Some notes on convex F-contraction in b-metric spaces

In this paper, a generalized version of convex F-contraction, where F is not necessarily strictly increas-ing, in b-metric spaces is introduced. A weaker form of the continuity, called nearly continuity, is presented and it is also shown, by providing an example, that it is a real generalization of the continuity. A new version of the convex F-contraction, by comparing the old one, in order to establish an existence result of a fixed point for a self-mapping in the setting of b-metric spaces, has been offered. By using this version of the convex F-contraction, being a Cauchy sequence of the Pica′rd s iteration is stated which by applying this result an affirmative answer to the first question raised in [(MDPI) Axioms, DOI:10.3390/axioms10020071] is given. By suitable conditions and replacing the strictly increasing by the quasi-convexity of the function F, the existence and uniqueness of fixed points for self-mappings which are satisfied in F-Kannan type contraction is proved. The results of the article improve the main results in this area as well answer to the open problems raised in the paper [On Convex F-Contraction in b-Metric Spaces, (MDPI) Axioms, DOI:10.3390/axioms10020071]. Moreover, all the fixed point theorems are obtained for F-contraction map-pings whose F are strictly increasing instead of being quasi-convex.