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

HTML articles powered by AMS MathViewer

- by Jean Mawhin and Katarzyna Szymańska-Dȩbowska
- Proc. Amer. Math. Soc.
**145**(2017), 2023-2032 - DOI: https://doi.org/10.1090/proc/13569
- Published electronically: January 26, 2017
- PDF | 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.## References

- Jifeng Chu and Zhongcheng Zhou,
*Positive solutions and eigenvalues of nonlocal boundary-value problems*, Electron. J. Differential Equations (2005), No. 86, 9. MR**2162247** - Ivar Ekeland and Roger Témam,
*Convex analysis and variational problems*, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR**1727362**, DOI 10.1137/1.9781611971088 - Robert E. Gaines and Jean L. Mawhin,
*Coincidence degree, and nonlinear differential equations*, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR**0637067** - Chaitan P. Gupta,
*A second order $m$-point boundary value problem at resonance*, Nonlinear Anal.**24**(1995), no. 10, 1483–1489. MR**1327929**, DOI 10.1016/0362-546X(94)00204-U - Chaitan P. Gupta,
*Existence theorems for a second order $m$-point boundary value problem at resonance*, Internat. J. Math. Math. Sci.**18**(1995), no. 4, 705–710. MR**1347059**, DOI 10.1155/S0161171295000901 - Chaitan P. Gupta, S. K. Ntouyas, and P. Ch. Tsamatos,
*Existence results for $m$-point boundary value problems*, Differential Equations Dynam. Systems**2**(1994), no. 4, 289–298. MR**1386275** - Chaitan P. Gupta and Sergei Trofimchuk,
*Solvability of a multi-point boundary value problem of Neumann type*, Abstr. Appl. Anal.**4**(1999), no. 2, 71–81. MR**1810319**, DOI 10.1155/S1085337599000093 - G. B. Gustafson and K. Schmitt,
*A note on periodic solutions for delay-differential systems*, Proc. Amer. Math. Soc.**42**(1974), 161–166. MR**326109**, DOI 10.1090/S0002-9939-1974-0326109-3 - Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal,
*Convex analysis and minimization algorithms. I*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305, Springer-Verlag, Berlin, 1993. Fundamentals. MR**1261420** - G. L. Karakostas and P. Ch. Tsamatos,
*Positive solutions for a nonlocal boundary-value problem with increasing response*, Electron. J. Differential Equations (2000), No. 73, 8. MR**1801638** - G. L. Karakostas and P. Ch. Tsamatos,
*On a nonlocal boundary value problem at resonance*, J. Math. Anal. Appl.**259**(2001), no. 1, 209–218. MR**1836454**, DOI 10.1006/jmaa.2000.7411 - G. L. Karakostas and P. Ch. Tsamatos,
*Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem*, Appl. Math. Lett.**15**(2002), no. 4, 401–407. MR**1902271**, DOI 10.1016/S0893-9659(01)00149-5 - G. L. Karakostas and P. Ch. Tsamatos,
*Positive solutions and nonlinear eigenvalue problems for retarded second order differential equations*, Electron. J. Differential Equations (2002), No. 59, 11. MR**1911926** - Jean Leray and Jules Schauder,
*Topologie et équations fonctionnelles*, Ann. Sci. École Norm. Sup. (3)**51**(1934), 45–78 (French). MR**1509338** - Xiaojie Lin,
*Existence of solutions to a nonlocal boundary value problem with nonlinear growth*, Bound. Value Probl. , posted on (2011), Art. ID 416416, 15. MR**2739198**, DOI 10.1155/2011/416416 - J. Mawhin,
*Topological degree methods in nonlinear boundary value problems*, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR**525202** - Katarzyna Szymańska-Dȩbowska,
*The solvability of a nonlocal boundary value problem*, Math. Slovaca**65**(2015), no. 5, 1027–1034. MR**3433052**, DOI 10.1515/ms-2015-0070

## Bibliographic 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