A coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations in the framework of nonlocal and nonsingular derivatives

• Published in: Advances in difference equations Vol. 2020; no. 1; pp. 1 - 30
• Main Authors: Baleanu, D., Alzabut, J., Jonnalagadda, J. M., Adjabi, Y., Matar, M. M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 27.05.2020; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Derivatives; Difference and Functional Equations; Differential equations; Existence; Existence theorems; Fixed point theorems; Fixed points (mathematics); Functional Analysis; Integral equations; Langevin equation; Mathematical analysis; Mathematics; Mathematics and Statistics; Non-singular fractional derivatives; Nonlinear equations; Nonlinear systems; Ordinary Differential Equations; Partial Differential Equations; Solutions dependence; Sturm–Liouville problem; 34C15; 34B15; Stability; 34B24; Langevin equation; 26A33; Non-singular fractional derivatives; Fixed point theorems; Solutions dependence; Sturm–Liouville problem; Existence; 34A08
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-020-02690-1

In this paper, we study a coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations described by Atangana–Baleanu–Caputo (ABC for short) derivatives whose formulations are based on the notable Mittag-Leffler kernel. Prior to the main results, the equivalence of the coupled system to a nonlinear system of integral equations is proved. Once that has been done, we show in detail the existence–uniqueness and Ulam stability by the aid of fixed point theorems. Further, the continuous dependence of the solutions is extensively discussed. Some examples are given to illustrate the obtained results.