Matrix systems of differential equations with the quasimonotonicity condition
The author considers initial value problems with quasimonotone increasing right-hand sides in the space of symmetric \(n\times n\)-matrices, ordered by the cone of positive semidefinite matrices. He proves existence of lower and upper solutons, comparison results, monotone dependence on the initial value, and stability, boundedness and invariance results by means of differential inequalities for matrix-valued Lyapunov functions. There seems to be an error in Lemma 4, where it is stated that a quasimonotone increasing function leads to a monotone function by adding a suitable positive multiple of the identity. This does not hold in general, for example, for some of the linear mappings considered in Lemma 10.6.
scientific article; zbMATH DE number 1758686