Generalized Hopf Bifurcation for Neutral Functional Differential Equations

Here we employ the Lyapunov–Schmidt procedure to investigate bifurcations in a general neutral functional differential equation (NFDE) when the infinitesimal generator has, for a critical value of the parameter, a pair of nonsemisimple purely imaginary eigenvalues with multiplicity [Formula: see text]. We derive criteria, explicitly in terms of the system's parameter values, for the existence of two branches of bifurcating periodic solutions and for the description of the bifurcation direction of these branches. The general result is illustrated by a detailed case study of an oscillator.

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Asymptotic behaviour of solutions to neutral functional differential equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017366 ◽

1989 ◽

Vol 40 (3) ◽

pp. 345-355

Author(s):

Shaozhu Chen ◽

Qingguang Huang

Keyword(s):

Differential Equation ◽

Difference Equation ◽

Differential Equations ◽

Asymptotic Behaviour ◽

Functional Differential Equation ◽

Necessary Conditions ◽

Functional Differential Equations ◽

Neutral Functional Differential Equation ◽

Neutral Functional Differential Equations ◽

Functional Differential

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

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Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4035267 ◽

2017 ◽

Vol 12 (3) ◽

Cited By ~ 2

Author(s):

Xiao-Li Ding ◽

Juan J. Nieto

Keyword(s):

Differential Equation ◽

Integral Operator ◽

Differential Equations ◽

Numerical Method ◽

Functional Differential Equation ◽

Functional Differential Equations ◽

Waveform Relaxation ◽

Neutral Functional Differential Equation ◽

Neutral Functional Differential Equations ◽

Functional Differential

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

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Oscillation of Second-Order Neutral Functional Differential Equations with Mixed Nonlinearities

Abstract and Applied Analysis ◽

10.1155/2011/927690 ◽

2011 ◽

Vol 2011 ◽

pp. 1-15 ◽

Cited By ~ 6

Author(s):

Shurong Sun ◽

Tongxing Li ◽

Zhenlai Han ◽

Yibing Sun

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Functional Differential Equation ◽

Functional Differential Equations ◽

Second Order ◽

Neutral Functional Differential Equation ◽

Neutral Functional Differential Equations ◽

Mixed Nonlinearities ◽

Functional Differential

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, whereγ>α>β>0,∫t0∞(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).

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Positive periodic solutions for singular high-order neutral functional differential equations

Mathematica Slovaca ◽

10.1515/ms-2017-0109 ◽

2018 ◽

Vol 68 (2) ◽

pp. 379-396 ◽

Cited By ~ 2

Author(s):

Fanchao Kong ◽

Zhiguo Luo ◽

Shiping Lu

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Periodic Solutions ◽

Functional Differential Equations ◽

High Order ◽

Positive Periodic Solutions ◽

Neutral Functional Differential Equation ◽

Neutral Functional Differential Equations ◽

Small Force ◽

Functional Differential

Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.

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Ak-Dimensional System of Fractional Neutral Functional Differential Equations with Bounded Delay

Abstract and Applied Analysis ◽

10.1155/2014/524761 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Dumitru Baleanu ◽

Sayyedeh Zahra Nazemi ◽

Shahram Rezapour

Keyword(s):

Differential Equations ◽

Functional Differential Equations ◽

Dimensional System ◽

Neutral Functional Differential Equation ◽

Initial Value ◽

Neutral Functional Differential Equations ◽

Krasnoselskii’S Fixed Point Theorem ◽

One Dimensional ◽

Bounded Delay ◽

Functional Differential

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 ofk-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.

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