Flows on centre manifolds for scalar functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 193-201 ◽  
Author(s):  
Jack K. Hale
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Imaginary Axis ◽  
Functional Differential Equations ◽  
Zero Eigenvalue ◽  
Functional Differential ◽  
Centre Manifolds ◽  
Scalar Functional ◽  
Linear Scalar

SynopsisBy assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

2012 ◽  
Vol 616-618 ◽  
pp. 2137-2141
Author(s):  
Zhi Min Luo ◽  
Bei Fei Chen
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Nonlinear Functional Differential Equation ◽  
Functional Differential

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

1986 ◽  
Vol 102 (3-4) ◽  
pp. 259-262 ◽  
Author(s):  
J. G. Dos Reis ◽  
R. L. S. Baroni
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Retarded Functional Differential Equations ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

scholarly journals Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jun Zhou ◽  
Jun Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Functional Differential Equation ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
State Dependence ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

scholarly journals Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Josef Rebenda ◽  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Recurrence Relation ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Neutral System ◽  
Initial Value ◽  
Differential Transformation ◽  
Functional Differential

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.

scholarly journals Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

2016 ◽  
Vol 8 (2) ◽  
pp. 255-270
Author(s):  
Mouataz Billah Mesmouli ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Completely Continuous ◽  
Continuous Map ◽  
Functional Differential ◽  
Large Contraction

Abstract In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation $${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

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