A coincidence theory involving Fredholm operators of nonnegative index
This article deals with the analysis of the problem \(F\circ p=q\), where \(\Gamma\) is a metric space, \(X\) and \(Y\) Banach spaces, \(p:\Gamma\to X\) is a Vietoris mapping (``a proper surjection with acyclic fibres''), \(F: X\to Y\), \(q:\Gamma\to Y\) a Fredholm operator of index \(i(F)\geq 0\), \(q: \Gamma\to Y\). The basic construction is connected with the passing from the equation \(F\circ p(\xi)= q(\xi)\) to the inclusion \(F(x)\in q(p^{-1}(x))\); then the generalized degree \(\deg((p,q), U,0)\) is defined as an element of \(\pi^n(S^m)\) in the finite-dimensional case and as an element of the corresponding stable homotopic group \(\Pi_k\) in the infinite-dimensional one; this degree has natural properties, in particular, the properties of homotopy invariance, additivity, and nontriviality; in general, the generalized degree coincides with the ``algebraic count'' of solutions to the inclusion \(F(x)\in q(p^{-1}(x))\). The considerations cover the set-valued setting yet.
scientific article; zbMATH DE number 1545386