On the topological degree of planar maps avoiding normal cones
Keywords
Poincaré-Bohl, topological degree, avoiding cones conditionAbstract
The classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones.
We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.
References
J.P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997), 4159–4179.
A. Ćwiszewski and W. Kryszewski, Equilibria of set-valued maps: a variational approach, Nonlinear Anal. 48 (2002), 707–746.
A. Ćwiszewski and W. Kryszewski, The constrained degree and fixed-point index theory for set-valued maps, Nonlinear Anal. 64 (2006), 2643–2664.
A. Denjoy, Mémoire sur les nombres dérivés des fonctions continues, J. Math. Pures Appl. 7 (1915), no. 1, 105–240.
A. Fonda and P. Gidoni, Generalizing the Poincaré–Miranda Theorem: the avoiding cones condition, Ann. Mat. Pura Appl. 195 (2016), 1347–1371.
H. Hopf, Über die Drehung der Tangenten und Sehnen ebener Kurven, Compos. Math. 2 (1935), 50–62.
W. Kryszewski, On the existence of equilibria and fixed points of maps under constraints, Handbook of Topological Fixed Point Theory, Springer, Berlin, 2005, pp. 783–866.
R.T. Rockafellar and R.J. Wets, Variational Analysis, Springer, Berlin, 1998.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
