Fractional differential equations of Riemann–Liouville of variable order with anti-periodic boundary conditions

Purpose The purpose of this paper is to investigate the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO). The study utilizes standard fixed point theorems (FiPoTh) to establish the existence and uniqueness of solutions. Additionally, the Ulam-Hyers-Rassias (Ul-HyRa) stability of the considered problem is examined. The obtained results are supported by an illustrative example. This research contributes to the understanding of fractional differential equations with variable order and anti-periodic boundary conditions, providing valuable insights for further studies in this field. Design/methodology/approach This paper (1) defines the Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO); (2) discusses the existence and uniqueness of solutions to these equations using standard FiPoTh; (3) investigates the stability of the considered problem using the Ul-HyRa stability concept (Ul-HyRa); (4) provides a detailed explanation of the design and methodology used to obtain the results and (5) supports the obtained results with a relevant example. Findings The authors confirm that no funds, grants or any other form of financial support were received during the preparation of this manuscript. Originality/value The originality/value of our paper lies in its contribution to the field of fractional differential equations. Specifically, we address the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order. By utilizing standard FiPoTh and investigating Ul-HyRa stability, we provide novel insights into this problem. The results obtained are supported by an example, further enhancing the credibility and applicability of your findings. Overall, our paper adds to the existing knowledge and understanding of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions, making it valuable to the scientific community.