The coincidence index for fundamentally contractible multivalued maps with nonconvex values. (English) Zbl 0969.47041

Gabor, Dorota

Ann. Pol. Math. 75, No. 2, 143-166 (2000).

The author studies coincidence problems of the form \(A(x)\in\phi(x)\) where \(A:E\to F\) is a linear Fredholm operator of nonnegative index between Banach spaces \(E,F\) and \(\phi\) is an upper semicontinuous map that need not be compact but satisfies a rather technical condition (i.e., \(\phi\) is assumed to be “\(A\)-fundamentally contractible” — a condition that is fulfilled by condensing or ultimately compact maps). The author then uses W. Kryszewski’s approach [Homotopy properties of set-valued mappings, Wyd. Uniwersytetu Mikolaja Kopernika, Toruń, 1997] to define a coincidence index as an element of the \(k\)-th stable homotopy group of spheres. This in turn is applied to a boundary value problem in Banach spaces.

Reviewer: Christian Fenske (Gießen)

MSC:

47H11	Degree theory for nonlinear operators
55M20	Fixed points and coincidences in algebraic topology
34G20	Nonlinear differential equations in abstract spaces
34B15	Nonlinear boundary value problems for ordinary differential equations
47A53	(Semi-) Fredholm operators; index theories
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05	Equations involving nonlinear operators (general)

Keywords:

coincidence point; Fredholm operator; condensing map; boundary value problem; upper semicontinuous; ultimately compact maps; coincidence index; \(k\)-th stable homotopy group of spheres; boundary value problem in Banach spaces