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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convex sets and second order systems with nonlocal boundary conditions at resonance
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by Jean Mawhin and Katarzyna Szymańska-Dȩbowska PDF
Proc. Amer. Math. Soc. 145 (2017), 2023-2032 Request permission

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
  • Received by editor(s): January 20, 2016
  • Published electronically: January 26, 2017
  • Communicated by: Yingfei Yi
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2023-2032
  • MSC (2010): Primary 34B10; Secondary 34B15, 47H11
  • DOI: https://doi.org/10.1090/proc/13569
  • MathSciNet review: 3611317