Comparison Theorems for Matrix Riccati Equations

Comparison theorems for the scalar Riccati system (1) x' + x² + q₁ ( t ) = 0,w' + w^2 + q( t ) = 0$ are generalized to the matrix Riccati system $(2) X' + X^2 + Q_1 ( t ) = 0\[ W' + W² + Q₁ ( t ) = 0. \] By assuming thatX' + X^2 + Q_1 ( t ) = 0$ has scalar matrix solutions and imposing integral conditions on the matrices $Q_1 ( t )$and $Q( t )$, a recently established comparison theorem for (1) is extended to the matrix system (2) and A. Yu. Levin’s comparison theorem for (1) is generalized to the matrix case. From these comparison theorems sufficient conditions are obtained for the existence of solutions of $W' + WP( t )W + Q( t ) = 0$.