On the relation between absolute stability with respect to a set of linear time-varying elements and a set of nonlinear time-varying elements

Consider two feedback systems with identical plants. System
L
has a time-varying linear gain and system
N
has a time-varying nonlinear gain; both gains satisfy the same sector condition. The absolute stability of
L
implies that of
N
in the case of stability in the sense of Lagrange and quasi-asymptotic stability in the large; the implication does not hold in the case of stability in the sense of Lyapunov, as is shown by an example.