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A fixed point theorem for a class of multivalued continuously differentiable maps. (English) Zbl 0663.58006

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

MSC:

58C30 Fixed-point theorems on manifolds
58C25 Differentiable maps on manifolds
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