Nonlinear Neutral Differential Equations Research Articles

Asymptotic stability in totally nonlinear neutral difference equations

In this paper we use fixed point method to prove asymptotic stability results of the zero solution of the totally nonlinear neutral difference equation with variable delay∆ x (n) = —a (n) f (x (n — τ (n))) + ∆g (n, x (n — τ (n))).An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Raffoul (2006) , Yankson (2009), Jin and Luo (2009) and Chen (2013).

Existence of nonoscillatory solutions to second-order nonlinear neutral difference equations

Existence of nonoscillatory solutions to second-order nonlinear neutral difference equations

OSCILLATION OF FIRST ORDER NON-LINEAR NEUTRAL DIFFERENCE EQUATIONS OF RETARDED AND ADVANCED TYPE

In this paper, we establish sufficient conditions for the oscillations for the all solutions of the first order non-linear neutral difference equations of retarded and advanced type. Our approach is to reduce the oscillation of neutral difference equations to the nonexistence of eventually positive solutions of non-neutral difference inequalities. Our results extend and improve several known results in the literature.

On implicit abstract neutral nonlinear differential equations

In this paper we continue our developments in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernandez and O'Regan (J Funct Anal 261:3457---3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial "nonlinear" neutral differential equations. Some applications involving partial neutral differential equations are presented.

Oscillation criteria for a class of higher odd order neutral difference equations with continuous variable

In this paper, we are mainly concerned with oscillatory behavior of solutions for a class of higher odd order nonlinear neutral difference equations with continuous variable. By converting the above difference equations to the corresponding differential equations and inequalities, the oscillatory criteria are obtained. In addition, examples are given to illustrate the obtained criteria, respectively.

Convergence of solutions of higher order neutral difference equations with quasi-differences

Abstract A higher order nonlinear neutral difference equation with quasi-differences is studied. Sufficient conditions are established under which for every real constant there exists a solution of considered equation convergent to this constant.

Oscillation criteria for a class of nonlinear neutral differential equations

In this paper, we deal with the oscillation of the second order nonlinear neutral differential equations of the form $$\bigl(a(t) \bigl(x(t)+ \delta p(t)x(t-\tau)\bigr)' \bigr)'+f\bigl(t, x(t-\sigma)\bigr)-g\bigl(t,x(t-\rho)\bigr)=0. $$ The oscillation criteria for these equations have been obtained. Furthermore, examples are given to illustrate the criteria, respectively.

Oscillatory behavior of second order nonlinear neutral differential equations with distributed deviating arguments

In this article, we shall consider second order nonlinear neutral differential equation of certain type. Some oscillation criteria are established for second-order neutral differential equation of the form[r(t)|z′(t)|γ−1z′(t)]′+∫cdf(t,x(σ(t,ξ)))dξ=0,where z(t)=x(t)+∫abp(t,ξ)x(τ(t,ξ))dξ. An example is given to show the applicability of our results.

Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions

This paper deals with the approximate controllability of impulsive neutral fuzzy stochastic differential equations with nonlocal conditions in a Banach space by using the concept of fuzzy numbers whose values are normal, convex, upper semicontinuous and compact. The hypotheses are obtained by Schauder’s and Banach fixed point theorems. The results are obtained by the evolution operator.

Oscillation criteria of third order nonlinear neutral difference equations

In this paper we consider the third order nonlinear neutral difference equation of the form$$\Delta\left(r_n\left(\Delta^2\left(x_n \pm p_n x_{\sigma(n)}\right)\right)^\alpha\right)+f\left(n, x_{\tau(n)}\right)=0,$$we establish some sufficient conditions which ensure that every solution of this equation are either oscillatory or converges to zero. Examples are provided to illustrate the main results.

Stability in totally nonlinear neutral differential equations with variable delay using fixed point theory

The totally nonlinear neutral differential equation (d/ dt) (x(t))=−a(t)g(x(t−τ (t))) + (d/ dt)( G(t,x(t−τ (t)))), with variable delay &#964(t) &#8805 0 is investigated. We find suitable conditions for t, a, g and G so that for a given continuous initial function 0 a mapping P for the above equation can be defined on a carefully chosen complete metric space S0ψ and in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Becker and Burton [6]. An example is given to illustrate our main result.

Study of the Periodic or Nonnegative Periodic Solutions of Functional Differential Equations via Krasnoselskii–Burton’s Theorem

In this paper, we study the existence of periodic or nonnegative periodic solutions of the nonlinear neutral differential equation $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}[x(t)-Q(t,\,x(t-\tau (t)))]=-a(t)h(x(t-\tau (t))) +G(t,\,x(t),\,x(t-\tau (t))). \end{aligned}$$ We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the modification of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

Convergence of solutions of nonlinear neutral differential equations with multiple delays

In this paper, on the basis of a new Lyapunov–Krasovskii functional and inequality technique, a set of novel sufficient conditions is derived for all solutions of a kind of nonlinear neutral differential equations of first order to approach zero as \(t\rightarrow \infty \). The derived conditions are delay independent. An example is discussed to illustrate the efficiency of the result. Our result extends and improves some related ones in the literature.

Periodic solutions for a system of nonlinear neutral functional difference equations with two functional delays

In this paper, we study the existence and uniqueness of periodic solutions of the system of nonlinear neutral difference equations ∆x (n) = A(n) x (n - t (n)) + ∆Q(n; x (n - g (n))) + G(n; x (n) ; x (n - g (n))). By using Krasnoselski's fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An example is given to illustrate our result. Our results extend and generalize the work [13].

Stability of Runge-Kutta Methods for Neutral Delay Differential Equations

This paper is concerned with the numerical stability of a class of nonlinear neutral delay differential equations. The numerical stability results are obtained for(k,l)-algebraically stable Runge-Kutta methods when they are applied to this type of problem. Numerical examples are given to confirm our theoretical results.

Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term

In this paper, we obtained some sufficient conditions for the oscillation of all solutions of the second order neutral differential equation of the form where , and . Examples are provided to illustrate the main results.

Oscillation criteria for odd higher order nonlinear neutral differential equations with positive and negative coefficients

Oscillation criteria for odd higher order nonlinear neutral differential equations with positive and negative coefficients

Oscillation of Second Order Nonlinear Neutral Differential Equations

The oscillation criteria are investigated for all solutions of second order nonlinear neutral delay differential equations. Our results extend and improve some results well known in the literature see ( [14] theorem 3.2.1 and theorem 3.2.2 pp.385-388). Some examples are given to illustrate our main results.

Oscillation theorems for second order nonlinear neutral difference equations

Abstract This paper deals with the oscillation of second order neutral difference equations of the form △ ( a n ( △ ( x n + p n x τ ( n ) ) ) α ) + q n x σ ( n ) β = 0 . The oscillation of all solutions of this equation is established via comparison theorems. Examples are provided to illustrate the main results. MSC:39A10.

Oscillation Criteria of First Order Nonlinear Neutral Differential Equations

A class of general nonlinear neutral differential equations for its special cases are discussed. sufficient conditionsfor all solutionsof the equations to be oscillatory are obtained. It can be treated in the extreme limite theorem.