Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 14 2 2023 02 01 Existence of solution for a fractional differential equation via a new type of $(\psi, F)$-contraction in $b$-metric spaces 87 100 7132 10.22075/ijnaa.2022.23232.2500 EN Francis Akutsah School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa Akindele Adebayo Mebawondu School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa Abass Hammed Anuoluwapo DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa Kazeem Olalekan Aremu School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa Narain Ojen Kumar School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa Journal Article 2021 04 23 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, 2016, Article ID 1907676, 1--6) and $(\psi, F)$-contraction mappings introduced in ( \cite{wad1}, D. Wardowski, \emph{Solving existence problems via $F$-contractions,} Proceedings of the American Mathematical Society, 146 (4), (2018), 1585--1598), 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,wad1, wad} and several well-known results on fixed point theory and its applications. https://ijnaa.semnan.ac.ir/article_7132_3bf531b51d39a63eabef8256ea4fda6d.pdf