scholarly journals Analytic Solutions of an Iterative Functional Differential Equations Near Regular Points

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Lingxia Liu
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Differential Equations ◽  
Existence Theorem ◽  
Unknown Function ◽  
Functional Differential Equations ◽  
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.

scholarly journals Analytic Solutions of an Iterative Functional Differential Equation near Resonance

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Tongbo Liu ◽  
Hong Li
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Unknown Function ◽  
Analytic Solutions ◽  
Complex Field ◽  
Second Order Differential Equations ◽  
Near Resonance ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function in the complex field . By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function + = , we get its local invertible analytic solutions.

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>

Non-linear functional differential equations and abstract integral equations

1979 ◽  
Vol 84 (1-2) ◽  
pp. 71-91 ◽  
Author(s):  
F. Kappel ◽  
W. Schappacher
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Space Setting ◽  
Non Linear ◽  
Functional Differential ◽  
Order Spline ◽  
The Given

SynopsisThe equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.

scholarly journals Periodic Perturbations of a Class of Functional Differential Equations

2021 ◽  
Author(s):  
Marco Spadini
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Functional Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Bounded Solutions ◽  
Differentiable Manifolds ◽  
Periodic Perturbations ◽  
Functional Differential

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.

scholarly journals Smooth Solutions of a Class of Iterative Functional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Houyu Zhao
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Functional Differential Equations ◽  
Smooth Solutions ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

By Faà di Bruno’s formula, using the fixed-point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equationx′(t)=1/(c0x[0](t)+c1x[1](t)+⋯+cmx[m](t)).

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.

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