Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference equations

Abstract

The aim of this note is to describe the continuation theorem of [J. Mawhin, <i>Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces</i>, J. Differential Equations 12 (1972), 610–636, J. Mawhin, <i>Topological Degree Methods in Nonlinear Boundary Value Problems</i>, CBMS Reg. Conf. in Math., No 40, American Math. Soc., Providence, RI, 1979] directly in the context of Brouwer degree, providing in this way a simple frame for multiple applications to nonlinear difference equations, and to show how the corresponding reduction property can be seen as an extension of the well-known reduction formula of Leray and Schauder [J. Leray, J. Schauder, <i>Topologie et équations fonctionnelles</i>, Ann. Scient. Ecole Normale Sup. (3) 51 (1934), 45–78], which is fundamental for their construction of Leray-Schauder's degree in normed vector spaces.