[Paper Review] Global attractivity for some classes of Riemann--Liouville fractional differential systems

Hoang The Tuan, Adam Czornik+2 more|arXiv (Cornell University)|Sep 1, 2017

Fractional Differential Equations Solutions6 references3 citations

TL;DR

This paper establishes existence and global attractivity of solutions for multidimensional Riemann–Liouville fractional differential systems using a Bielecki-type norm and Banach fixed point theorem. By leveraging properties of Mittag-Leffler functions, it proves that solutions of linear systems with time-varying perturbations converge to zero under specific decay and boundedness conditions on the system matrix and forcing terms.

ABSTRACT

In this paper, we present some results for existence of global solutions and attractivity for mulidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and Banach fixed point theorem, we prove a Picard-Lindelöf type theorem on the existence and uniqueness of solutions. Then, applying the properties of Mittag-Leffler functions, we describe the attractivity of solutions to some classes of Riemann--Liouville linear fractional differential systems.

Motivation & Objective

• To establish the existence and uniqueness of global solutions for multidimensional Riemann–Liouville fractional differential systems on (0, ∞).
• To analyze the asymptotic behavior (attractivity) of solutions, particularly their convergence to zero as t → ∞.
• To extend classical Picard–Lindelöf theory to fractional-order systems with Riemann–Liouville derivatives.
• To characterize conditions under which solutions of linear fractional systems with time-varying coefficients and forcing terms are globally attractive.

Proposed method

• Uses a Bielecki-type norm on the space $ C_{1-eta}([0, ∞), \mathbb{R}^d) $ to handle the singularity at t = 0.
• Applies the Banach fixed point theorem to a nonlinear integral operator derived from the variation-of-parameters formula.
• Employs the Mittag-Leffler function $ E_{\alpha,\alpha}(\cdot) $ to represent the fundamental solution of the linearized system.
• Establishes contractivity of the solution operator by bounding the norm of the integral operator involving $ |||E_{\alpha,\alpha}((t-\tau)^\alpha A)||| $.
• Introduces a weighted norm $ \|\cdot\|_w $ to control growth and ensure convergence in the fixed point argument.
• Derives sufficient conditions on the system matrix A, perturbation Q(t), and forcing term g(t) for attractivity via integral estimates.

Experimental results

Research questions

• RQ1 Under what conditions does a Riemann–Liouville fractional differential system admit a unique global solution on (0, ∞)?
• RQ2 When do solutions of linear fractional systems with time-varying coefficients and forcing terms converge to zero as t → ∞?
• RQ3 How can the asymptotic behavior of such systems be analyzed using Mittag-Leffler functions and fixed point techniques?
• RQ4 What role does the Bielecki-type norm play in proving existence and uniqueness in singular fractional systems?
• RQ5 Can the attractivity of solutions be guaranteed even when the perturbation Q(t) is not small, but decays appropriately?

Key findings

• A unique global solution exists for the initial value problem $ D_{0+}^\alpha x(t) = f(t,x(t)) $ on $ (0, \infty) $ under a Lipschitz condition with a bounded function $ L(t) $.
• For linear systems of the form $ D_{0+}^\alpha x(t) = Ax(t) + Q(t)x(t) + g(t) $, attractivity to zero is guaranteed if $ \|Q(t)\| $ is bounded and $ g(t) $ decays sufficiently fast.
• The solution operator is contractive in the weighted space $ C_{1-\alpha}^0([0,\infty), \mathbb{R}^d) $ under the condition $ \frac{K 2^{2-\alpha} \Gamma(\alpha)}{\gamma^\alpha} \leq \frac{1}{4} $, ensuring a unique fixed point.
• Explicit estimates are derived for the norm of the Mittag-Leffler function $ E_{\alpha,\alpha}((t-\tau)^\alpha A) $, which are crucial for bounding the solution operator.
• In Example 5.1, the condition $ |Q(t)| \leq \frac{1}{\sup_{t \geq 0} t^{1/2} E_{1/2}(-t^{1/2})} $ ensures that the integral operator norm is less than 1, guaranteeing attractivity.
• Example 5.2 confirms attractivity for a piecewise-defined Q(t) that decays as $ t^{-1} $, showing the theory applies even to non-smooth perturbations.