Continuous dependence of a solution of a neutral functional differential equation on the right-hand side and initial data taking into account perturbations of variable delays

2016 ◽  
Vol 23 (4) ◽  
pp. 519-535
Author(s):  
Nika Gorgodze ◽  
Ia Ramishvili ◽  
Tamaz Tadumadze
Keyword(s):  
Initial Data ◽  
Continuous Dependence ◽  
Functional Differential Equations ◽  
Initial Vector ◽  
Initial Moment ◽  
Neutral Functional Differential Equations ◽  
Variable Delays ◽  
Right Hand ◽  
Functional Differential ◽  
The Right

AbstractTheorems on the continuous dependence of a solution of the Cauchy problem with respect to the nonlinear term of the right-hand side and initial data are proved for neutral functional differential equations whose right-hand sides are linear with respect to the prehistory of the phase velocity. Under the initial data we imply a collection of initial moment, variable delays entering in the phase coordinates, initial vector and initial functions. In this paper, an essential novelty is that perturbations of variable delays are taken into account in proving the main theorems.

scholarly journals Resolvent of nonautonomous linear delay functional differential equations

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Joël Blot ◽  
Mamadou I. Koné
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Differentiable Function ◽  
Functional Differential Equations ◽  
Initial Value ◽  
Complete Proof ◽  
Right Hand ◽  
Linear Delay ◽  
Functional Differential ◽  
The Right

AbstractThe aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

scholarly journals Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
J. Diblík ◽  
M. Kúdelčíková
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Positive Solutions ◽  
Linear Functional ◽  
Linear Equations ◽  
Functional Differential Equations ◽  
Right Hand ◽  
Functional Differential ◽  
The Right

Solutions of the equationy˙(t)= −f(t,yt)are considered fort→∞. The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.

scholarly journals CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

2018 ◽  
Vol 54 (2) ◽  
pp. 193-209
Author(s):  
I. V. Kachan
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Hölder Continuous ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Right Hand ◽  
The Right

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

scholarly journals Some neutral equations with a control parameter

1981 ◽  
Vol 23 (3) ◽  
pp. 383-394
Author(s):  
Vasil G. Angelov
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Continuous Dependence ◽  
Control Parameter ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Accretive Operators ◽  
Neutral Functional Differential Equations ◽  
Neutral Equations ◽  
Functional Differential

This paper presents sufficient conditions, involving accretive operators, for the existence, uniqueness and continuous dependence on a control parameter of the solutions of some initial and boundary value problems for neutral functional differential equations.

scholarly journals Functional Differential Inequalities with Unbounded Delay

2005 ◽  
Vol 12 (2) ◽  
pp. 237-254
Author(s):  
Zdzisław Kamont ◽  
Adam Nadolski
Keyword(s):  
Differential Inequality ◽  
Continuous Dependence ◽  
Functional Differential Equations ◽  
Uniqueness Result ◽  
Classical Solutions ◽  
Differential Inequalities ◽  
Unbounded Delay ◽  
Several Variables ◽  
Functional Differential ◽  
Continuous Dependence Of Solutions

Abstract We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perron-type uniqueness result and a result on continuous dependence of solutions on given functions for partial functional differential equations with unbounded delay. We consider classical solutions on the Haar pyramid.

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