Convex sets and second order systems with nonlocal boundary conditions at resonance

Abstract:

The solvability of the resonant nonlocal boundary value problem \[ x'' =f(t,x,x’),\quad x(0)=0, \quad x’(1)=\int _{0 }^{1}x’(s)dg(s),\] with $f : [0,1] \times \mathbb {R}^n \times \mathbb {R}^n \to \mathbb {R}^n$ continuous, $g = \mbox {diag}(g_1,\ldots ,g_n)$, $g_i : [0,1] \to \mathbb {R}$ of bounded variation, $\int _0^1dg_i(s)=1$ $(i=1,\dots ,n)$, is studied using the Leray-Schauder continuation theorem. The a priori estimates follow from the existence of an open bounded convex subset $C \subset \mathbb {R}^n$, such that, for each $t \in [0,1]$ and $x \in \overline C$, the vector fields $f(t,x,\cdot )$ satisfy suitable geometrical conditions on $\partial C$. The special cases where $C$ is a ball or a parallelotope are considered.