On the Correctness of the Cauchy Problem for a Linear Differential System on an Infinite Interval

The conditions ensuring the correctness of the Cauchy problem dx dt = P(t)x + q(t), x(t0) = c0 on the nonnegative half-axis R+ are found, where P : R+ → Rn×n and q : R+ → Rn are locally summable matrix and vector functions, respectively, t0 ∈ R+ and c0 ∈ Rn. Consider the differential system dx dt = P(t)x + q(t) (1) with the initial condition x(t0) = c0 (2) where P : R+ → Rn×n and q : R+ → Rn are, respectively, the matrix and the vector functions with the components summable on every finite interval, t0 ∈ R+ and c0 ∈ Rn. It is known [1,2] that problem (1), (2) for arbitrarily fixed a ∈ ]0, +∞[ is correct on the interval [0, a], i.e., its solution on this interval is stable with respect to small, in an integral sense, perturbations P and q and small perturbations t0 and c0. In the paper under consideration we establish sufficient conditions for problem (1), (2) to be correct on R+. We shall use the following notation: R is the set of real numbers, R+ = [0, +∞[; 1991 Mathematics Subject Classification. 34B05, 35D10.