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On the Novel Ulam–Hyers Stability for a Class of Nonlinear \(\psi \)-Hilfer Fractional Differential Equation with Time-Varying Delays

  • Danfeng Luo
  • Kamal Shah
  • Zhiguo LuoEmail author
Article
  • 14 Downloads

Abstract

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.

Keywords

Ulam–Hyers–Rassias stability Ulam–Hyers stability Uniqueness \(\psi \)-Hilfer fractional derivative Time-varying delays 

Mathematics Subject Classification

26A33 34D20 

Notes

Acknowledgements

The authors thank the referees for the helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11471109).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MalakandLower DirPakistan
  3. 3.Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China

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