This study addresses the exponential stabilization of second-order impulsive delay differential equations and impulsive integro-delay differential equations with finitely many constant delays via Lyapunov-Krasovskii functionals. Throughout the paper, four new theorems are established, providing sufficient conditions for the exponential stabilization of the considered impulsive problems. The findings in this paper generalize and strengthen known results, which are available in the relevant literature, thereby contributing to the ongoing development of the field. Finally, two illustrative examples are provided to highlight the applicability, efficacy, and advantages of the proposed results.

Using a version of the degree of nondensifiability in a generalized Banach space in the sense of Perov, we establish the existence of solutions to a system of tempered fractional differential equations subject to three-point boundary conditions. Furthermore, we investigate the Ulam stability of the system. Finally, we illustrate our theoretical results with an example.

In this article, we investigate a nonlinear ψ-Hilfer fractional order Volterra integro-delay differential equation (ψ-Hilfer FRVIDDE) and a nonlinear ψ-Hilfer fractional Volterra delay integral equation (ψ-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the ψ-Hilfer FRVIDDE and ψ-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered ψ-Hilfer FRVIDDE and ψ-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature.

This paper examines the asymptotic behavior of solutions to linear fractal differential equations within the framework of F^α -calculus. We identify the conditions that determine whether the solutions remain stable, grow, or decay. These dynamics are further explored through comprehensive examples and theoretical findings, emphasizing the self-similar characteristics of solutions, including first- and second-order higher α -order fractal differential equations.

In this study, we addressed a higher-order iterative Volterra integro-delay differential equation (HOIVIDDE) involving two variable time delays. Our primary focus was on establishing the uniqueness of solutions and analyzing Ulam-type stability properties of the considered HOIVIDDE. We presented three novel results concerning Ulam-Hyers-Rassias (U-H-R), sigma-semi-Ulam-Hyers (sigma- semi-U-H), and Ulam-Hyers (U-H) stability for HOIVIDDE, along with uniqueness results for the associated initial value problem (IVP). The analysis was conducted using the properties of iterative functions, the Banach fixed point theorem, and the Bielecki metric. Notably, this was the first study that extended and enhanced these qualitative properties to an nth-order HOIVIDDE. To illustrate the applicability of the results obtained here, we provided an example verifying the requirements of the new theorems.

The paper deals with a nonlinear Caputo fractional order neutral differential equation (CFrNDE) such that nonlinear terms in the CFrNDE contain multiple variable delays. Some new results with regard to the existence, uniqueness of solutions and the Ulam–Hyers–Mittag–Leffler (UHML) stability for the considered CFrNDE are presented by using the Banach fixed point theorem, the Chebyshev norm, the Bielecki norm and the Picard operator. Two numerical examples are also provided to display the validation of our new results.

We present an analysis to establish the stability results for a general non-linear Caputo integro-fractional delay coupled system with multiple variable time delays in the sense of Ulam–Hyers–Rassias on finite and infinite intervals employing a fixed point method. A theoretical example is presented to demonstrate our findings.

In this paper, we develop the theory of beta,gH-calculus for interval-valued functions by combining the beta-functions with the generalized Hukuhara difference. Within this framework, we establish various properties related to beta,gH-differentiation and beta,gH-integration. Furthermore, based on two distinct and nonequivalent convexity structures, such as CW-convexity and subset of-convexity, several beta-Hermite-Hadamard type inequalities are established for interval-valued functions under suitable convexity conditions. A statistical motivation for the restricted beta,gH-integral is also presented.

We investigate two concepts of Ulam's type stability for an impulsive delay differential equation (DDE) with several time-varying delays. By applying Banach contraction mapping principle (Banach CMP), the integral inequality of Gronwall type for piecewise continuous functions and abstract Gronwall lemma, Ulam-Hyers stability and Ulam-Hyers-Rassias stability results for impulsive DDE with several time-varying delays are obtained. We also provide an example to demonstrate how our findings might be applied. In addition, this article offers some novel complementary findings related to the qualitative theory of impulsive DDEs with multiple time-varying delays.

We investigate the existence of solutions for a class of semilinear integrodifferential evolution equations with infinite delay and infinite state-dependent delay in Banach spaces. We employ a new fixed-point theorem that relies on the degree of nondensifiability. To illustrate our findings, we present an illustrative example that demonstrates the key outcomes of our analysis.

This study explores an SEIR epidemic model, aiming to achieve rapid stabilization of infectious disease dynamics. The model's dynamic behavior is analyzed with an emphasis on both local and global stability of equilibria using a Lyapunov function. The existence and uniqueness of the model are confirmed. The theoretical findings are validated, and the controller's effectiveness is illustrated through numerical simulations conducted in MAT-LAB/Simulink.

In this paper, we discuss the existence of a unique solution and Ulam-Hyers-Mittag-Leffler (UHML) stability for a nonlinear Hammerstein-type psi-Hilfer delay differential equation of fractional order (psi-HFDDE) in the configuration of several variable time-dependent delays. Our study is based on fixed-point methodology. The results established in this work are novel and enhance the literature on the topic. We also present examples to illustrate the application of the obtained results. The findings of this paper provide new contributions to the qualitative theory of Hilfer fractional delay differential equations.

. This paper delves into the exponential stabilization of delay differential equations (DDEs) and integro-delay differential equations (IDDEs) of second order including multiple constant retardations. By means of impulsive controls (ICs), we prove that the considered DDEs and IDDEs of second order can be exponentially stabilized. By defining two new Lyapunov-Krasovskii functionals (LKFs), we prove four new results on the exponential impulsive stabilization of the equations. Two examples are given to show numerical applications of the results. The results of this paper generalize and enhance some recent and related results in the literature.

In this article, we conduct a rigorous analysis of the Ulam-type stability of first-order impulsive delay differential equations (IP-D-D-Es) with multiple time-dependent delays. Employing a Gronwall-type integral inequality tailored for piecewise continuous functions, we derive two new theorems concerning the generalized Ulam–Hyers–Rassias (G-U-H-R) stability of the first-order IP-D-D-E incorporating several constant time delays. To illustrate the applicability of the theoretical results, a concrete example is presented. The outcomes of this study offer significant and complementary contributions to the qualitative theory of the IP-D-D-Es with multiple constant delays.

In this paper, we extend the principles of Nambu mechanics by incorporating fractal calculus. This extension introduces Hamiltonian and Lagrangian mechanics that incorporate fractal derivatives. By doing so, we broaden the scope of our analysis to encompass the dynamics of fractal systems, enabling us to capture their intricate and self-similar properties. This novel approach opens up new avenues for understanding and modeling complex fractal structures, thereby advancing our comprehension of these intricate phenomena.

This paper establishes sufficient conditions for the stochastic asymptotic stability (SAS) and uniform stochastic boundedness (USB) of solutions to a class of fourth-order stochastic delay differential equations. By defining an appropriate Lyapunov-Krasovskii functional (LKF), two new theorems are proved that guarantee these stability and boundedness properties. As an application of the theoretical findings, two illustrative examples are presented to demonstrate the effectiveness of the established criteria. Finally, the results provide a novel and meaningful contribution to the qualitative analysis of higher-order stochastic delay differential equations and are expected to support further theoretical developments and applied investigations.

We study a nonlinear Hilfer fractional‐order delay integro‐differential equation ( Hilfer FrODIDE) that incorporates multiple variable time delays. Utilizing the Hilfer fractional derivative ( Hilfer‐FrD), we investigate the Ulam–Hyers––Rassias (U–H–R), semi‐Ulam–Hyers–Rassias (semi‐U–H–R) and Ulam–Hyers (U–H) stability of the considered Hilfer FrODIDE through the fixed‐point method. Throughout this work, using Banach's fixed‐point theorem and the Bielecki norm, we establish three new theorems related to these qualitative concepts. The theorems presented in this work are novel and contribute to the existing literature on the Ulam‐type stability of Hilfer FrODIDEs.

In the present work, we pay attention to a number of nonlinear Volterra integro-differential equations (VIDEs) with constant time-lag. We define three new Lyapunov functionals (LFs) and employ them to get specific conditions guaranteeing the uniform exponential asymptotic stability (UEAS) of the trivial solutions of the (VIDEs) considered. The results obtained generalize, compliment and improve the existing results in the literature from the cases of the without delay to the more general cases with time-lag.

Diverse fields of modern science, including mathematical modeling and physics, plasma physics, atmospheric sciences, marine sciences, hydrodynamics, nonlinear mechanics, and other complex nonlinear physical phenomena, are expressed through Nonlinear Partial Differential Equations (NLPDEs). Exact solutions play an important role in understanding the behavior of solitary wave solutions and the dynamical properties of significant outcomes for higher-dimensional NLPDEs. As an integrable extension of the nonlinear Schr & ouml;dinger equation, the nonlocal Kundu-Eckhaus model comprehends higher-order nonlinearities and nonlocal effects. The stability properties of dark, bright, periodic multiple, and singular soliton solutions to the nonlocal Kundu-Eckhaus (KE) equation within the Parity-Time (PT) symmetry model are studied here, whereas past studies have neglected the effects of nonlocal interactions and PT symmetry. AG ' GE is a well-established method, particularly for depicting wave dynamics in nonlinear optical and quantum systems. We attain exact traveling wave solutions evolved in hyperbolic and trigonometric forms, representing several classes of solitons by applying this method. The findings elucidate that the obtained soliton solutions are stable under small perturbations, highlighting their robustness and the persistence of localized waveforms in nonlocal circumstances. We plotted 3D, 2D, and contour graphs for simulating our findings using MATLAB. This research sheds light on the mysterious understanding of nonlocal nonlinear wave behaviors and offers analytical methodologies for future scrutiny in fields such as optical physics and complex nonlinear systems.