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In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild random solutions. Next, we prove that our problems are controllable. Finally, an example is given to validate the theory part. [ABSTRACT FROM AUTHOR]

In this article, we study the η-stability in qth moment (η.s.q.m) of hybrid neutral stochastic differential equations with infinite delay (HNSDEID) using the Lyapunov techniques and the method of M-matrix. Finally, we apply the main result to some examples. [ABSTRACT FROM AUTHOR]

(iii) The interaction between prey and predator is of Holling type-II functional response: HT ht Graph (2.7) Here, 2 and represent natural death rate and coefficient of consumption rate, respectively, of the predator population, b 2 is the half saturation constant, a 1 denotes the conversion factor. So, the decline rate is given by HT ht and the growth rate is I bN i SB 1 sb , both depend on the present population size. [Extracted from the article]

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation ... oscillates or tends to zero as t → to ∞, where, n ≥ 1 is any positive integer, Pi, ri ∈ C (n) ([0, ∞), ℝ) and pi are bounded for each i = 1, 2,...,k. Further, f ∈ C([0, ∞), ℝ), g, h, v, u ∈ C([0, ∞), [0, ∞)), G and H ∈ C(ℝ, ℝ). The functional delays ri(t) ≤ t, g(t) ≤ t and h(t) ≤ t and all of them approach ∞ as t → ∞. The results hold when u = 0 and f (t) = 0. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature. [ABSTRACT FROM AUTHOR]

Considering some linear autonomous differential equation with distributed delay that has a positive fundamental solution, we develop a method for obtaining effective exponential stability tests and two-sided estimates of the fundamental solution in the form of two exponential functions with exponents and coefficients determined exactly. A few examples illustrate the use of the method. [ABSTRACT FROM AUTHOR]

In this paper, we shall apply symmetry analysis to second order functional differential equations with constant coefficients. The determining equations of the admitted Lie group are constructed in a manner different from that of the existing literature for delay differential equations. We define the standard Lie bracket and make a complete classification of the second order linear functional differential equations with constant coefficients, to solvable Lie algebras. We also classify some second order nonlinear functional differential equations with constant coefficients, to solvable Lie algebras. [ABSTRACT FROM AUTHOR]

Uncertain differential equations with time-varying delay are a special type of uncertain functional differential equation. Stability in measure and stability in mean for this type of differential equation have been investigated. For the sake of completeness, this article explores the other two types of stability. Firstly, the concepts of the almost sure stability and the pth moment stability of uncertain differential equations with time-varying delay are proposed. Next, sufficient conditions, in the form of theorems, for uncertain differential equations with time-varying delay to be stable almost surely and in pth moment are deduced. Finally, the relationships between several types of stability for uncertain differential equations with time-varying delay are discussed. [ABSTRACT FROM AUTHOR]

For the perturbed controlled nonlinear delay functional differential equation with the mixed initial condition a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of main interval. In the formula the effects of perturbations of the delay parameters containing in the phase coordinates and controls, the initial vector, the initial and control functions are detected. [ABSTRACT FROM AUTHOR]

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales. [ABSTRACT FROM AUTHOR]

We study existence and local uniqueness of periodic solutions of nonlinear functional differential equations of first order with small delays. Bifurcations of periodic and bounded solutions of particular periodically forced second-order equations with small delays are investigated as well. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior of even-order neutral delay differential equation (a ⋅ (u + ρ ⋅ u ∘ τ) (n − 1) ) ′ (ℓ) + h (ℓ) u (g (ℓ)) = 0 , ℓ ≥ ℓ 0 , {(a\cdot {(u+\rho \cdot u\circ \tau)}^{(n-1)})}^{^{\prime} }(\ell)+h(\ell)u(g(\ell))=0,\hspace{1.0em}\ell \ge {\ell }_{0}, where n ≥ 4 n\ge 4 , and in noncanonical case, that is, ∫ ∞ a − 1 (s) d s < ∞. \mathop{\int }\limits^{\infty }{a}^{-1}\left(s){\rm{d}}s\lt \infty. To the best of our knowledge, most of the previous studies were concerned only with the study of n n -order neutral equations in canonical case. By using comparison principle and Riccati transformation technique, we obtain new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Examples are presented to illustrate our new results. [ABSTRACT FROM AUTHOR]

In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of even-order nonlinear neutral differential equations with both retarded and advanced arguments. The arithmetic-geometric mean inequality is employed. Examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

We study the local and global existence and uniqueness of solution, the well-posedness and the existence of global attractor for abstract functional differential equations with state-dependent delay. Some examples concerning partial differential equations with state dependent delay are presented. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with systems determined by partial differential equations with infinite and state-dependent delay. To study the system, we model the equation as an abstract retarded functional differential equations with infinite and state-dependent delay. We establish the existence of mild solutions, and the existence of S-asymptotically periodic solutions and asymptotically periodic solutions using local Lipschitz conditions on the involved functions. We apply our results to study the existence of asymptotically periodic solutions of a partial differential equation with infinite and state-dependent delay. [ABSTRACT FROM AUTHOR]

The delay PDEs are called partial functional differential equations as their unknown solutions are used in these equations as functional arguments. On the other hand, a neutral delay PDE is an especial case when the equation depends on the derivative(s) of the solution at some past stage(s). The current paper concerns to find an accurate and robust numerical solution for solving neutral delay time-space distributed-order fractional damped diffusion-wave equation based on the Galerkin meshless method. The test and trial functions for the used Galerkin method are constructed from the shape functions of reproducing kernel particle method (RKPM). For this aim, time derivative is approximated by a finite difference formula with convergence order O (τ 3 − α) where 1 < α < 2. The stability and convergence of the time-discrete formulation are studied. For the next stage and to derive a fully-discrete scheme, we employ the Galerkin RKPM method. Moreover, the error estimate of the full-discrete scheme of new technique is discussed. Finally, an example is examined to check the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions of some classes of perturbed partial functional hyperbolic differential equations with state-dependent delay involving the Caputo fractional derivative by using a nonlinear alternative of Avramescu in Frtchet spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtaining a new oscillation criterion for a class of neutral delay differential equations of the type u(x)ym−1(x)κ′+q(x)zκγ(x)=0, where m ≥ 4 is an even integer and y(x):=z(x)+g(x)zξ(x). It should be noted here that the results obtained improve and complement those given by Xing et al. (2016). An example illustrating the applicability of the new criterion is presented. [ABSTRACT FROM AUTHOR]

In this work, we aim at studying the asymptotic and oscillatory behavior of even-order neutral delay noncanonical differential equations. To the best of our knowledge, most of the related previous works are concerned only with neutral equations in the canonical case. Our new oscillation criteria essentially improve, simplify, and complement related results in the literature, especially those from a paper by Li and Rogovchenko (Abstr. Appl. Anal. 2014:395368, 2014). Some examples are presented that illustrate the importance of the new criteria. [ABSTRACT FROM AUTHOR]

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of neutral time-delay control systems. The relative controllability of the linear system is given by constructing the corresponding neutral delay Gramian matrix and adopting the classical analysis idea. Next, we use Krasnoselskii's fixed point theorem to show that the semilinear system is relatively controllable under the mild condition. Finally, we give two numerical examples to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, nonlinear differential evolution equations of fractional order in Banach spaces involving unbounded delays are investigated. We aim to prove the existence of mild solutions and demonstrate its polynomial decay by the fixed point principle for condensing maps. An example of the application of abstract results is given for illustration. [ABSTRACT FROM AUTHOR]

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out. [ABSTRACT FROM AUTHOR]

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results. [ABSTRACT FROM AUTHOR]

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples. [ABSTRACT FROM AUTHOR]

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson's blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied. [ABSTRACT FROM AUTHOR]

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories. [ABSTRACT FROM AUTHOR]

This paper provides a mathematical characterization of the robust (strong) H-infinity norm of an uncertain linear time-invariant system with discrete delays in terms of the robust distance to instability of an associated characteristic matrix. The considered class of uncertainties consists of real-valued, structured, Frobenius norm-bounded matrix uncertainties that act on the coefficient matrices. The robust H-infinity norm, defined as the worst-case value of the H-infinity norm over all admissible uncertainty values, is an important measure of robust performance to quantify the worst-case disturbance rejection of an uncertain dynamical system. For the considered system class, this robust H-infinity norm is however a fragile measure, as for a particular instance of the uncertainties, the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the uncertainties on the system matrices and infinitesimal delay perturbations. This quantity is a continuous function of both the elements of the system matrices and the delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the robust distance to instability of an associated characteristic matrix. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems. [ABSTRACT FROM AUTHOR]

We propose and investigate a Chebyshev spectral collocation method for solving mixed functional differential equations. One can usually not solve these equations analytically, and hence one must employef numerical methods. Our method is for boundary value problems which include delay and advance terms in the solution or a derivative. Even though, in general, solutions are not very smooth, spectral collocation methods are well suited for these types of problems as they allow easy and accurate evaluation of the approximated solution at points which are not grid points. We present numerical results and examine smoothness related convergence behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of a class of fourth-order Emden–Fowler delay differential equations with neutral term. Using the Riccati transformation and comparison method, we establish several new oscillation conditions. These new conditions complement a number of results in the literature. We give examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Using Krasnoselskii's fixed‐point theorem, we obtain some results for the existence of periodic solutions to the third‐order neutral differential equations with delay. The results obtained essentially improve, include, and complement the results in the literature. [ABSTRACT FROM AUTHOR]

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: y (t) ≤ E (t , τ) ‖ y τ ‖ + ∫ τ t K 1 (t , s) ‖ y s ‖ d s + ∫ t ∞ K 2 (t , s) ‖ y s ‖ d s + ρ , t ≥ τ ≥ 0. As a simple example of application, the retarded scalar functional differential equation x ˙ = − a (t) x + B (t , x t) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system x ˙ = F 0 (t , x) + ∑ i = 1 m F i (t , x (t − r i (t))) on R n with superlinear nonlinearities F i (0 ≤ i ≤ m). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions. The third example for application concerns the study of the dynamics of the functional cocycle system d u d t + A u = F (θ t p , u t) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium solution Γ is obtained under the hyperbolicity assumption on operator A and some additional hypotheses, and the global asymptotic stability of Γ is also addressed. [ABSTRACT FROM AUTHOR]

Summary: In this article, we study the delay‐dependent stability of a class of hybrid neutral stochastic differential equations (NSDEs) with multiple delays which is highly nonlinear. Some novel stability criteria are investigated according to Lyapunov functions and generalized Itô's formula. In particular, we generalize and improve the results in Shen et al, Systems & Control Letters (2018) from a single delay to multiple delays. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of nonlinear neutral functional differential equations with state-dependent delays. We study well-posedness and differentiability of the solution with respect to the parameters in a pointwise sense and also in the supremum norm. [ABSTRACT FROM AUTHOR]

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation x ′ = A x (t) + L x t + f (t , x t , λ) , with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ. [ABSTRACT FROM AUTHOR]

This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results. [ABSTRACT FROM AUTHOR]

This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results. [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]

This paper concerns a model of hierarchical neural network with D n × D n -symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model. [ABSTRACT FROM AUTHOR]

Neutral Delay Differential Equation (NDDE) of pantograph type has been solved by developing a fifth order explicit multistep block method. NDDE has become of great interest among researchers in its industrial applications. In finding the solution for the problem, a two-point explicit multistep block method has been modelled by applying Taylor Series interpolation polynomial. The proposed method will solve pantograph NDDE at two points concurrently with the strategy of consistent step-size. The implementation is based on multistep method Adam Bashforth formula in predictor mode. In handling pantograph delay, Lagrange interpolation polynomial needs to be applied to find the solution of delay terms that are larger than the initial value given. The delay derivatives are estimated using divided difference formula. The order and convergence have been determined to ensure the reliability of the proposed explicit block method. The stability analysis has been constructed using test equation for NDDE. Numerical results obtained have shown that the suggested method is suitable and applicable for solving pantograph equation. [ABSTRACT FROM AUTHOR]

For the nonlinear controlled functional differential equations with several constant delays, the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected. [ABSTRACT FROM AUTHOR]

In this paper, we shall apply symmetry analysis to some first order functional differential equations with constant coefficients. The approach used in this paper accounts for obtaining the inverse of the classification. We define the standard Lie bracket and make a complete classification of some first order linear functional differential equations with constant coefficients to solvable Lie algebras. We also classify some nonlinear functional differential equations with constant coefficients to solvable Lie algebras. [ABSTRACT FROM AUTHOR]