Generalized Leray–schauder Principles for Compact Admissible Multifunctions

We establish the Leray–Schauder type theorems for very general classes of multifunctions, which are called admissible. Our admissible classes contain compositions of important multifunctions in nonlinear analysis and algebraic topology. Moreover, our arguments are elementary, without using the concept of degree of maps or theory of homotopy extensions. The Leray–Schauder principle [LS], one of the most important theorems in nonlinear analysis, was first proved for a Banach space in the context of degree theory. In [N], Nagumo extended the degree theory to locally convex topological vector spaces and his results can be used to generalize the Leray–Schauder principle. Variations of the principle were due to Browder [B] for Banach spaces and to Schaefer [Sc2] for locally convex topological vector spaces without using degree theory. Schaefer’s version has important applications to integral equations. Later Potter [Po] generalized the results of Browder and Schaefer. However, those authors considered single-valued maps and adopted boundary conditions particular to the so-called Leray–Schauder condition (LS). There are many other authors who obtained generalized versions of the Leray–Schauder type theorems (see the references).