Appl Math Optim (2013) 67:453–477
Asymptotic Growth of Solutions of Neutral Type
G.M. Sklyar · P. P olak
Published online: 7 March 2013
© The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract We consider a differential system of neutral type with distributed delay.
We obtain a precise norm estimation of semigroup generated by the operator corre-
sponding to the system in question. Our result is based on a spectral analysis of the
operator and some uniform estimation of norms of the exponentials of matrices. We
also discuss the stability properties of corresponding solutions and the existence of
the fastest growing solution.
Keywords Delay systems · Neutral type systems · Asymptotic behaviour of
solutions · Maximal asymptotics
One of the important problems in the theory of functional differential equations is
the estimation of the asymptotic behaviour of their solutions. It is also related to sta-
bility analysis of those equations. Even in the case when stability is studied, there
remains the question about the rate of growth or decay of individual solutions and
their dependence on initial states. All those questions pertain directly to the equa-
tions with delay. Among the works devoted to this problem, we can single out the
works of D.A. Medvedev and V.V. Vlasov , W.E. Brumley , J.K. Hale, S.M.
Verduyn Lunel [4, 5], D.A. O’Connor, T.J. Tarn , R. Rabah, G.M. Sklyar [12,
13], R. Rabah et al. , S.M. Verduyn Lunel, D.V. Yakubovich . The funda-
mental approach for estimation of asymptotic growth of solutions is interpretation of
This work was partially supported by Polish National Science Centre grant No. N N514 238438.
G.M. Sklyar (
) · P. P ola k
Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
P. P ol ak