Characterising small solutions in delay differential equations through numerical approximations

This paper discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.Manchester Centre for Computational Mathematic

A functional model approach to linear neutral functional differential equations

We consider a linear neutral functional equation of the form d dt Mx t = Lx t ; x(0) = c; x 0 = '; where M;L : L 2 \Gamma [\Gammah; 0]; C n \Delta ! C n are unbounded linear operators of certain type and x t (`) = x(t+`), \Gammah ` 0. Under certain conditions, the solution semigroup fT (t)g t0 is well-defined on the space Z = C n \Theta L 2 \Gamma [\Gammah; 0]; C n \Delta of initial data. Let A be the infinitesimal generator of this semigroup. We shall establish the following model representation for fT (t)g t0 and A: there exists a half-plane \Pi \Gamma = fRe z ! ag, an n \Theta n matrix function ffi in H 1 \Gamma \Pi \Gamma ; C n\Thetan \Delta and a linear isomorphism U of Z onto the quotient space H 2 \Gamma \Pi \Gamma ; C n\Thetan \Delta =ffiH 2 \Gamma \Pi \Gamma ; C n\Thetan \Delta that transforms the operators T (t) and A into multiplication operators by e tz and z, respectively. We explain the relations between this result and special factorizatio..

A Functional Model Approach to Linear Neutral Functional Differential Equations

We consider a linear neutral functional equation of the form d dt Mx t = Lx t ; x(0) = c; x 0 = '; where M;L : L 2 \Gamma [\Gammah; 0]; C n \Delta ! C n are unbounded linear operators of certain type and x t (`) = x(t+`), \Gammah ` 0. Under certain conditions, the solution semigroup fT (t)g t0 is well-defined on the space Z = C n \Theta L 2 \Gamma [\Gammah; 0]; C n \Delta of initial data. Let A be the infinitesimal generator of this semigroup. We shall establish the following model representation for fT (t)g t0 and A: there exists a half-plane \Pi \Gamma = fRe z ! ag, an n \Theta n matrix function ffi in H 1 \Gamma \Pi \Gamma ; C n\Thetan \Delta and a linear isomorphism U of Z onto the quotient space H 2 \Gamma \Pi \Gamma ; C n\Thetan \Delta =ffiH 2 \Gamma \Pi \Gamma ; C n\Thetan \Delta that transforms the operators T (t) and A into multiplication operators by e tz and z, respectively. We explain the relations between this result and special factorizatio..