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100 Jahre Mathematisches Seminar der Karl-Marx-Universitaet by H.; Beckert, H. Schumann PDF

By H.; Beckert, H. Schumann

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Liapunov-Razumikhin theorems for Volterra integro-differential equations have been given by Seifert (1973, 1974). CHAPTER V. EXISTENCE OF PERIODIC SOLUTIONS Periodic solutions of FDE's frequently arise by bifurcation from an equilibrium point that becomes unstable at a critical value of a parameter in the equation. The method of Hopf bifurcation can often be used to show the existence of such solutions for equations that are perturbations of linear autonomous equations. here. For lack of space, this method will not be discussed See Hale (1977, 1978), Stech (1979), and Kazarinoff, van den Driessche , and Wan (1978).

It is known that stability and asymptotic stability properties of such a difference V(~) = ~ TB~, ~V(~) = V(~+l) equation can be obtained from a Liapunov function of the form where - B is a positive definite symmetric matrix. = ~ T(ATBA-B)~. V(~) ATBA-B Given any positive definite matrix S, the equation = -S can be solved for a unique positive definite symmetric so lution B. Then found In fact, S, Infante and Walker chose a suitable B explicitly. 1). Following this formal procedure, the authors then show rigorously that this functional has the desired properties.

S(t);t ~ O} is a strongly continuous semigroup and has a closed, densely defined infinitesimal generator (i) The domain of A is A such that: D(A) = {(n, ~): $ is p 39 absolutely continuous, ep(0) (ii) If (iii) c (7~30) where There exist constant Z and z(t) 6 f then z(t) 0, £ > in If (n, $) c n, Lp(O,t = S(t)(n,$)+ of constants formula in the space and y and f: and and $. define l), (L($),~). 5. then A(n,$) = (n, $) E D(A), for every fixed in p n}, spectrum of For (n,$) E L (-r,O), ~ ~:[0,t1) ~ Z by S(t-u)(f(u),8)du, L (-r,O).

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100 Jahre Mathematisches Seminar der Karl-Marx-Universitaet Leipzig by H.; Beckert, H. Schumann

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