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


Authors: Jean Mawhin and Katarzyna Szymańska-Dȩbowska
Journal: Proc. Amer. Math. Soc. 145 (2017), 2023-2032
MSC (2010): Primary 34B10; Secondary 34B15, 47H11
DOI: https://doi.org/10.1090/proc/13569
Published electronically: January 26, 2017
MathSciNet review: 3611317
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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.


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Additional Information

Jean Mawhin
Affiliation: Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium
MR Author ID: 121705
Email: jean.mawhin@uclouvain.be

Katarzyna Szymańska-Dȩbowska
Affiliation: Institute of Mathematics, Lódź University of Technology, 90-924 Lódź, ul. Wólczańska 215, Poland
Email: katarzyna.szymanska-debowska@p.lodz.pl

Keywords: Nonlocal boundary value problem; boundary value problem at resonance; Leray-Schnauder degree
Received by editor(s): January 20, 2016
Published electronically: January 26, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society