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Cone compression and expansion fixed point theorems in Fréchet spaces with applications. (English) Zbl 0989.47046

The authors present a new kind of fixed point theory for multivalued maps between Fréchet spaces.

From the text: “This paper is concerned with the existence of single and multiple fixed points for multivalued maps between Fréchet spaces. There are two main sections. In Section 2, the fixed point theory of Krasnosel’skij and Leggett and Williams and Petryshyn [see R. P. Agarwal and Donal O’Regan [J. Differ. Equations 160, No. 2, 389-403 (2000; Zbl 1008.47055); Nonlinear Anal., Theory Methods Appl. 42A, No. 6, 1091-1099 (2000; Zbl 0969.47038)] and their references) in Banach spaces are extended to the Fréchet space setting. Existence of fixed points will be established by means of a diagonal process together with a result on hemicompact maps [K. K. Tan and X.-Z. Yuan, J. Math. Anal. Appl. 185, No. 2, 378-390 (1994; Zbl 0856.47036)]. Section 3 shows how the fixed point theory in Section 2 can be applied naturally to obtain general existence results for nonlinear integral inclusions”.

47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H04Set-valued operators
47H09Mappings defined by “shrinking” properties
47G20Integro-differential operators
46A04Locally convex Fréchet spaces, etc.