초록
Consider the system (i) x'=Ax. Let ${\Phi}$ be its fundamental matrix solution. If there is w>0 such that A(t+w)-A(t) commutes with ${\Phi}$ for all t, then we call this system a "generalized" Floquet system or a "G. F. system". We show that $A(t+w)-A(t)=B_1$=constant if and only if $A(t)=C+B_1t/w+Q(t)$, Q is periodic of period w>0. For this A(t) We prove that if all eigenvalues of $B_1$ have negative real parts, then the origin is asumptotically stable. We find a growth condition for a continuous D(t) which guarantees that all solutions of z'=[A(t)+D(t)]z are bounded if all solutions of the G. F. system (i) are bounded. Combining the foregoing results yield a class of perturbed G. F. operators all of whose solutions are bounded.