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The Pečarić Journal of Mathematical Inequalities (PJMI)

ISSN: xxxx-xxxx (Online) xxxx-xxxx (Print)
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  • Research article

Superstability of extended fractional differential equations

Abdellatif Ben Makhlouf1,2, Lassaad Mchiri3
1Mathematics Education Section, Faculty of Education and arts, Sohar University, P.O. Box 44, Postal Code 311, Sohar, Sultanate of Oman
2Department of Mathematics, Faculty of Sciences, Sfax University, Sfax BP 1171, Tunisia
3Department of Mathematics, Galilee institute, University of Paris 13, Campus de Villetaneuse 99, avenue Jean-Baptiste Clement, 93430, Villetaneuse, France
Copyright © Abdellatif Ben Makhlouf, Lassaad Mchiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received: 05/05/2025
  • Accepted: 01/08/2025
  • Published: 05/10/2025

Abstract

This work advances the stability theory of fractional differential equations by establishing superstability criteria for a significant class of problems involving the Caputo-Katugampola derivative. Utilizing a generalized Taylor series as a foundational tool, we prove that these equations exhibit superstable behavior under specific conditions. Our results generalize a wide range of existing stability theorems, creating a unified framework that encompasses systems governed by the Caputo fractional derivative as special cases of the more general Katugampola operator.

Keywords: superstability, Taylor’s formula, fractional differential equations

A cornerstone of functional equations theory is the study of Hyers-Ulam stability (HUS). This area of inquiry originated with a foundational question posed by Ulam in 1940 concerning the stability of homomorphisms, specifically asking when an approximate linear mapping can be closely approximated by an exact one. A year later, Hyers provided a definitive solution to this problem [1]. The concept has since been significantly extended beyond functional equations to differential equations, leading to a substantial body of literature establishing HUS results for various classes of differential equations [2, 3].

Within this broader context, fractional differential equations (FDEs) have emerged as a particularly vibrant field of research. While the concept of the fractional derivative dates back to the correspondence between L’Hospital and Leibniz in 1695, its rigorous application has seen explosive growth in recent decades. Contemporary investigations into FDEs encompass a wide range of topics, including stability analysis [4– 7], finite-time stability [8], stabilization [9], observer design [9, 10], and fault estimation [11]. A specific and actively researched aspect is the Hyers-Ulam stability of FDEs, to which numerous significant contributions have been made [12– 15].

A specialized form of stability, known as superstability (SS), was introduced in [8] as a particular case of HUS. Superstability describes a situation where any approximate solution to a given equation is either an exact solution or is uniformly bounded by a constant multiple of the perturbation parameter. Initial research on this concept for differential equations can be found in [16, 17]. In [18], the author investigated the SS for fractional differential equations with Caputo derivative.

Our work focuses on establishing superstability for the following initial value problem. \[\label{fracdiffeqnb} {}^CD_{r}^{p\lambda,\rho} E(x)+A(x) E(x)=0, \tag{1}\] with initial conditions (IC): \[\label{it} E(r)={}^CD_{r}^{\lambda,\rho} E(r)={}^CD_{r}^{2\lambda,\rho} E(r)=…={}^CD_{r}^{(p-1)\lambda,\rho} E(r)=0, \tag{2}\] where \(p\in \mathbb{N}^*\), \({}^CD_{r}^{s\lambda,\rho} E \in C\big([r,r+u] \big)\), for each \(s\in \{0,1,…,p\}\), \(A\in C\big([r,r+u] \big)\), \(u>0\) and \({}^CD_{r}^{s\lambda,\rho} ={}^CD_{r}^{\lambda,\rho}.{}^CD_{r}^{\lambda,\rho}…{}^CD_{r}^{\lambda,\rho}\) (\(s-\)times).

Motivated by [16– 18], we introduce the following definition.

Definition 1. Suppose that \(E\) satisfies: \[\label{diffeqn} |\psi\big(A, E, {}^CD_{r}^{\lambda,\rho}E,{}^CD_{r}^{2\lambda,\rho}E,…,{}^CD_{r}^{p\lambda,\rho}E \big)|\leq \nu,\ \forall \omega\in [r,r+u], \tag{3}\] for some \(\nu\geq 0\) with IC therefore either \[|E(\omega)|\leq \vartheta \nu,\ \forall \omega\in [r,r+u],\] where \(\vartheta>0\), or \[\psi\big(A, E, {}^CD_{r}^{\lambda,\rho}E,{}^CD_{r}^{2\lambda,\rho}E,…,{}^CD_{r}^{p\lambda,\rho}E \big)=0.\]

Then, we say that (1) has SS with IC.

Definition 2.[19] Given \(0<l<1\), \(\rho>0\). The Caputo-Katugampola fractional derivative of an absolutely continuous function \(f\) is defined as, \[\label{dercap} ^CD_{c}^{l,\rho} f(s)=\displaystyle \frac{\rho^l}{\Gamma(1-l)} \displaystyle \int_{c}^s (s^\rho-\tau^\rho)^{-l}f'(\tau)d\tau. \tag{4}\]

Theorem 1 ([20] Generalized Taylor’s formulat).Let \(0<\eta< 1\), \(\rho>0\). Assume that \({}^CD_{r_1}^{t \eta,\rho} h \in C\big([r_1,r_2] \big)\), for each \(t\in \{0,1,…,s\}\), with \(s\in \mathbb{N}^*\), then we have \[h(x)=\displaystyle \sum_{t=0}^{s-1}{}^CD_{r_1}^{t\eta,\rho} h(r_1) \frac{(\frac{x^\rho-r_1^\rho}{\rho})^{t \eta}}{\Gamma(t \eta+1)}+{}^CD_{r_1}^{s\eta,\rho} h(c) \frac{(\frac{x^\rho-r_1^\rho}{\rho})^{s \eta}}{\Gamma(s \eta+1)},\] with \(c\in [r_1,x]\), for each \(x\in (r_1,r_2]\).

1. Main theorem

The main contribution of this work is outlined in this section.

Theorem 2. Assume that \(\displaystyle\sup_{\chi\in [r,r+u]}|A(\chi)|< \displaystyle\frac{\Gamma(p \lambda+1)}{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}\). Then, (1) has the SS with IC (2).

Proof. Let \(\nu>0\), and \(E\in C\big([r,r+u] \big)\) with \({}^CD_{r}^{t\lambda,\rho} E \in C\big([r,r+u] \big)\) for each \(t\in \{ 0,1,…p \}\), if \[|{}^CD_{r}^{p\lambda,\rho} E(x)+A(x) E(x)|\leq \nu,\] and \[E(r)={}^CD_{r}^{\lambda,\rho} E(r)={}^CD_{r}^{2\lambda,\rho} E(r)=…={}^CD_{r}^{(p-1)\lambda,\rho} E(r)=0.\]

It follows from Theorem 1 that \[E(x)=\displaystyle \sum_{t=0}^{p-1}{}^CD_{r}^{t\lambda,\rho} E(r) \frac{(\frac{x^\rho-r^\rho}{\rho})^{t \lambda}}{\Gamma(t \lambda+1)}+{}^CD_{r}^{p\lambda,\rho} E(c) \frac{(\frac{x^\rho-r^\rho}{\rho})^{p \lambda}}{\Gamma(p \lambda+1)},\] where \(c\in [r,x]\), for all \(x\in (r,r+u]\). Therefore \[\begin{aligned} |E(x)|&=&|{}^CD_{r}^{p\lambda,\rho} E(c) \frac{(\frac{x^\rho-r^\rho}{\rho})^{p \lambda}}{\Gamma(p \lambda+1)}| \notag \\ &\leq& \displaystyle\sup_{\chi\in [r,r+u]}|{}^CD_{r}^{p\lambda,\rho} E(\chi)| \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)}. \end{aligned} \tag{5}\]

Thus,

\[\begin{aligned} \displaystyle\sup_{\chi\in [r,r+u]}|E(\chi)|&\leq & \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)} \Big[ \displaystyle\sup_{\chi\in [r,r+u]}|{}^CD_{r}^{p\lambda,\rho}E(\chi)-A(\chi)E(\chi)|+\displaystyle\sup_{\chi\in [r,r+u]}|A(\chi)| \displaystyle\sup_{\chi\in [r,r+u]}|E(\chi)| \Big] \notag \\ &\leq& \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)} \nu + \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)} \displaystyle\sup_{\chi\in [r,r+u]}|A(\chi)| \displaystyle\sup_{\chi\in [r,r+u]}|E(\chi)| . \end{aligned} \tag{6}\]

Then, \[\displaystyle\sup_{\chi\in [r,r+u]}|E(\chi)|\Big( 1- \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)} \displaystyle\sup_{\chi\in [r,r+u]} |A(\chi)| \Big)\leq \frac{\big(\frac{(u+r)^\rho-r^\rho}{\rho}\big)^{p \lambda}}{\Gamma(p \lambda+1)} \nu.\]

Hence, there is \(K>0\) with \[|E(x)|\leq K \nu,\] for every \(x\in [r,r+u]\). ◻

Remark 1. It is important to note that in [17], the authors obtained SS results for DEs with integer-order derivatives. In [18], the SS is studied for the Caputo fractional derivative. In our work, however, the main result is obtained for the Caputo-Katugampola fractional-order derivative.

2. Conclusion

In conclusion, this work successfully establishes a unified superstability framework for fractional differential equations involving the Caputo-Katugampola derivative. By leveraging a generalized Taylor series, we have derived specific conditions under which these equations exhibit superstable behavior. Our findings not only generalize a wide spectrum of existing stability theorems but also consolidate systems governed by the standard Caputo derivative as special cases within this broader, more versatile theory.

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