J. Mawhin and K. Szymańska-Dębowska
abstract:
Recently, some extensions of results of M. A. Krasnosel'skii and
Gustafson-Schmitt for systems of the type $x' = f(t,x)$ with periodic boundary
conditions $x(0) = x(1)$ have been obtained for nonlocal boundary conditions of
the type $x(1) = \int\limits_0^1 dh(s)\,x(s)$ or $x(0) = \int\limits_0^1 dh(s)\,x(s)$,
where $h$ is a real non-decreasing function satisfying some conditions, and
containing the periodic boundary conditions as special cases. The situations
with periodic and nonlocal boundary conditions are compared through the use of
counterexamples, exhibiting the special character of the periodic case. Similar
counter\-examples also show, in the case of second order systems with some
nonlocal boundary conditions, that the sense of some inequalities in the
assumptions cannot be reversed.
Mathematics Subject Classification: 34B10, 34B15, 47H11
Key words and phrases: Nonlocal boundary value problem, boundary value problem at resonance, periodic solutions, Leray-Schauder degree, convex sets