Viability for time fractional functional differential equations driven by the fractional Brownian motion
Pub. online: 1 April 2026
Type: Research Article
Open Access
Open Access
Received
9 July 2025
9 July 2025
Revised
15 February 2026
15 February 2026
Accepted
21 February 2026
21 February 2026
Published
1 April 2026
1 April 2026
Abstract
Let ${B^{H}}$ be a fractional Brownian motion with Hurst index $\frac{1}{2}\lt H\lt 1$. In this paper, we consider the time fractional functional differential equation of the form
\[ \left\{\begin{array}{l@{\hskip10.0pt}l}{^{C}}{D_{t}^{\gamma }}x(t)=f(t,{x_{t}})+G(t,{x_{t}})\frac{d}{dt}{B^{H}}(t),\hspace{3.33333pt}\hspace{3.33333pt}\hspace{3.33333pt}\hspace{3.33333pt}\hspace{1em}& t\in (0,T],\\ {} {x_{0}}(t)=\eta (t),\hspace{3.33333pt}\hspace{3.33333pt}\hspace{3.33333pt}\hspace{3.33333pt}\hspace{3.33333pt}\hspace{1em}& t\in [-r,0],\end{array}\right.\]
where $\frac{3}{2}-H\lt \gamma \lt 1$, ${^{C}}{D_{t}^{\gamma }}$ denotes the Caputo derivative, and ${x_{t}}\in {\mathcal{C}_{r}}=\mathcal{C}([-r,0],\mathbb{R})$ with ${x_{t}}(u)=x(t+u)$, $u\in [-r,0]$. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.References
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