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

In this paper, we further develop the notion of cyclic (α, β)-admissible mappings introduced in ([14], S. Chandok, K. Tas, A. H. Ansari, Some fixed point results for TAC-type contractive mappings, J. Function spaces, 2016, Article ID 1907676, 1–6) and (ψ, F)-contraction mappings introduced in ([34], D. Wardowski, 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 (α, β) − S-admissible mappings and a new class of generalized (ψ, 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 [14, 34, 35] and several well-known results on fixed point theory and its applications.