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In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained. [ABSTRACT FROM AUTHOR]

In this article, we test whether solutions of second-order delay functional differential equations oscillate. The considered equation is a general case of several important equations, such as the linear, half-linear, and Emden-Fowler equations. We can construct strict criteria by inferring new qualities from the positive solutions to the problem under study. Furthermore, we can incrementally enhance these characteristics. We can use the criteria more than once if they are unsuccessful the first time thanks to their iterative nature. Sharp criteria were obtained with only one condition that guarantees the oscillation of the equation in the canonical and noncanonical forms. Our oscillation results effectively extend, complete, and simplify several related ones in the literature. An example was given to show the significance of the main results. [ABSTRACT FROM AUTHOR]

In this article, we explore the question of existence and finite time stability for fuzzy Hilfer-Katugampola fractional delay differential equations. By using the generalized Gronwall inequality and Schauder's fixed point theorem, we establish existence of the solution, and the finite time stability for the presented problems. Finally, the effectiveness of the theoretical result is shown through verification and simulations for an example. [ABSTRACT FROM AUTHOR]

In this article, we explore the question of existence and finite time stability for fuzzy Hilfer-Katugampola fractional delay differential equations. By using the generalized Gronwall inequality and Schauder's fixed point theorem, we establish existence of the solution, and the finite time stability for the presented problems. Finally, the effectiveness of the theoretical result is shown through verification and simulations for an example. [ABSTRACT FROM AUTHOR]

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them. [ABSTRACT FROM AUTHOR]

Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented. [ABSTRACT FROM AUTHOR]

Uncertain fractional-order delay differential equation is a class of fractional-order functional differential equations driven by Liu process. This paper devotes to studying linear uncertain fractional-order delay differential equation. The explicit representation and iterative formula of the solution to linear uncertain fractional-order delay differential equations are obtained. Meanwhile, the inverse uncertainty distribution of the solution to linear uncertain fractional-order delay differential equation by the α -path is presented. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form (r (t) (y (t) + p (t) y (t − τ)) ″) ″ + q (t) G (y (t − α)) − h (t) H (y (t − σ)) = 0 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0 ∞ t r (t) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$ [ABSTRACT FROM AUTHOR]

Under consideration is the possibility of constructing an almost periodic solution for one inhomogeneous system of linear differential equations with delay that is a linear function of the argument (time) under some assumptions about the right-hand side of the system. This solution is proved to be asymptotically stable. Also, we study the existence of an almost periodic solution to another system without neutral terms; in this event the solution is stable. [ABSTRACT FROM AUTHOR]

This article concerns the oscillatory behavior of solutions to second-order half-linear delay differential equations with mixed neutral terms. The authors present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

In this work, an attempt is made to discuss the existence of nonoscillatory solutions of second order nonlinear neutral differential equations with variable delays. The main tools are Lebesgue's dominated convergence theorem and Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions. This problem is considered in various ranges of the neutral coefficient. Further, two illustrative examples showing applicability of the new results are included. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form (x ⁢ (t) + r ⁢ (t) ⁢ x ⁢ (t - τ) ) ′ + ∑ i = 1 m ϕ i ⁢ (t) ⁢ H ⁢ (x ⁢ (t - σ i)) = f ⁢ (t) \bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t) is oscillatory or tends to zero as t → ∞. {t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results. [ABSTRACT FROM AUTHOR]

Abstract A new criterion for the existence of positive solutions of the second-order delayed differential equation y ̈ (t) = f (t , y t , y ̇ t) , t ∈ [ t 0 , ∞) is given with applications to linear equations. Open problems for future research are formulated. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout. [ABSTRACT FROM AUTHOR]

Monotonicity of solutions is an important property in the investigation of oscillatory behaviors of differential equations. A number of papers provide some existence criteria for eventually positive increasing solutions. However, relatively little attention is paid to eventually positive solutions that are also eventually decreasing solutions. For this reason, we establish several new and sharp oscillatory criteria for impulsive functional differential equations from this viewpoint. [ABSTRACT FROM AUTHOR]

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$, where D is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay. [ABSTRACT FROM AUTHOR]

Asymptotic properties and estimate of singular solutions (either defined on a finite interval only or trivial in a neighbourhood of ∞) of the second order delay differential equation with p-Laplacian are investigated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a delayed predator–prey system with diffusion effect. First, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the distribution of the eigenvalues. Next the direction and the stability of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results. [Copyright &y& Elsevier]

Abstract: Based on the stability analysis of the impulsive functional differential equation, the exponential synchronization of the complex dynamical network with a coupling delay and impulses is investigated in the paper. The criteria for the exponential synchronization are derived by the geometrical decomposition of network states and linear matrix inequality method. Two examples are given to show the effectiveness of the proposed criteria. [Copyright &y& Elsevier]

Abstract: The Letter studies the projective synchronization of a class of delayed chaotic systems. The drive-response system can be synchronized to within a desired scaling factor via impulsive control. Some sufficient conditions are derived by the stability analysis of the impulsive functional differential equations. An illustrative example is provided to show the effectiveness and feasibility of the proposed method and results. [Copyright &y& Elsevier]

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question. [ABSTRACT FROM AUTHOR]

Abstract: A symmetric BAM neural network model with delay is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. The existence of multiple periodic solutions is established using a symmetric Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society 350 (12) (1998) 4799–4838]. [Copyright &y& Elsevier]

Abstract: In this paper, we establish some sufficient conditions for the persistence of system without the condition that is monotonously decreasing with respect to . This partly answers the open problems proposed by Teng. [Copyright &y& Elsevier]

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

We study a semi-discretisation scheme for stochastic optimal control problems whose dynamics are given by controlled stochastic delay (or functional) differential equations with bounded memory. Performance is measured in terms of expected costs. By discretising time in two steps, we construct a sequence of approximating finite-dimensional Markovian optimal control problems in discrete time. The corresponding value functions converge to the value function of the original problem, and we derive an upper bound on the discretisation error or, equivalently, a worst-case estimate for the rate of convergence. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that the non-impulsive equations can be stabilized by the imposition of proper impulsive control. Some recent results are extended and improved. We give two examples to illustrate the efficiency of our results. [Copyright &y& Elsevier]

Abstract: In this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form where and are certain constants and . Here is a constant delay. In proving our results we make use of Pontryagin''s theory for quasi-polynomials. [Copyright &y& Elsevier]

Abstract: In this article, a nonlinear impulsive neutral differential equation with time-varying delays is considered. By establishing a singular impulsive delay differential inequality and transforming the -dimensional impulsive neutral delay differential equation to a -dimensional singular impulsive delay differential equation, some sufficient conditions ensuring the global exponential stability in of the zero solution of an impulsive neutral differential equation are obtained. The results extend and improve on earlier publications. An example is given to illustrate the theory. [Copyright &y& Elsevier]

Abstract: A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe. [Copyright &y& Elsevier]

Abstract: In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458]. [Copyright &y& Elsevier]

We derive from a structured population model a system of delay differential equations describing the interaction of five subpopulations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector borne disease with particular reference to West Nile virus, and we also incorporate spatial movements by considering the analogue reaction-diffusion systems with nonlocal delayed terms. Specific conditions for the disease eradication and sharp conditions for the local stability of the disease-free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semiflows. A formal calculation of the minimal wave speed for the traveling waves is given and compared with field observation data. [ABSTRACT FROM AUTHOR]

A delayed periodic ratio-dependent predator–prey model with prey dispersal and stage structure for predator is investigated. It is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age, and that immature predators don''t have the ability to attack prey, and that predator species is confined to one of the patches while the prey species can disperse between two patches. We first discuss the uniform persistence and impermanence of the model. By using Gaines and Mawhin''s continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model. Numerical simulations are presented to illustrate the feasibility of our main results. [Copyright &y& Elsevier]

Delayed duopoly model with gradient dynamics is considered. Linear stability constraints are derived. It is shown that delay increases stability. Hence firms using duopoly model with gradient method in optimized theory with delay has a higher chance of reaching Nash equilibrium. Cycles and chaos regions are determined. [Copyright &y& Elsevier]

In this paper, we discuss the quasi-periodic logistic delay differential equations. As a corollary, we give a more sharp result than that in [G. Seifert, J. Differential Equations 164 (2000) 451–458] for the periodic logistic delay differential equations. [Copyright &y& Elsevier]

Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system's past, allowing for more accurate and efficient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufficient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results. [ABSTRACT FROM AUTHOR]

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results. [ABSTRACT FROM AUTHOR]

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic and oscillatory properties of a certain class of third-order neutral delay differential equations with middle term. We obtain new characterizations of oscillation of the third-order neutral equation in terms of oscillation of a related, well-studied, second-order linear equation without damping. An Example is provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle. [ABSTRACT FROM AUTHOR]

In the paper, we study the oscillatory and asymptotic properties of solutions to a class of third-order linear neutral delay differential equations with noncanonical operators. Via the application of comparison principles with associated first and second-order delay differential inequalities, we offer new criteria for the oscillation of all solutions to a given differential equation. Our technique essentially simplifies the process of investigation and reduces the number of conditions required in previously known results. The strength of the newly obtained results is illustrated on the Euler type equations. [ABSTRACT FROM AUTHOR]

In this work, oscillatory behaviour of the solutions of a class of non-linear first-order neutral differential equations with several delays of the form ... and ... are studied under various ranges of p(t). Sufficient conditions are obtained for existence of bounded positive solutions of (E1). [ABSTRACT FROM AUTHOR]