A general coincidence theory for set-valued maps
The author derives coincidence theorems for multivalued maps admitting continuous selections by translating this problem into a fixed point problem for multivalued maps with acyclic values. His main fixed point result reads as follows: Let \(E\) be a Banach space and \(Q\ni 0\) a closed convex subset of \(E\). Choose a retraction \(r:E\to Q\) and let \(G\) be a map which assigns to each \(x\in Q\) a nonempty closed acyclic subset of \(E\). Assume that \(G(Q)\) is bounded and that \(G\circ r\) is condensing with closed graph. Assume moreover that for each sequence \((x_n,\lambda_n)\) in \(\partial Q\times[0,1]\) which converges to \((x,\lambda)\in\partial Q\times[0,1)\) with \(x\in\lambda G(x)\) there is an \(n_0\) such that \(\lambda_nG(x_n)\subset Q\) whenever \( n\geq n_0\). The conclusion is that \(G\) has a fixed point in \(Q\).
fixed point problem for multivalued maps with acyclic values
scientific article; zbMATH DE number 1356404