Asymptotic Stability for a Functional Differential Equation in Hilbert Space

2003 ◽  
pp. 333-340
Author(s):  
Miklavž Mastinšek
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Asymptotic Stability ◽  
Functional Differential Equation ◽  
Functional Differential

scholarly journals 19.—Asymptotic Stability for Some Functional Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 253-255 ◽  
Author(s):  
Hans-Otto Walther ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Characteristic Equation ◽  
Population Biology ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Linear Functional Differential Equation ◽  
Functional Differential

SynopsisFor a non-linear functional differential equation from population biology, a result on asymptotic stability is obtained by investigating the zeros of the characteristic equation of the linearised functional differential equation.

scholarly journals ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

2020 ◽  
pp. 7-26
Author(s):  
Vera Malygina ◽  
◽  
Kirill Chudinov ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Initial Data ◽  
Exponential Stability ◽  
Functional Differential Equation ◽  
Initial Function ◽  
Neutral Type ◽  
Cauchy Function ◽  
Functional Differential ◽  
Equation Of Neutral Type

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

Asymptotic Stability of an Abstract Delay Functional-Differential Equation

2006 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
J. M. Tchuenche
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Functional Differential Equation ◽  
Chaotic Dynamics ◽  
Integral Form ◽  
Transform Method ◽  
Deviating Arguments ◽  
Interesting Connection ◽  
Exponential Asymptotic Stability ◽  
Functional Differential

We study the exponential asymptotic stability of an abstract functional-differential equation with a mixed type of deviating arguments, namely: c which might represent the gestation period of the population and r(u(t)), a suitably defined function. The equation is reduced to its equivalent integral form and solved via Laplace transform method. An interesting connection of our study is with generalizations of populations with potentially complex (chaotic) dynamics.

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