Abstract:
This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.

Abstract:
This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.

Abstract:
Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results. 1. Introduction Stochastic modelling has played an important role in many areas of science and engineering for a long time. Some of the most frequent and most important stochastic models used when dynamical systems not only depend on present and past states but also involve derivatives with functionals are described by the following neutral stochastic functional differential equation: The conditions imposed on their studies are the standard uniform Lipschitz condition and the linear growth condition. The classical result is described by the following well-known Mao's test see [1, page 202, Theorem ]. Theorem 1.1. Assume that there exist positive constants , and such that for all Then there exists a unique solution to (1.1) with initial data (i.e., is an measurable -valued random variable such that ). Theorem 1.1 requires that the coefficients and satisfy the Lipschitz condition and the linear growth condition. However, there are many NSFDEs that do not satisfy the linear growth condition. For example, the following nonlinear NSFDE: where coefficients and do not obey the linear growth condition although they are Lipschitz continuous. To the authors' best knowledge, there is so far no result that shows that (1.3) has a unique global solution for any initial data. On the other hand, we still encounter a new problem when we attempt to deduce the exponential decay of the solution even if there is no problem with the existence of the solution. For example, Mao [2] initiated the following study of exponential stability for NSFDEs employing the Razumikhin technique. Theorem 1.2. Let be all positive numbers and for any and assume that there exists a function such that for all and also for all provided satisfying for all Then for all where It is very difficult to verify the conditions of Theorem 1.2, and it is clear that does not hold for many NSFDEs. In fact, for

Abstract:
We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation $$ frac{d}{dt}x(t) = -a(t)x(t)+ frac{d}{dt}Q(t, x(t-g(t))) +G(t,x(t), x(t-g(t))). $$ In the process we use integrating factors and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution provided that $Q(0,0)= G(t, 0,0) = 0$.

Abstract:
in this paper, we improve some boundedness results, which have been obtained with respect to nonlinear differential equations of fifth order without delay, to a certain functional differential equation with constant delay. we give an illustrative example and also verify our main result by means of liaponov tecnique.

Abstract:
A parabolic initial boundary value problem and an associated elliptic Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations are considered. It is shown that the solutions of the parabolic problem is asymptotically stable and the limit of the solution of the parabolic problem as $t \to \infty$ is the solution of the associated elliptic problem. The result is based on the monotone methods.

Abstract:
We consider fifth-order nonlinear dispersive $K(m,n,p)$ type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons - the compact solitary waves free of exponential tails. This interaction also generates many other solitary wave structures like cuspons, peakons, tipons etc. which are otherwise unattainable with linear dispersion. Various self similar solutions of these higher order nonlinear dispersive equations are also obtained using similarity transformations. Further, it is shown that, like the third-order nonlinear $K(m,n)$ equations, the fifth-order nonlinear dispersive equations also have the same four conserved quantities and further even any arbitrary odd order nonlinear dispersive $K(m,n,p...)$ type equations also have the same three (and most likely the four) conserved quantities. Finally, the stability of the compacton solutions for the fifth-order nonlinear dispersive equations are studied using linear stability analysis. From the results of the linear stability analysis it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability conditions are satisfied for arbitrary values of the nonlinear parameters.

Abstract:
By constructing a Lyapunov function, a new instability result is established, which guarantees that the trivial solution of a certain nonlinear vector differential equation of the fifth order is unstable. An example is also given to illustrate the importance of the result obtained. By this way, our findings improve an instability result related to a scalar differential equation in the literature to instability of the trivial solution to the afore-mentioned differential equation.

Abstract:
In this paper, we present sufficient conditions for all solutions of a fifth-order nonlinear differential equation to converge. In this context, two solutions converge to each other if their difference and those of their derivatives up to order four approach zero as time approaches infinity. The nonlinear functions involved are not necessarily differentiable, but satisfy certain increment ratios that lie in the closed sub-interval of the Routh-Hurwitz interval.

Abstract:
This paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations (SDDEs). We derive sufficient conditions for the stability, contractivity and asymptotic contractivity in mean square of the solutions for nonlinear SDDEs. The results provide a unified theoretical treatment for SDDEs with constant delay and variable delay (including bounded and unbounded variable delays). Then the stability, contractivity and asymptotic contractivity in mean square are investigated for the backward Euler method. It is shown that the backward Euler method preserves the properties of the underlying SDDEs. The main results obtained in this work are different from those of Razumikhin-type theorems. Indeed, our results hold without the necessity of constructing of finding an appropriate Lyapunov functional.