By J. L. Dupont, I. H. Madsen

Collage of Aarhus, 50. Anniversary, eleven September 1978

**Read or Download Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978 PDF**

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**Extra info for Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978**

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5]). (a) If F is continuous and satisﬁes the Palais–Smale condition, then I F is 0-closed. F / is compact, then F satisﬁes the Palais–Smale condition. 33. 35. Let Y be a real Hilbert space, F W K ! t /; t 2 Kg: If a continuous map N W M ! N / is a compact Rı -set. The following result is useful in applications since it is more adapted to Volterra integral equations. 36. 0; r / D fu 2 E j kuk Ä r g be the closed ball in E. Assume that F W X ! E is a compact map and f W X ! t / for each t 2 Œ0; ".

X; X/. Using the fact that X is convex, then X 2 AR. x/. 68), to get a sequence of continuous maps fn W X ! Gr f /. xn /. x/. 71. 70. 72 (Kakutani’s theorem). E/; and F W X ! X/ compact. Then F has a ﬁxed point in X. Proof. E/. 46 and the compactness of the multi-valued operator F; we deduce that the multi-valued map F W C ! c. y/. Now, we present the classical multi-valued version of the nonlinear alternative of Leray and Schauder. 73. Let E be a normed linear space, F W E ! E/ an upper semicontinuous multi-valued map and completely continuous multi-valued map.

2) To do this, we additionally deﬁne the mappings rn W Œ0; a ! 1 Case of single-valued mappings It is easily seen that Fn is a continuous and compact mapping for n D 1; 2; : : :. 1 uniformly in x 2 X: Now we shall prove that fn is a one-to-one map. t /; for every t 2 Œ0; a=n. t /; for every t 2 Œ0; a. Therefore, fn is a one-to-one map and the proof is complete. 36, we deduce a Szuﬂa type result [449–452]. 37. 36. F / is an Rı -set. 38. E; k k/ be a Banach space, and f W X ! E be a proper map.