Lyapunov Exponents for Linear Delay Equations in Arbitrary Phase Spaces

TY - JOUR

T1 - Lyapunov Exponents for Linear Delay Equations in Arbitrary Phase Spaces

AU - Riedle, Markus

PY - 2006/2

Y1 - 2006/2

N2 - A linear integral equation with infinite delay is considered where the admissible function space B of initial conditions is as usually only described axiomatically. Merely using this axiomatic description, the long time behavior of the solutions is determined by calculating the Lyapunov exponents. The calculation is based on a representation of the solution in the second dual space of B and on a connection between the asymptotic behavior of the solutions of the integral equation under consideration and its adjoint equation subject to the spectral decomposition of the space of initial functions. We apply the result to an example of a stochastic differential equation with infinite delay.

AB - A linear integral equation with infinite delay is considered where the admissible function space B of initial conditions is as usually only described axiomatically. Merely using this axiomatic description, the long time behavior of the solutions is determined by calculating the Lyapunov exponents. The calculation is based on a representation of the solution in the second dual space of B and on a connection between the asymptotic behavior of the solutions of the integral equation under consideration and its adjoint equation subject to the spectral decomposition of the space of initial functions. We apply the result to an example of a stochastic differential equation with infinite delay.

KW - Lyapunov exponents

KW - differential equations with infinite delay

KW - weak-integral

KW - abstract phase space

KW - variation of constants formula

KW - stochastic differential equations with delay

U2 - 10.1007/s00020-004-1351-3

DO - 10.1007/s00020-004-1351-3

M3 - Article

SN - 0378-620X

VL - 54

SP - 259

EP - 278

JO - INTEGRAL EQUATIONS AND OPERATOR THEORY

JF - INTEGRAL EQUATIONS AND OPERATOR THEORY

IS - 2

ER -