
 The appreciation of inequalities in convexity is critical for fractional calculus and its application in a variety of fields. In this paper, we provide a unique analysis based on Hermite-Hadamard inequalities in the context of newly defined class of convexity which is known as left and right harmonically $ {h} $-convex IVF (left and right $ \mathcal{H}$-$ {h} $-convex IVF), as well as associated integral and fractional inequalities, are addressed by the suggested technique. Because of its intriguing character in the numerical sciences, there is a strong link between fractional operators and convexity. There have also been several exceptional circumstances studied, and numerous well-known Hermite-Hadamard inequalities have been derived for left and right $ \mathcal{H}$-$ {h} $-convex IVF. Moreover, some applications are also presented in terms of special cases which are discussed in this study. The plan's outcomes demonstrate that the approach may be implemented immediately and is computationally simple and precise. We believe, our findings, generalize certain well-known new and classical harmonically convexity discoveries from the literature.


New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities

This paper is devoted to the exploration of the global asymptotic synchronization (GAS) and global Mittag–Leffler synchronization (GMLS) issues on bidirectional associative memory reaction-diffusion neural networks (BAMRDNNs), which include a Caputo’s fractional partial differential operator , leakage and discrete delays. By means of the Green’s theorem, Jensen’s integral inequality and Lyapunov functional approach, several synchronization criteria are established by using the hybrid controllers . The presented results are described in the algebraic inequalities form, which can immensely reduce the computational complexity. Moreover, these results reveal the impact of the network system coefficient matrices on GAS and GMLS. Finally, in the three-dimensional space, the effectiveness and practicability of these synchronization criteria are confirmed by choosing different derivative orders and diffusion coefficients . 

Global synchronization for BAM delayed reaction-diffusion neural networks with fractional partial differential operator

• The F-TS of FOQVBAMNNs with time-varying delays is explored in details. • A new lemma is established based on the Caputo fractional derivative. In addition, the fixed time is predicted, which is determined by the FO and the parameters involved in this new lemma. • The existence of the solution is proved by using the Banach contractive mapping principle. By virtue of several inequality techniques and a newly proved lemma, the criterion of achieving F-TS is derived, which is characterized as algebraic inequalities and convenient to be rationally verified. • The appropriate controllers are designed to realize the F-TS. In this paper, an estimate scheme for derivative order-dependent fixed-time synchronization (F-TS) on Caputo quaternion-valued BAM neural networks (QVBAMNNs) with time-varying delays is implemented. A novel fractional-order (FO) F-T stability lemma is established by the synthetic properties of fractional calculus. The appropriate feedback controllers are designed to realize the F-TS between the master-slave systems. Based on the Lyapunov functional method and multiple inequality techniques, the F-TS criterion of FOQVBAMNNs is obtained. Furthermore, the upper bound of settling time is predicted and controlled by the Caputo derivative order and system parameters, which is independent of the initial state. Finally, two simulation examples manifest the feasibility and validity of the results on F-TS and the control strategies. 

Yuhong Cheng

Papers

Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects

Estimate scheme for fractional order-dependent fixed-time synchronization on Caputo quaternion-valued BAM network systems with time-varying delays

Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays