On Some Two-point Boundary Value Problems for Two-dimensional Systems of Ordinary Differential Equations

Sufficient conditions for the solvability of two-point boundary value problems for the system xi = fi(t, x1, x2) (i = 1, 2) are given, where f1 and f2 : [a1, a2]×R → R are continuous functions. 1. Statement of the problems and formulation of the main results Consider the system of ordinary differential equations xi = fi(t, x1, x2) (i = 1, 2) (1.1) with boundary conditions λi1x1(ai) + λi2x2(ai) + gi(x1, x2) = 0 (i = 1, 2) (1.2) or λi1x1(ai) + λi2x2(ai) + hi(x1(ai), x2(ai)) = 0 (i = 1, 2), (1.3) where −∞ < a1 < a2 < +∞, λij ∈ R (i, j = 1, 2), the functions fi : [a1, a2] × R2 → R, hi : R2 → R (i = 1, 2) are continuous and gi : C([a1, a2];R) → R (i = 1, 2) are the continuous functionals. The problems of the forms (1.1),(1.2) and (1.1),(1.3) have been studied earlier in [1-10]. In the present paper new criteria for solvability of these problems are established which have the nature of one-sided restrictions imposed on f1 and f2. We use the following notation: R is the set of all real numbers; R+ = [0, +∞[, D = [a1, a2]×R, D1 = [a1, a2]× (R\{0})×R; D2 = [a1, a2]×R× (R\{0}), C(A,B) is the set of continuous maps from A to B. 1991 Mathematics Subject Classification. 34B15.