Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability

• Main Authors: Amar Benkerrouche, Mohammed Said Souid, Sina Etemad, Ali Hakem, Praveen Agarwal, Shahram Rezapour, Sotiris K. Ntouyas, Jessada Tariboon
• Format: Article
• Language: English
• Published: MDPI AG 2021-09-01
• Series: Fractal and Fractional
• Subjects: boundary value problem; Hadamard derivatives of variable order; piecewise constant functions; fixed point theorems
• Online Access: https://www.mdpi.com/2504-3110/5/3/108

In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example.