Fournier, G.; Violette, D. A fixed point theorem for a class of multivalued continuously differentiable maps. (English) Zbl 0663.58006 Ann. Pol. Math. 47, 381-402 (1987). Let E, F be real Banach spaces and U an open subset of E. Let T: \(U\to F\) be a multivalued map such that T(x) is compact for any \(x\in U\). We say T is differentiable at \(x\in U\), if there is an upper-semi continuous multivalued map \(S_ x\) from T(x)\(\times E\) into F having the following properties. (i) For every z in T(x), \(S_ x(z,.)\) is a homogeneous and semi-linear positive multivalued maps from E into F. (ii) For any \(\epsilon >0\) there exists \(\delta (\epsilon,x)>0\) such that the Hausdorff distance \(d_ H(T(x,h),\quad \cup_{z\in T(x)}(z+S_ x(z,h)))\leq \epsilon \| h\|.\) Under this concept of differentiability, the authors obtain a mean theorem and a remainder theorem for multivalued maps and study the fixed point index of continuously differentiable multivalued maps which are eventually condensing and have the acyclic decompositions. Reviewer: Duong Minh Duc Cited in 3 ReviewsCited in 3 Documents MSC: 58C30 Fixed-point theorems on manifolds 58C25 Differentiable maps on manifolds Keywords:set-valued mappings; differentiation theory PDFBibTeX XMLCite \textit{G. Fournier} and \textit{D. Violette}, Ann. Pol. Math. 47, 381--402 (1987; Zbl 0663.58006) Full Text: DOI