Existence and κ-Mittag-Leffler-Ulam-Hyers stability results for implicit coupled (κ,ϑ)-fractional differential systems

Abstract

In this paper, we delve into the analysis of the existence and stability concerning the $\kappa$-Mittag-Leffler-Ulam-Hyers within a particular class of coupled systems related to boundary value problems. These problems entail implicit nonlinear fractional differential equations and $\kappa$-generalized ϑ-Hilfer fractional derivatives. To achieve our objectives, we employ the Mönch fixed point theorem, complemented by the application of the measure of noncompactness technique and an extension of the well-established Gronwall inequality. Furthermore, we include an illustrative example to showcase the practical utility of our results. The significance of our research lies in examining a comprehensive problem involving coupled systems, which serves as a generalization encompassing all the works mentioned in the introduction. It is viewed as a logical extension and continuation within the framework of this continually evolving theory.

Keywords:

• Banach spaces
• coupled systems
• existence
• generalized Gronwall inequality
• generalized Hilfer fractional derivative
• measure of noncompactness
• Mittag-Leffler function
• Ulam-Hyers stability

AMS (MOS) SUBJECT CLASSIFICATIONS:

• 26A33
• 34A12
• 34G20

Acknowledgement

J. Alzabut would like to thank Prince Sultan University and OSTIM Technical University for their endless support.

Author’s contributions

The study was carried out in collaboration of all authors. All authors read and approved the final manuscript.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.