Using the definitions from [1] on the $q_e-$calculus we prove $q_e-$analogues of the Fixed Point Theorem and Inverse Function Theorem.

This study examines a class of fractional stochastic pantograph differential equations involving the Hilfer fractional derivative and non-instantaneous impulses. The existence of solutions is established using topological degree theory, while Banach's contraction principle is employed to demonstrate uniqueness. Additionally, an illustrative example and graphical analysis are provided to validate the findings.

In this paper, we are concerned with the output stabilization of infinite dimensional bilinear systems with delayed observation. We first establish the well-posedness of such systems and then we provide sufficient conditions for strong and weak output stabilization. Through the analysis of the wave equation and beam equation, we demonstrate the applicability and effectiveness of our proposed methods.

This paper tackles some existence results for semilinear integro-differential equations with non-instantaneous impulsions on a finite interval via resolvent operators. Our criteria, obtained by applying a new fixed point theorem with respect to Meir-Keeler condensing operators. The obtained result is illustrated by an example at the end.

This article proposes and analyzes a fear-induced predator-prey model incorporating prey density-dependent predator intra-specific competition. The study focuses on how non-constant predator intra-specific competition affects the dynamics of the system. The investigation explores various aspects, including positivity, boundedness, local and global stability, uniform persistence, and Hopf bifurcation. Numerical simulation supports theoretical results, offering practical insights into the model behaviour.

This work investigates the effects of thermal modulation on Darcy-Brinkman bio-convection in a porous medium saturated with a Newtonian fluid containing gyrotactic microorganisms. We investigate the oscillatory bioconvection with low modulation amplitude using a weak nonlinear stability analysis. The heat transfer is measured by the mean Nusselt number governed by a complex Ginzburg-Landau equation (CGLE). The CGLE is derived by solvability condition at lowest order of perturbed parameter. The results are presented graphically, illustrating the impact of the system parameters on heat transfer. The results show that both Vadaszs number and modulation amplitude have a progressive effect on heat transfer. On the other hand, upon increase in the modified bio-Rayleigh number and cell eccentricity leads to a decrease in heat transfer. It is found that only OPM/LBM are effective on controlling heat transfer than IPM. This highlights the effectiveness of external modulation in controlling heat transport within the system. Further, it is found that the convective heat transfer process may be delayed, due to asymmetries and irregularities ($\alpha \neq 0$) of microorganisms than spherical-shaped microorganisms ($\alpha = 0$).

This paper investigates the controllability of linear fractional-order control systems featuring both state and control delays, using the Riemann-Liouville fractional derivative framework. An explicit representation of the system's solution is derived, enabling the formulation of a controllability criterion based on the rank condition. We establish necessary and sufficient conditions ensuring controllability of the considered class of fractional systems. To validate the theoretical results, a numerical example is provided and discussed. In particular, a numerical simulation is carried out to illustrate the behavior of the system under the designed control law. The evolution of the state trajectories and the applied control input are plotted, showing the impact of the delay and the fractional-order dynamics on the system performance.

This work delves into the non-fragile projective synchronization (PS) of fractional-order neural networks (FONNs) with proportional and mixed-delay under a memory-based sampled-data (MSD) control framework. We build an MSD controller with norm-bounded uncertainty to accomplish synchronization in the addressed FONN systems. A suitable Lyapunov–Krasovskii functional (LKF) is developed, taking sampling instants into account when mixed delays are present. To provide adequate criteria for the asymptotic stability of the synchronization error system(ES) using linear matrix inequalities (LMIs), the fractional integral inequality is employed. Numerical modeling is provided to present the benefits and applicability of the suggested procedure.