Nonlocal boundary vector value problems for ordinary differential systems of higher order (Q1849044)

The authors study the existence of solutions to a \(n\)th-order \(m\)-dimensional ordinary differential system of the type \[ x^{(n)}=f(t, x, x', \ldots , x^{(n-1)}),\qquad t\in I, \tag{1} \] where \(I\) stands for the interval \([0, 1]\) and \(f: I\times (\mathbb{R}^{n})^{n} \to \mathbb{R}^{m}\) is a Carathéodory function, satisfying the following nonlocal boundary value conditions \[ x^{(j)} (0) = 0, \qquad j = 0, 1, \ldots , k-1, \tag{2} \] \[ x^{(j)} (1) = \int \limits^{1}_{0} [d G_{n-j}(s)]x^{j}(s), \qquad j = k, k+1, \ldots , n-1, \tag{3} \] for a certain fixed number \(k \in {1, 2, \ldots , n-1}\), where the integral is in the sense of Riemann-Stieltjes. In this article, the authors apply the Leray-Schauder continuation theorem to derive sufficient conditions for the existence of solutions to the boundary value problem (1)--(3).