WELL-POSEDNESS OF SECOND-ORDER DEGENERATE DIFFERENTIAL EQUATIONS WITH FINITE DELAY IN VECTOR-VALUED FUNCTION SPACES

We give necessary and sufficient conditions of the LpLp-well-posedness (respectively, B-p,q(s)) for the second-order degenerate differential equation with finite delay: (Mu')'(t)+alpha u'(t)=Au(t)+Gu'(t)+Fu(t)+f(t)(Mu')'(t)+alpha u'(t)=Au(t)+Gu(t)'+Fu(t)+f(t), (t is an element of[0,2 pi]) with periodic boundary conditions u(0)=u(2 pi) (Mu')(0)=(Mu')(2 pi), (Mu')(2 pi), where A and M are closed linear operators on a Banach space XX satisfying D(A)subset of D(M) and F and G are bounded linear operators from L-p([-2 pi,0];X) (respectively, B-p,q(s)([-2 pi,0];X)B-p,(s)(q)([-2 pi,0];X)) into X.