# Algebraic Topology by P. Hoffman, R. Piccinini, D. Sjerve

By P. Hoffman, R. Piccinini, D. Sjerve

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Sketch of proof. For the proof we consider a sequence of completely continuous mappings gn W C ! 2 The case of multi-valued mappings 27 where x0 is a ﬁxed point in C and n D 1; 2; : : :. 46 are satisﬁed. Without compactness of the nonexpansive map, we have a weaker result owed to G. 64. Let X be a Banach space and f W X ! X/ is bounded. Then the set of ﬁxed points of f is a nonempty connected set. 2 The case of multi-valued mappings In this section we shall present needed classes of multi-valued mappings.

Let X be a Banach space and K W X ! 0; r /. 20. H/ is an a priori estimate assumption. (b) Corollary is equivalent to the Schauder ﬁxed point theorem. 21 (Schaefer’s theorem, 1955). Let X be a Banach space and K W X ! X be a compact map. x/ D xg is unbounded. 22 (Rothe’s theorem, 1957). Let B be an open ball in a Banach space X and K W X ! @ / B. Then K has a ﬁxed point in B. 23 (Schauder’s domain invariance). Let U be an open subset of a normed space E and f W U ! E be an injective completely continuous ﬁeld.

X/. 4 Fixed point sets of multi-valued contractions Below we shall concentrate our considerations on the topological structure of the set of ﬁxed points of contraction mappings. 153). 82. Let F W R ! x/ D A; for every x 2 R; where A we have: R is a nonempty set. Then F as a constant map is a contraction. F / of a contraction F may have many elements, it is interesting to look for its topological properties. 83. Let X be a complete metric space and F W X ! X/ be a multivalued contraction. F / is compact.