Summary: The authors consider the system
if , . are measurable essentially bounded functions, , , , implies , where is the Lebesgue measure, , , . Let be the fundamental matrix of (1) satisfying the condition and the Cauchy matrix.
The authors obtain sufficient conditions for the validity of exponential estimates of the form
with positive numbers and . The estimates (2) are based on results on the nonnegativity of the entries of the Cauchy matrix [see A. I. Domoshnitsky and M. V. Sheina, Differ. Equations 25, No. 2, 145-150 (1989); translation from Differ. Uravn. 25, No. 2, 201-208 (1989; Zbl 0694.34060)]. Furthermore, the authors’ method allows one to estimate .