Existence of solution for a fractional differential equation via a new type of (\psi, F)-contraction in b-metric spaces

In this paper, we further develop the notion of cyclic (\alpha, \beta)-admissible mappings introduced in (\cite{tac}, S. Chandok, K. Tas, A. H. Ansari, \emph{Some fixed point results for TAC-type contractive mappings,} J. Function spaces, ۲۰۱۶, Article ID ۱۹۰۷۶۷۶, ۱--۶) and (\psi, F)-contraction mappings introduced in ( \cite{wad۱}, D. Wardowski, \emph{Solving existence problems via F-contractions,} Proceedings of the American Mathematical Society, ۱۴۶ (۴), (۲۰۱۸), ۱۵۸۵--۱۵۹۸), in the framework of b-metric spaces. To achieve this, we introduce the notion of (\alpha,\beta)-S-admissible mappings and a new class of generalized (\psi, F)-contraction types and establish a common fixed point and fixed point results for these classes of mappings in the framework of complete b-metric spaces. As an application, we establish the existence and uniqueness of the solutions to differential equations in the framework of fractional derivatives involving Mittag-Leffler kernels via the fixed point technique. The results obtained in this work provide extension as well as substantial generalization and improvement of the fixed point results obtained in \cite{tac,wad۱, wad} and several well-known results on fixed point theory and its applications.

Keywords:

Fixed point , (alpha , \beta)-S-admissible mappings , Generalized (\psi , F)-contraction , b-metric space , Differential equation