In Banach spaces, first we present a new geometric constant, $C_Z^{(q)}(t,B)$, which is closely related to the Zb $\check{a}$ganu constant. We prove that $\frac{(t+2{)}^q}{2^{q-1}\left(2^{q-1}+t^q\right)}$ and $\frac{4}{3}$ are respectively the lower and upper bounds for $C_Z^{(q)}(t,B)$. Furthermore, we also derive that $C_Z^{(q)}(t,B)=C_Z^{(q)}(t,\tilde{B})$, where $\tilde{B}$ denotes the ultrapower spaces of $B$. Then, we can establish certain sufficient conditions that ensure a Banach spaces possesses a normal structure, which involve different constants such as the Zb $\check{a}$ganu constant and Domínguez–Benavides coefficient.

To study the Mittag-Leffler projective synchronization (MLPS) problem of fractional-order fuzzy neural networks (FOFNNs), in this work we introduced the FOFNNs model. On this basis, we discussed the MLPS of uncertain fractional-order fuzzy complex valued neural networks (FOFCVNNs) with distributed and time-varying delays. Utilizing Banach contraction mapping principle, we proved the existence and uniqueness of the model solution. Moreover, employing the construction of a new hybrid controller, an adaptive hybrid controller, and the fractional-order Razumikhin theorem, algebraic criteria was obtained for implementing MLPS. The algebraic inequality criterion obtained in this article improves and extends the previously published results on MLPS, making it easy to prove and greatly reducing the computational complexity. Finally, different Caputo derivatives of different orders were given, and four numerical examples were provided to fully verify the accuracy of the modified criterion.