Analysis of Hilfer type coupled implicit (μ,σ)-order differential equations with Riemann–Liouville fractional integrable conditions via Topological degree method

Abstract

The article stated the class of $(\mu ,\sigma )$-order coupled implicit Hilfer fractional differential equations associated with integral type initial conditions, which is the generalization of Riaz and Zada (2021); Albidah (2023) and other well-known models, i.e. coupled competition species model, coupled oscillation model and coupled elastic beam model having transverse vibrations. The existence solution of the proposed coupled system can be found with the help of the Laplace transform, which is the fundamental method of differential equations. Banach contraction principle and the topological degree method will be used as tools for the uniqueness and at least one solution of the considered coupled problem. Stability of the problem is one of the important issues for the real wold problem. Stability in the sense of Hyers–Ulam and its types of the proposed coupled problem can be proved if the inequality F $>$ 0. For the illustration of results, an example will be presented, and the graphs of the system and its respective perturb system clearly show the differences when the order is changing.