Fixed Point Theory and Algorithms for Sciences and Engineering Janardhanan et al. Fixed Point Theory Algorithms Sci Eng (2024) 2024:7 https://doi.org/10.1186/s13663-024-00763-4 RESEARCH Open Access Solution of a nonlinear fractional-order initial value problem via a C ∗ -algebra-valued R-metric space Gopinath Janardhanan 1 , Gunaseelan Mani 1 , Edwin Antony Raj Michael 2 , Sabri T.M. Thabet 3* and Imed Kedim 4 * Correspondence: th.sabri@yahoo.com 3 Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen Full list of author information is available at the end of the article Abstract In this article, we prove new common fixed-point theorems on a C ∗ -algebra-valued R-metric space. An example is given based on our obtained results. To enhance our results, a strong application based on the fractional-order initial value problem is provided. Mathematics Subject Classification: 47H10; 54H25; 46J10; 46J15 Keywords: Common fixed point; R-metric space; C ∗ -algebra; C ∗ -algebra-valued R-metric space 1 Introduction The concept of C ∗ -AVMS was outlined by Ma et al. in 2014, [1] and they proved some fixed-point results with a new contraction type. Many authors and researchers have gen- eralized with a new type of outcome (see [2–5]). Let B be the unital algebra with unit I . The conjugate linear map δ  → δ ∗ on B is such that δ ∗∗ = δ and (δη) ∗ = η ∗ δ ∗ for all δ, η ∈ B. The set of all bounded linear operators on a Hilbert space H, under the norm topology L(H), is a C ∗ -algebra. The concept of a cone metric space was outlined by Huang and Zhang in 2007 [6] and they replaced the set of real numbers by an ordered Banach space. The CFP for commuting mappings in metric space was investigated by Jungck in 1966 [7]. Likewise, many fixed and CFP results were obtained in different types like cone met- ric space [8], uniform space [9], noncommutative Banach space [10], fuzzy metric space [11] and so on. Hussain et al. proved Suzuki–Berinde-type fixed-point theorems and the CFP theorem on a cone b-metric space in these works [12, 13], respectively. Khalehoghli, Rahimi and Gordji introduced the R-metric space to prove the fixed-point theorem [14]. Wardowski proposed a new Banach contraction principle in a complete metric space to prove the fixed-point theorem [15]. Astha, Deepak and Choonkil proposed a C ∗ algebra- valued R-metric space to prove a unique fixed-point theorem [16]. Afshari and Khosh- vaghti proved a unique fixed-point theorem in an operator equation on the ordered Ba- nach space [17]. Afshari et al. [18], used a fixed-point theorem to study a boundary value © The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.