Existence and stability of solution for a coupled system of Caputo–Hadamard fractional differential equations

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2024; no. 1; pp. 17 - 19
• Main Authors: Teshome Beyene, Mesfin, Daba Firdi, Mitiku, Temesgen Dufera, Tamirat
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 02.12.2024; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Applications of Mathematics; Boundary conditions; Caputo–Hadamard fractional derivative; Differential equations; Differential Geometry; Equation; Exact solutions; Existence and uniqueness; Fixed point theorems; Fixed points (mathematics); Fractional calculus; Laplace transform; Laplace transforms; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Metric Fixed Point Theory and Its Applications; Mittag-Leffler functions; Stability criteria; Topology; Caputo–Hadamard fractional derivative; Fixed point theorems; Equation; Mittag-Leffler functions; Laplace transform; Existence and uniqueness
• ISSN: 2730-5422; 2730-5422
• DOI: 10.1186/s13663-024-00773-2

In our present work, we study a coupled system of Caputo–Hadamard fractional differential equations supplemented with a novel set of initial value conditions involving the η = ( t d d t ) derivatives. We provided sufficient criteria for the existence and stability of the solutions for a coupled system of fractional differential equations by applying the Hyers–Ulam stability theory, the fixed point theorems of Banach, Krasnoselskii, and the Leray–Schauder nonlinear alternative. When computing priori bounds in Leray–Schauder nonlinear alternative and stability of the solutions, a novel Gronwall type inequality related to Hadamard integral is employed. This study investigates the properties of a solution, such as existence, uniqueness, and stability, to a given problem without attempting to solve the exact solution, and its theoretical applications are illustrated by providing an example.