Some mappings with topological degree zero

Abstract

is sometimes proved by determining the topological degree of M at q. If the degree is different from zero, the equation has a solution. However if the degree is zero, then purely topological considerations give practically no information about solutions. The advantage of the topological degree approach is that comparatively little information about the mapping is needed. In this note we study locally certain mappings which arise in the study of functional equations and whose topological degree can be computed. By taking advantage of working in the small, we obtain sufficient conditions for the existence or nonexistence of solutions if the topological degree of the mapping is zero. Verifying that these conditions hold requires little information beyond that needed to compute the topological degree. The results we obtain can be applied to obtain existence and nonexistence theorems for the nonlinear integral equations and nonlinear elliptic differential equations studied in [3] and [4]. We study mappings in Euclidean 2-space. Our methods apply to similar mappings in n-space (n> 2), but if n> 2 the problem of computing the topological degree or verifying the conditions derived here is hampered by algebraic complications.