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Fixed point principles for cones of a Banach space for the multivalued maps differentiable at the origin and infinity. (English) Zbl 0761.47035

This paper contains fixed point principles for upper semi-continuous (u.s.c.) multivalued maps, which are \(k\)-set contractions, \(0<k<1\), and differentiable at the origin/infinity. One of the results is (Theorem 1.12):
Let \(E\) be a real Banach space and \(C\subset E\) be a cone. Let \(F: C\to C\) be a u.s.c. multivalued map with convex values such that \(F(0)=0\), \(F\) is a \(k\)-set contraction and differentiable at infinity. Let \(S\) be a differential of \(F\) at infinity such that \(S(h)\) is convex for all \(h\), and the eigenvalues of \(S\) belong to the interval \([0,1)\). Then the fixed point index, \(\text{ind}(N_ r(0),F,C)=1\) if \(r\) is big enough, where, for \(A\subset E\), \(N_ r(A)=\{x\in E\mid\;d(x,A)<r\}\).

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
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