this paper, by considering the Wardowski's technique, we present fixed point results for multivalued mapping on a space with two metrics. Also, taking into account α-admissibility of a multivalued mapping, we provide some more general results.