The finite-time synchronization (FTS) is studied for a class of fractional-order delayed fuzzy cellular neural networks (FODFCNNs) with parameter uncertainties. A linear fractional-order finite time inequality (FOFTI) ${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V(t)-b$ is extended to the nonlinear case ${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V^{-\eta }(t)-b$ , $\eta \ge 1$ , which plays a vital role in the FTS of fractional-order systems. However, for the case of $0< \eta < 1$ , a theoretical cornerstone justifying its use is still missing. To fill this research gap, on the basis of the $C_{p}$ inequality and the rule for fractional-order derivative of composite function, a nonlinear FOFTI ${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V^{-\eta }(t)-b$ , $0< \eta < 1$ is developed. Furthermore, a nonlinear FOFTI ${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -bV^{-\xi }(t)-a V^{-\eta }(t)$ is also established. These two novel inequalities provide the new tools for the research on the finite time stability and synchronization of fractional-order systems and can greatly extend the pioneer ones. Next, on the basis of these novel inequalities, the feedback controller is designed and two novel FTS criteria of FODFCNNs with parameter uncertainties are obtained. Finally, two examples are presented to verify the effectiveness of the derived results.

In this article, the finite-time stability (FTS) of fractional-order Hopfield neural networks with time delays (FHNNTDs) is studied. A widely used inequality in investigating the stability of the fractional-order neural networks is fractional-order Gronwall inequality related to the Mittag-Leffler function, which cannot be directly used to study the stability of the factional-order neural networks with time delays. In the existing works related to fractional-order Gronwall inequality with time delays, the order λ>0 was divided into two cases: λ∈(0,0.5] and λ∈(0.5,+∞). In this article, a new fractional-order Gronwall integral inequality with time delay and the unified form for all the fractional order λ>0 is developed, which can be widely applied to investigate FTS of various fractional-order systems with time delays. Based on this new inequality, a new criterion for the FTS of FHNNTDs is derived. Compared with the existing criteria, in which fractional order λ∈(0,1) was divided into two cases, λ∈(0,0.5] and λ∈(0.5,1), the obtained results in this article are presented in the unified form of fractional order λ∈(0,1) and convenient to verify. More importantly, the criteria in this article are less conservative than some existing ones. Finally, two numerical examples are given to demonstrate the validity of the proposed results.

In this paper, we examine the relation between monotonicity and convexity for nabla fractional (q, h)-differences. In particular we prove that Theorem A Theorem A: Assume $x:{\tilde{\mathbb{T}}}_{(q,h)}^{\mathit{\sigma }(a)}\to \mathbb{R}$, $a{\mathrm{\nabla }}_{(q,h)}^{\mathit{\alpha }}x(t)\ge 0$ for each $t\in {\tilde{\mathbb{T}}}_{(q,h)}^{\mathit{\sigma }(a)}$, with $1<\mathit{\alpha }<2$. Then ${\mathrm{\nabla }}_{(q,h)}x(t)\ge 0$ for $t\in {\tilde{\mathbb{T}}}_{(q,h)}^{{\mathit{\sigma }}^2(a)}$. Theorem B Theorem B: Assume $x:{\tilde{\mathbb{T}}}_{(q,h)}^{\mathit{\sigma }(a)}\to \mathbb{R}$, $a{\mathrm{\nabla }}_{(q,h)}^{\mathit{\alpha }}x(t)\ge 0$ for each $t\in {\tilde{\mathbb{T}}}_{(q,h)}^{\mathit{\sigma }(a)}$, with $2<\mathit{\alpha }<3$. Then ${\mathrm{\nabla }}_{(q,h)}^{2}x(t)\ge 0$ for $t\in {\tilde{\mathbb{T}}}_{(q,h)}^{{\mathit{\sigma }}^3(a)}$. This shows that, in some sense, the positivity of the $\mathit{\alpha }$th order (q, h)-fractional difference has a strong connection to the monotonicity and convexity of x(t).