A study on -fractional iterative differential equations with a state-dependent nonlocal condition

• Published in: Journal of applied mathematics & computing Vol. 71; no. 4; pp. 6231 - 6250
• Main Author: Vu, Ho
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2025; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Differential equations; Fixed points (mathematics); Fractional calculus; Integral equations; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theory of Computation; 34K40; fractional calculus; 47H10; Iterative differential equation; 47H09; Nonlocal condition; 34K14; Fractional iterative differential equations; 45G05
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-025-02542-9

This paper investigates a new class of -fractional iterative differential equations subject to a state-dependent nonlocal condition, introduced by Hernández and O’Regan [1]. By reformulating the problem into an equivalent -fractional integral equation, we establish sufficient conditions for the existence and uniqueness of solutions using the fixed-point techniques, notably Schauder’s and Banach’s fixed-point theorems. We also demonstrate the continuous dependence on data of the solution of the problem setting. Furthermore, we explore both Ulam–Hyers and generalized Ulam–Hyers stability, providing explicit criteria. Two illustrative examples are presented to validate the theoretical findings.