Global existence for an abstract Riccati initial-value problem with possibly unbounded coefficients (Q1191774)

The paper deals with the operator-valued Riccati initial-value problem of the form (1a) \(R'(t)+TR(t)+R(t)T=TA(t)+TB(t)R(t)+R(t)TC(t)+R(t)TD(t)R(t)\), (1b) \(R(0)=R_ 0\). Suppose (A1) \(A(t)\) through \(D(t)\) have values as nonnegative bounded linear operators in \(L^ 1(\mu)\), where \(\mu\) is a \(\sigma\)-finite measure, and are nondecreasing in \(t\). Moreover they are bounded by \(c/2\) for some positive \(c\leq 1\). Let \(T\) be a closed positive operator in \(X=L^ 1(\mu)\) satisfying certain additional technical conditions. The authors define the notion of strongly mild solutions and establish, via an iterative method, the global existence of a strongly mild solution of (1) for any \(c<1\) with sufficiently small initial data \(R_ 0\) and for \(c=1\) with zero initial data. The equation (1a) models a type of problem arising in transport theory. The results extend those of the authors' previous work [Math. Methods Appl. Sci. 12, No. 1, 53-67 (1990; Zbl 0679.34072)].