Asma Alharbi

This article investigates the stochastic asymptotic stability, boundedness, and square integrability of solutions to a class of second-order nonlinear stochastic integro-differential equations with multiple variable delays. The analysis is conducted through the construction of an appropriate Lyapunov–Krasovskii functional (L-KF), tailored to handle the combined effects of stochastic perturbations, time-varying delays, and integro-differential memory terms. Unlike many existing studies, our framework accommodates all these complexities simultaneously, thereby generalizing and extending recent contributions in the field while relaxing several restrictive assumptions. To validate the theoretical results and illustrate their practical applicability, numerical simulations are provided.

We establish a complete classification of Hyers–Ulam stability on the whole real line for the second‐order constant‐coefficient equation m > 0, b ∈ ℂ , k ∈ ℝ . The equation is Hyers–Ulam stable on ℝ if and only if its characteristic polynomial P ( λ ) = m λ ² + b λ + k has no roots on the imaginary axis. More explicitly, writing b = u + i v , the stability criterion is k ≠ 0 and [ u ≠ 0 or ( u = 0 and v ² + 4 m k < 0)]. This criterion covers undamped oscillators, positive damping, antidamping, negative stiffness, purely imaginary damping, and genuinely complex damping. In the undamped case, stability holds precisely for k < 0, and the sharp Hyers–Ulam constant C _{H U} = 1/| k | is obtained from a Green‐kernel upper estimate and a matching lower bound. If the characteristic spectrum intersects the imaginary axis, an explicit bounded resonant forcing generates a secular particular solution: A simple imaginary‐axis root produces linear amplitude growth, whereas a double root produces quadratic growth. By contrast, every fixed nonresonant forcing frequency yields a bounded beat response, although the corresponding bound becomes singular as the forcing frequency approaches resonance. Whenever the spectrum avoids the imaginary axis, bounded corrections are constructed through causal, anticausal, or two‐sided Green operators, according to the spectral configuration. These results distinguish whole‐line Hyers–Ulam stability from forward asymptotic stability and provide a unified spectral classification of second‐order linear oscillators under uniformly bounded perturbations.

This paper explores the problem of analyzing the Hyers–Ulam stability of quaternion‐valued neural networks incorporating time‐varying delays through the utilization of the Caputo–Fabrizio fractional differential operator. A lemma has been generalized from the existing results which will be used to establish the desired stability behavior. We examine the existence and uniqueness of the solution for the considered quaternion‐valued neural networks by employing Banach and Darbo's fixed point techniques. Furthermore, we evaluate the Hyers–Ulam stability behavior of our model within the framework of the Caputo–Fabrizio fractional operator by establishing some sufficient conditions. Lastly, two numerical examples have been presented to validate the stability of the proposed model by demonstrating the effectiveness of the derived results.

This paper presents a mathematical model to investigate the effects of radiation on tumor growth, which is modified using piecewise differential operators. The model incorporates classical derivatives in one interval, fractional derivatives (such as Caputo and Caputo–Fabrizio) in another, and stochastic components in yet another. The incorporation of fractional derivatives allows for the modeling of memory effects and non-local interactions, while the stochastic elements capture random fluctuations and uncertainties in tumor behavior. The study explores both deterministic and stochastic behaviors of the model under these different derivative types, focusing on how fractional and stochastic components influence the model’s dynamics. The effects of deterministic, fractional, and stochastic approaches on the model’s output are thoroughly analyzed. The piecewise model is solved using the sequential midpoint method, and simulations are provided to illustrate the model’s predictions and the impact of these advanced derivative techniques on tumor growth dynamics under radiation treatment.

This study explores the Hyers–Ulam stability and controllability of nonlinear integro-differential equations involving the Caputo fractional derivative. Fixed point methods are employed to establish the existence and uniqueness of solutions. The Hyers–Ulam stability is analyzed within the context of the Caputo fractional operator by formulating the necessary conditions. Furthermore, the controllability of the obtained solutions is investigated, providing a deeper understanding of their qualitative properties. The study also presents illustrative examples and practical applications of the proposed equations.

This study investigates the intricate dynamics of hybrid nanofluid (HNF) flow within the annular space between two concentric cylinders. The hybrid nanofluid consists of a precise blend of ethylene glycol (80%) and water (20%) combined with copper oxide (CuO) and silver (Ag) nanoparticles. This research aims to delineate the intricate flow, temperature, and pressure profiles of the hybrid nanofluid influenced by the rotation of the outer cylinder, with the inner cylinder remaining stationary, mitigating joule heating and fluid velocity through the application of a radial magnetic field. A unique deep neural network (DNN) technique enhances model prediction accuracy by combining diverse influencing variables. Numerical simulations meticulously examine the distribution of fluid velocity, temperature, and pressure, corroborated by measurements. As the strength of the magnetic field increases, fluid velocity decreases, whereas the Brinkman number and magnetic field intensity exhibit a positive correlation with fluid temperature. Intense magnetic fields diminish fluid velocity and elevate pressure. Magnetic properties and Brinkman numbers substantially influence temperature gradients within the system. The model effectively replicates fluid dynamics conditions owing to the strong connection between projected and actual values. This work investigates the flow characteristics of hybrid nanofluids within concentric cylinders, focusing on nanoparticle concentration, magnetic field parameters, and Brinkman numbers. The results elucidate fluid dynamics in intricate systems, establishing a foundation for subsequent research and technological applications.

The multifaceted burden of HIV infection encompasses significant health, social, economic, and psychological challenges. Mitigating these impacts, improving the well-being of those affected, and eliminating the HIV epidemic are of paramount importance. In this research, a mathematical model is constructed to study the dynamics of immune response and HIV with the effects of antiretroviral drugs through the application of fractional derivatives. The foundational theory of fractional calculus is introduced for the analysis of the system. This research focuses on the qualitative and quantitative examination of the proposed system of HIV. The existence and uniqueness of solutions are examined using the fixed-point theorems of Banach and Schaefer. Furthermore, the solutions of the system are examined for Ulam–Hyers’ stability. The impact of different input parameters on the dynamics is investigated numerically. The system’s key parameters are identified to facilitate the control and effective management of the infection.

We study the number of limit cycles of the following planar differential system: [ ] where [ ] are positive integers, for every [ ] the polynomial [ ] has degree [ ] with [ ] [ ] and [ ] are generalized trigonometric functions and [ ] is a small parameter. We provide an accurate upper bound of the maximum number of limit cycles such that the above system can have bifurcation from an unperturbed Hamiltonian center, using the averaging theory of the first order. These findings contribute to the broader understanding of nonlinear dynamics and have potential applications in various fields such as control systems, signal processing, and artificial intelligence.