This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results.

A newly proposed p-Laplacian nonperiodic boundary value problem is studied in this research paper in the form of generalized Caputo fractional derivatives. The existence and uniqueness of solutions are fully investigated for this problem using some fixed point theorems such as Banach and Schauder. This work is supported with an example to apply all obtained new results and validate their applicability.

In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions; additionally, a numerical scheme based on the the Lagrange's interpolation polynomial is established and applied to a dynamical system for the applications. We also study the uniqueness and Hyers-Ulam stability for the solutions of the presumed mABC-FDEs system. Such a system has not been studied for the mentioned mABC-operator and this work generalizes most of the results studied for the ABC operator. This study will provide a base to a large number of dynamical problems for the existence, uniqueness and numerical simulations. The results are compared with the classical results graphically to check the accuracy and applicability of the scheme.

In this paper, we provide a new version for the Gronwall inequality in the frame of the generalized proportional fractional derivative. Prior to the main results, we introduce the generalized proportional fractional derivative and expose some of its features. As an application, we accommodate the newly defined derivative to prove the uniqueness and obtain a bound in terms of Mittag-Leffler function for the solutions of a nonlinear delay proportional fractional system. An example is presented to demonstrate the applicability of the theory.

We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order \(\alpha \in (1,2]\). Indeed, it is shown that if the boundary value problem $$ \bigl(\textbf{T}_{\alpha }^{c} x\bigr) (t)+r(t)x(t)=0,\quad t \in (c,d), x(c)=x(d)=0 $$ has a nontrivial solution, where r is a real-valued continuous function on \([c,d]\), then $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}(d-c)^{ \alpha -1}}. $$ (1) Moreover, a Lyapunov type inequality of the form $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{3\alpha -1}{(d-c)^{2\alpha -1}} \biggl( \frac{3 \alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }},\quad \frac{1}{2}< \alpha \leq 1, $$ (2) is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.