A reduction theorem for the topological degree for mappings of class (𝑆+)

Abstract

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class ( S + ) (S_+) from a real separable reflexive Banach space X X into the dual space X ∗ X^* can be reduced into the calculation of degree of mapping from a closed subspace V ⊂ X V\subset X into V ∗ . V^*. Since the Leray-Schauder mappings are acting from X X to X X and we are dealing with mappings from X X to X ∗ , X^*, the standard ‘invariant subspace’ condition must be replaced by a less obvious one.