Coincidence points for perturbations of linear Fredholm maps of index zero
Let \(X\) and \(E\) be Fréchet spaces, \(\operatorname {dom} L\) a vector subspace of \(X\) and \(L: \operatorname {dom} L\to E\) a linear Fredholm map of index zero. For \(U\) an open subset of \(X\) containing 0, let \(G:\overline{U}\to E\) be a continuous map satisfying a technical condition (Mönch-Precup). Hypotheses are presented that imply the existence of a coincidence of \(L\) and \(G\), that is, a solution to the equation \(Lx= G(x)\). Such coincidences correspond, for instance, to solutions to systems of first-order ordinary differential equations. The key requirement is for the existence of an essential map in the sense of \textit{A. Granas} [C. R. Acad. Sci., Paris, Sér A 282, 983-985 (1976; Zbl 0348.47039)] which is combined with homotopy techniques to produce the coincidence. The authors extend their results to the setting of multivalued maps with a view toward applications to differential inclusions.
scientific article; zbMATH DE number 1455562