Document Type : Special issue editorial

Authors

• Francis Akutsah ¹
• Akindele Adebayo Mebawondu ^{1, 2}
• Abass Hammed Anuoluwapo ²
• Kazeem Olalekan Aremu ¹
• Narain Ojen Kumar ¹

¹ School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

² DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa

Abstract

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.

Keywords

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