On the Novel Ulam–Hyers Stability for a Class of Nonlinear \(\psi \)-Hilfer Fractional Differential Equation with Time-Varying Delays
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In this paper, we present some alternative results concerning the uniqueness and Ulam–Hyers stability of solutions for a kind of \(\psi \)-Hilfer fractional differential equations with time-varying delays. Under some updated criteria along with the generalized Gronwall inequality, the new constructive results have been established in the literature. The derived analysis has the ability to generalize and improve some other results from the literature. As an application, two typical examples are delineated to demonstrate the effectiveness of our theoretical results.
KeywordsUlam–Hyers–Rassias stability Ulam–Hyers stability Uniqueness \(\psi \)-Hilfer fractional derivative Time-varying delays
Mathematics Subject Classification26A33 34D20
The authors thank the referees for the helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11471109).
- 1.Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, vol. 207. Elsevier, Amsterdam (2006)Google Scholar
- 2.Miller, K., Rose, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993). (ISBN: 0-471-58884-9)Google Scholar
- 4.Podlubny, I., Thimann, K.: Fractional Differential Equation: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999). (ISBN: 0125588402)Google Scholar
- 5.Chen, F., Liu, Z.: Asymptotic stability results for nonlinear fractional difference eqautions. J. Appl. Math. 2012, 155–172 (2012)Google Scholar