Vol. 15, No. 3 (2026): Regular Issue

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.

Biological and behavioral strategies that enhance species survival and population stability play a crucial role in predator–prey interactions within ecological food webs. In this study, we develop and analyze a mathematical model involving two prey species and a single predator, incorporating key ecological mechanisms such as herd behavior, fear effects, prey refuge, intraspecific competition among predators, and the Allee effect in one of the prey species. The predator–prey dynamics are governed by a Holling type II functional response that accounts for herd behavior, where the prey capture rate is proportional to the square root of prey density. We derive equilibrium points to examine system stability and establish the non-negativity and boundedness of solutions. Local stability conditions at the interior equilibrium are analyzed, and bifurcation analyses including transcritical and Hopf bifurcations are carried out to explore the system's dynamic behavior. Numerical simulations support the analytical results, offering insights into species interactions through time-series analysis and one-parameter bifurcation diagrams. The findings emphasize the roles of prey refuge, the Allee effect, and other ecological factors in shaping predator–prey dynamics, contributing to a deeper understanding of ecosystem stability and species persistence.

This paper investigates a nonlinear biharmonic equation involving variable exponent source terms, which arise naturally in models with non-standard growth conditions. The primary aim is to analyze the qualitative behavior of solutions, with particular focus on the existence and finite-time blow-up phenomena under certain energy constraints. We are especially interested in understanding how the spatial variability of the exponent influences the dynamics of the solution. We first establish the existence of weak solutions within an appropriate functional framework. The main objective, however, is to explore the conditions leading to the finite-time blow-up of solutions when the initial energy is negative. To this end, we demonstrate that such solutions cannot remain globally bounded and, in fact, blow up in finite time. Moreover, we derive sharp upper and lower bounds for the blow-up time, highlighting the dependence of these estimates on the initial data and the structure of the variable exponent. We also provide a precise characterization of the blow-up rate.

In this paper, we study the existence and uniqueness of solutions for mixed fractional integro-differential equations combined with integral boundary conditions of slit-strips type. Uniqueness results are proved using the classical contraction mapping principle and D.O'Regan's fixed point theorem is used to establish the existence results. To illustrate our main findings, we present an example.

This work explores novel integral inequalities for $(h,m)$-convex functions using key mathematical techniques, including Hölder's, Young's and Power Mean inequalities. By leveraging these classical results, we establish improved bounds and extend Hermite-Hadamard type inequalities. The findings contribute to a deeper understanding of $(h,m)$-convex functions and their role in mathematical analysis. Additionally, we discuss particular cases to emphasize the relevance of our contributions.