Research Article Common Fixed Point Results on b-Metric Spaces for Generalized Rational Type ðϑ; ψ ; φÞ-Weakly Contractive Mappings With Applications Albray Gebremariam Kalo , 1 Kidane Koyas Tola , 2 and Haider Ebrahim Yesuf 1 1 Department of Mathematics, Arba Minch University, Arba Minch, Ethiopia 2 Department of Mathematics, Jimma University, Jimma, Ethiopia Correspondence should be addressed to Kidane Koyas Tola; kidane.tola@ju.edu.et Received 3 September 2024; Accepted 15 January 2025 Academic Editor: Christopher Goodrich Copyright © 2025 Albray Gebremariam Kalo et al. Abstract and Applied Analysis published by John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. This paper focuses on existence and uniqueness of common fixed points for a pair of self-mapping satisfying generalized rational type ðϑ; ψ ; φÞ-weak contractive condition in which one of the mapping is α-admissible with respect to the other and weakly compatible mappings in the framework of b-metric spaces. The results presented herein generalize and improve some well-known results in the existing literature. Furthermore, we draw some corollaries from our results and provide an example for illustrating the validity of our findings. As an application of our result, we discuss the existence of a solution to a fractional order differential equation. Keywords: α s -admissible mapping; b-metric spaces; coincidence point; common fixed point; fractional order differential equation; lower semi-continuity; rational type (ϑ, ψ, φ)-weak contractive condition; weakly compatible mappings 1. Introduction Fixed point theory is a very important tool for proving solu- tions to mathematical models such as integral equations, ordinary and partial differential equations, fractional differ- ential equations, and so on. The first significant fixed point result for contraction mappings was the well-known Banach- Contraction Mapping Theorem which was introduced by Banach [1] in 1922. Since then, researchers have obtained numerous results related to mappings satisfying various types of contractive inequalities. In this direction, Dass and Gupta [2] and Jaggi and Dass [3] obtained fixed point results by extending Banach’s Theorem through rational expressions in the setup of a metric space. Czerwik [4, 5] introduced the concept of b-metric space as a generalization of Banach’s contraction principle by mod- ifying the triangular inequality in [1]. The fixed point the- orems in b-metric space were further explored by several researchers. For some recent significant developments in this area, the reader refers to the work of Boriceanu, Bota, and Petrusel [6], Latif et al. [7], Aydi, Bota, and Moradi [8], Zada, Sarwar, and Kumam [9], Afshari and Karapınar [10], Shatanawi, Pitea, and Lazovic [11], Abbas, Chemac, and Razani [12], and the references therein. The concept of α-admissible and ðα - ψ Þ-contractive mappings introduced by Samet, Vetro, and Vetro [13]. Then, La Rosa and Vetro [14] presented the notion of f - α-admis- sible mappings and obtained common fixed point theorems for ðα; ψ ; ϕÞ-contractions in generalized metric spaces. In 2018, Zoto and Vardhami [15] proved common fixed point results for generalized α s p contractive mappings on the setting of b-metric-like spaces using the concept of g - α s p -admissi- ble mapping. Numerous authors improved and modified the α-admissiblity condition in the setting of various abstract spaces; see [16, 17] and the references therein. Wiley Abstract and Applied Analysis Volume 2025, Article ID 4630792, 22 pages https://doi.org/10.1155/aaa/4630792