On Two-point Boundary Value Problems for Systems of Higher-order Ordinary Differential Equations with Singularities

The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form u(n) = f(t, u, u′, . . . , u(n−1)), where the vector function f :]a, b[×Rnl → Rl has nonintegrable singularities with respect to the first argument at the points a and b. § 1. Statement of the main results In this paper for an l-dimensional system of differential equations u(n) = f(t, u, u′, . . . , u(n−1)) (1.1) we consider the boundary value problem of Vallèe-Poussin u(a+) = · · · = u(m−1)(a+) = 0, u(b−) = · · · = u(n−m−1)(b−) = 0 (1.2) and that of Cauchy-Niccoletti u(a+) = · · · = u(m−1)(a+) = 0, u(m)(b−) = · · · = u(n−1)(b−) = 0, (1.3) where l ≥ 1, n ≥ 2, m is an integer part of the number 2 , −∞ < a < b < +∞, and the vector function f :]a, b[×Rnl → Rl satisfies the Caratheodory conditions on each compact contained in ]a, b[×Rnl. We are interested mainly in the singular case when f is nonintegrable with respect to the first argument on [a, b], having singularities at the ends of this interval. The above problems were investigated for l = 1 in [2-6]. 1991 Mathematics Subject Classification. 34B15. 31 32 I. KIGURADZE AND G. TSKHOVREBADZE The following notations will be used: In(a, b) = { ]a, b[ for n = 2m ]a, b] for n = 2m + 1 ;