On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations

Natasha Irshad; Rahim Shah; Zubaria Waqar

doi:10.15672/hujms.1731010

On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations

Abstract

In recent years, fractional–order derivatives and integrals have proven to be powerful tools for modeling complex systems with memory effects, mak- ing them well–suited for describing various real–world phenomena. Fractional– order models are often more effective than their integer–order counterparts in capturing the dynamics of materials such as polymers, biological systems, and other physical processes. Given these considerations, the study of qual- itative properties, particularly Ulam–type stabilities, of fractional integro– differential equations has gained significant attention in mathematical research due to their wide–ranging applications in aerodynamics, biology, physics, electrodynamics, polymer science, and other disciplines. Motivated by this, we investigate various types of Ulam–type stabilities, including Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and σ–semi–Ulam–Hyers stability, for mild solutions of noninstantaneous impulsive fractional–order integro– differential equations with local conditions in a Banach space. By introduc- ing a novel set of suitable assumptions, we establish stability results and obtain five new findings concerning these stability concepts. Our analysis considers both finite and infinite intervals. As an application, we present three illustrative examples that support the theoretical findings. The anal- ysis in this work relies on the Banach fixed–point theorem and the Bielecki metric, which are fundamental tools in nonlinear analysis with applications in functional equations, fractal theory, and optimization. Our findings con- tribute to the ongoing development of stability theories for fractional–order models and their applications in various scientific fields.

Keywords

References

[1] K. Balachandran and J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro–differential equations in Banach spaces, Nonlinear Anal. 72 (3–4), 4587–4593, 2010.

Details

Primary Language

English

Subjects

Approximation Theory and Asymptotic Methods

Journal Section

Research Article

Authors

Natasha Irshad
0009-0008-8166-6520
Pakistan

Rahim Shah ^*
0009-0001-9044-5470
Pakistan

Zubaria Waqar This is me
0009-0004-9798-2959
Pakistan

Early Pub Date

August 17, 2026

Publication Date

-

Submission Date

June 30, 2025

Acceptance Date

April 10, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.15672/hujms.1731010

IZ

https://izlik.org/JA26EL62UP

Cite

RIS / Bibtex

APA

Irshad, N., Shah, R., & Waqar, Z. (2026). On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations. Hacettepe Journal of Mathematics and Statistics, Advanced Online Publication. https://doi.org/10.15672/hujms.1731010

AMA

1.Irshad N, Shah R, Waqar Z. On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations. Hacettepe Journal of Mathematics and Statistics. 2026;(Advanced Online Publication). doi:10.15672/hujms.1731010

Chicago

Irshad, Natasha, Rahim Shah, and Zubaria Waqar. 2026. “On the Ulam–Hyers–Rassias Stability of Noninstantaneous Impulsive Fractional Integro–differential Equations”. Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication. https://doi.org/10.15672/hujms.1731010.

EndNote

Irshad N, Shah R, Waqar Z (August 1, 2026) On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations. Hacettepe Journal of Mathematics and Statistics Advanced Online Publication

IEEE

[1]N. Irshad, R. Shah, and Z. Waqar, “On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations”, Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication, Aug. 2026, doi: 10.15672/hujms.1731010.

ISNAD

Irshad, Natasha - Shah, Rahim - Waqar, Zubaria. “On the Ulam–Hyers–Rassias Stability of Noninstantaneous Impulsive Fractional Integro–differential Equations”. Hacettepe Journal of Mathematics and Statistics. Advanced Online Publication (August 1, 2026). https://doi.org/10.15672/hujms.1731010.

JAMA

1.Irshad N, Shah R, Waqar Z. On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations. Hacettepe Journal of Mathematics and Statistics. 2026. doi:10.15672/hujms.1731010.

MLA

Irshad, Natasha, et al. “On the Ulam–Hyers–Rassias Stability of Noninstantaneous Impulsive Fractional Integro–differential Equations”. Hacettepe Journal of Mathematics and Statistics, no. Advanced Online Publication, Aug. 2026, doi:10.15672/hujms.1731010.

Vancouver

1.Natasha Irshad, Rahim Shah, Zubaria Waqar. On the Ulam–Hyers–Rassias stability of noninstantaneous impulsive fractional integro–differential equations. Hacettepe Journal of Mathematics and Statistics. 2026 Aug. 1;(Advanced Online Publication). doi:10.15672/hujms.1731010