Periodic Perturbations of a Class of Functional Differential Equations

Bounded Solutions ◽

Differentiable Manifolds ◽

Periodic Perturbations ◽

AbstractWe study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026342 ◽

1986 ◽

Vol 102 (3-4) ◽

pp. 259-262 ◽

Cited By ~ 1

Author(s):

J. G. Dos Reis ◽

R. L. S. Baroni

Keyword(s):

Differential Equation ◽

Periodic Solution ◽

Differential Equations ◽

Periodic Solutions ◽

Continuous Functions ◽

Retarded Functional Differential Equations ◽

Existence Of Periodic Solutions ◽

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.

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

Complexity ◽

10.1155/2020/2854574 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Josef Rebenda ◽

Zuzana Pátíková

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Differential Equations ◽

Recurrence Relation ◽

Numerical Solutions ◽

Neutral System ◽

Initial Value ◽

Differential Transformation ◽

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.

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

Acta Universitatis Sapientiae Mathematica ◽

10.1515/ausm-2016-0017 ◽

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 ◽

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.

One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

Abstract and Applied Analysis ◽

10.1155/2011/843292 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 5

Author(s):

Ruyun Ma ◽

Yanqiong Lu

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Periodic Function ◽

Periodic Solutions ◽

Functional Differential Equation ◽

Global Bifurcation ◽

Periodic Functions ◽

First Order ◽

We study one-signed periodic solutions of the first-order functional differential equationu'(t)=-a(t)u(t)+λb(t)f(u(t-τ(t))),t∈Rby using global bifurcation techniques. Wherea,b∈C(R,[0,∞))areω-periodic functions with∫0ωa(t)dt>0,∫0ωb(t)dt>0,τis a continuousω-periodic function, andλ>0is a parameter.f∈C(R,R)and there exist two constantss2<0<s1such thatf(s2)=f(0)=f(s1)=0,f(s)>0fors∈(0,s1)∪(s1,∞)andf(s)<0fors∈(-∞,s2)∪(s2,0).

Existence of triple positive periodic solutions of a functional differential equation depending on a parameter

Abstract and Applied Analysis ◽

10.1155/s1085337504311085 ◽

2004 ◽

Vol 2004 (10) ◽

pp. 897-905 ◽

Cited By ~ 3

Author(s):

Xi-lan Liu ◽

Guang Zhang ◽

Sui Sun Cheng

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Differential Equations ◽

Periodic Solutions ◽

Functional Differential Equation ◽

Point Theorem ◽

Positive Periodic Solutions ◽

We establish the existence of three positive periodic solutions for a class of delay functional differential equations depending on a parameter by the Leggett-Williams fixed point theorem.

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 ◽

High Order ◽

Positive Periodic Solutions ◽

Neutral Functional Differential Equation ◽

Neutral Functional Differential Equations ◽

Small Force ◽

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.

Existence of Periodic Solutions to Multidelay Functional Differential Equations of Second Order

Abstract and Applied Analysis ◽

10.1155/2013/968541 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 2

Author(s):

Cemil Tunç ◽

Ramazan Yazgan

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Periodic Solutions ◽

Functional Differential Equation ◽

Functional Approach ◽

Second Order ◽

Nonlinear Functional ◽

Existence Of Periodic Solutions ◽

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

Analytic Solutions of an Iterative Functional Differential Equations Near Regular Points

Journal of Applied Mathematics ◽

10.1155/2013/726076 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Lingxia Liu

Keyword(s):

Differential Equation ◽

Functional Equation ◽

Differential Equations ◽

Existence Theorem ◽

Unknown Function ◽

Analytic Solutions ◽

Functional Differential ◽

Iterative Functional Differential Equation ◽

The Given

The existence of analytic solutions of an iterative functional differential equation is studied when the given functions are all analytic and when the given functions have regular points. By reducing the equation to another functional equation without iteration of the unknown function an existence theorem is established for analytic solutions of the original equation.