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In this paper, we investigate a class of a third-order neutral-type differential equation with time-varying delays. Some sufficient conditions on the existence of a periodic solution are established for the considered system. Different from the previously reported research results, by utilizing the properties of neutral operators and a special variable substitution, we transform a high-order neutral equation into a first-order three-dimensional nonneutral system. The existence of a periodic solution such as the high-order neutral equation has not been given much attention in past papers due to the difficulty of estimation of prior bounds of solutions. This paper is devoted to the use of properties of neutral-type operators with a variable parameter and Mawhin's continuation theorem for overcoming the above difficulties. The neutral term in the third-order neutral differential equation in this paper contains a variable parameter which is different from third-order neutral-type equations that have been studied. The third-order neutral-type equation studied in this paper is more general, and similar equations studied in the past are special cases of the equations studied in this paper. Finally, an example is given to elucidate the effectiveness and values of the present results. Our results are new and complement the related results of third-order functional differential equations. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with some qualitative analyses of fractional-order partial hyperbolic functional differential equations under the ψ -Caputo type. To be precise, we investigate the existence and uniqueness results based on the nonlinear alternative of the Leray-Schauder type and Banach contraction mapping. Moreover, we present two similar results to nonlocal problems. Then, the guarantee of the existence of solutions is shown by Ulam-Hyer's stability. Two examples will be given to illustrate the abstract results. Eventually, some known results in the literature are extended. [ABSTRACT FROM AUTHOR]

In this analysis, we develop a new approach to investigate the semianalytical solution of the delay differential equations. Mohand transform coupled with the homotopy perturbation method is called Mohand homotopy perturbation transform method (MHPTM) and performs the solution results in the form of series. The beauty of this approach is that it does not need to compute the values of the Lagrange multiplier as in the variational iteration method, and also, there is no need to implement the convolution theorem as in the Laplace transform. The main purpose of this scheme is to reduce the less computational work and the error analysis of the problems than others studied in the literature. Some illustrated examples are interpreted to confirm the accuracy of the newly developed scheme. [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 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]

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the proposed theories. [ABSTRACT FROM AUTHOR]

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD. [ABSTRACT FROM AUTHOR]

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods. [ABSTRACT FROM AUTHOR]

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

We study the second-order neutral delay differential equation [r(t)Φγ(z'(t))]' + q(t)Φβ(x(σ(t))) = 0, where Φα(t) = |t|α-1t, α ≥ 1 and z(t) = x(t) + p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫∞ r-1/γ (t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case. [ABSTRACT FROM AUTHOR]

A linearized compact difference scheme is provided for a class of variable coefficient parabolic systems with delay. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions. [ABSTRACT FROM AUTHOR]

In 2010, Agarwal et al. studied the existence of a one-dimensional fractional neutral functional differential equation. In this paper, we study an initial value problem for a class of k-dimensional systems of fractional neutral functional differential equations by using Krasnoselskii's fixed point theorem. In fact, our main result generalizes their main result in a sense. [ABSTRACT FROM AUTHOR]

By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations x(t) = -f(x(t - σ)), where σ ∊ (0,+8) and f ∊ C(R", R"). It is shown that all the periodic solutions derived here are brought about by the time delay. [ABSTRACT FROM AUTHOR]

This paper studies the following third order neutral delay discrete equation Δ(anΔ2(xn+pnxn-τ))+ f(n, xn-d1n, . . ., xn-dln) = gn, n ≥ n0, where τ, l ∈ N, n0 ∈ N ∪ {0}, {an} n∈Nn0, {pn} n∈Nn0, {gn} n∈Nn0 are real sequences with an≠0 for n ≥ n0, {din} n∈Nn0 ⊆ Z with limn→∞(n - din= +∞ for i ∈ {1, 2, . . ., l} and f ∈ C(Nn0 × Rl,R. By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper. [ABSTRACT FROM AUTHOR]

We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity. [ABSTRACT FROM AUTHOR]

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee the pth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results. [ABSTRACT FROM AUTHOR]

We study the following second-order neutral functional differential equation with mixed nonlinearities (r(t)|(u(t) + p(t)u(t - σ))'|α- 1(u(t) + p(t)u(t - σ))')' + q0(t)|u(τ0(t))|α-1u(τ0(t)) + q1(t)|u(τ1(t))|β-1u(τ1(t)) + q2(t)|u(τ2(t))|γ-1u(τ2(t)) = 0, where γ > α > β > 0, ∫Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.(1/r1/α(t))dt < ∞. Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010). [ABSTRACT FROM AUTHOR]

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments [a(t)[x"(t)]γ]' = q(t)f(x[τ(t)]+p(t)h(x[σ(t)]). Bothcases ∫∞ a-1/γ (s)ds = ∞ and ∫∞ a-1/γ (s)ds < ∞ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones. [ABSTRACT FROM AUTHOR]

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established. [ABSTRACT FROM AUTHOR]

The article presents information on a study which investigated the global behavior of the difference equation of higher order. It notes that although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the global behavior of their solutions. It describes difference equations as discrete analogues and as numerical solutions of differential and delay differential equations having applications in various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology.

The article presents information on a study which investigated the global behavior of the difference equation of higher order. It notes that although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the global behavior of their solutions. It describes difference equations as discrete analogues and as numerical solutions of differential and delay differential equations having applications in various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology.

The article investigates a delayed stage-structured Ivlev's functional response predator-prey model with continuous harvesting on predator and impulsive stocking on prey. It examines the condigions of the global attractivity of predator-extinction periodic solution and the permanence of the system. The findings indicate the important role of the behavior of impulsive stocking on prey for the performance of the system. According to the authors, the results offer reliable tactical basis for the biological resource management and enrich the theory of impulsive delay differential equations.