A Multidimensional Singular Boundary Value Problem of the Cauchy–nicoletti Type

A two-point singular boundary value problem of the Cauchy–Nicoletti type is studied by introducing a two-point boundary value set and using the topological principle. The results on the existence of solutions whose graph lies in this set are proved. Applications and comparisons to the known results are given, too. Introduction Consider the system of ordinary differential equations y′ = f(x, y), (1) where x ∈ I = (a, b), −∞ ≤ a < b ≤ ∞, y ∈ Rn and n > 1. We will study the following singular boundary value problem of the Cauchy–Nicoletti type: yi(a+) = Ai (i = 1, . . . ,m), yk(b−) = Ak (k = m + 1, . . . , n) (2) where Ai, i = 1, . . . , n, are some constants and 1 ≤ m < n. It is assumed that the vector-function f ∈ C(Ω,Rn), where Ω is an open set such that Ω ∩ {(x∗, y) : y ∈ Rn} 6= ∅ for each x∗ ∈ I and, moreover, f satisfies local Lipschitz condition in the variable y in Ω (f ∈ Lloc(Ω)). In this case the solutions of system (1) are uniquely determined by the initial data in Ω. We define the solution of problem (1), (2) as a vector-function y = (y1, . . . , yn) ∈ C1(I,Rn) which satisfies system (1) on I, (x, y1(x), . . . , yn(x)) ⊂ Ω if x ∈ I and yi(a+) = Ai (i = 1, . . . ,m), yk(b−) = Ak (k = m + 1, . . . , n). 1991 Mathematics Subject Classification. 34B10, 34B15.