[Paper Review] On Ulam type stability for nonlinear implicit fractional differential equations

D. B. Dhaigude, Sandeep P. Bhairat|arXiv (Cornell University)|Jul 24, 2017

Fractional Differential Equations Solutions15 references3 citations

TL;DR

This paper establishes existence, uniqueness, and four types of Ulam-type stability (Hyers-Ulam, generalized Ulam-Hyers, $E_{\alpha}$-Ulam-Hyers, and $E_{\alpha}$-Ulam-Hyers-Rassias) for nonlinear implicit fractional differential equations using the method of successive approximations. The key contribution is proving stability under mild Lipschitz-type conditions via integral inequalities and Mittag-Leffler functions, validated by a concrete example with $\alpha = \frac{5}{2}$ on $[1,e]$.

ABSTRACT

In present paper, we establish sufficient conditions for existence and stability of solutions for system of nonlinear implicit fractional differential equations. The main techniques are based on method of successive approximations. Finally, an illustrative example is given to show the applicability of our theoretical results.

Motivation & Objective

• To establish sufficient conditions for existence and uniqueness of solutions to a system of nonlinear implicit fractional differential equations with Caputo-Hadamard derivatives.
• To investigate four distinct types of Ulam-type stability—Hyers-Ulam, generalized Ulam-Hyers, $E_{\alpha}$-Ulam-Hyers, and $E_{\alpha}$-Ulam-Hyers-Rassias—for such equations.
• To apply the method of successive approximations as the core analytical technique to derive stability results.
• To validate theoretical findings with a concrete example demonstrating the applicability of the proposed stability criteria.

Proposed method

• Utilizes the method of successive approximations to construct a sequence of functions converging to the solution of the implicit fractional differential equation.
• Employs the Caputo-Hadamard derivative of order $\alpha \in (m-1, m]$, defined via logarithmic kernels and the $\delta^n = (t\frac{d}{dt})^n$ operator.
• Applies integral inequalities involving the Mittag-Leffler function $E_{\alpha}(z)$ to bound solution differences and establish stability bounds.
• Derives stability estimates using the $E_{\alpha}$-function and a parameter $\theta = \frac{M}{1-N}$, where $M$ and $N$ are Lipschitz constants.
• Introduces a weight function $\Phi(t) = C E_{\alpha}((\log t)^\alpha)$ to analyze $E_{\alpha}$-Ulam-Hyers-Rassias stability with variable bounds.
• Uses the Hadamard fractional integral and its properties to transform the differential equation into an equivalent integral form for analysis.

Experimental results

Research questions

• RQ1 Under what conditions does a solution exist and remain unique for a system of nonlinear implicit fractional differential equations with Caputo-Hadamard derivatives?
• RQ2 Can Ulam-type stability be established for such equations using successive approximations rather than fixed-point theorems?
• RQ3 How do the four types of Ulam-type stability—Hyers-Ulam, generalized, $E_{\alpha}$-Ulam-Hyers, and $E_{\alpha}$-Ulam-Hyers-Rassias—apply to implicit fractional equations?
• RQ4 What is the quantitative bound on the approximation error between an approximate solution and the exact solution under each stability type?
• RQ5 How does the choice of weight function $\Phi(t)$ affect the $E_{\alpha}$-Ulam-Hyers-Rassias stability condition?

Key findings

• The problem (33) with $\alpha = \frac{5}{2}$ on $[1,e]$ has a unique solution due to the Lipschitz condition with $M = \log(2+e) > 0$ and $N = \frac{1}{e^2} < 1$.
• The problem is Ulam-Hyers stable with the bound $||y(t) - x(t)|| \leq \left(\frac{E_{\frac{5}{2}}(\theta) - 1}{\theta}\right)\epsilon$, where $\theta = \frac{e^2 \log(2+e)}{e^2 - 1} \approx 0.779$.
• The problem is generalized Ulam-Hyers stable because $\psi(0) = 0$ for $\psi(\epsilon) = \frac{E_{\frac{5}{2}}(\theta) - 1}{\theta}\epsilon$.
• The problem is $E_{\frac{5}{2}}$-Ulam-Hyers stable with the bound $||y(t) - x(t)|| \leq \frac{1}{\theta} E_{\frac{5}{2}}(\theta)$, independent of $\epsilon$.
• The problem is $E_{\frac{5}{2}}$-Ulam-Hyers-Rassias stable under the condition $||\mathfrak{D}_{1}^{\frac{5}{2}}y(t) - f(t,y(t),\mathfrak{D}_{1}^{\frac{5}{2}}y(t))|| \leq \epsilon \Phi(t)$, with $\Phi(t) = C E_{\frac{5}{2}}((\log t)^{\frac{5}{2}})$, yielding $||y(t) - x(t)|| \leq \epsilon \frac{1}{1 - \theta} \Phi(t)$.
• The problem is generalized $E_{\frac{5}{2}}$-Ulam-Hyers-Rassias stable, as confirmed by Remark 3, with the bound $||y(t) - x(t)|| \leq \epsilon \frac{2}{1 - \theta} \Phi(t) E_{\frac{5}{2}}((\log t)^{\frac{5}{2}})$.