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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 {\it R. P. Agarwal} and {\it 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 [{\it K. K. Tan} and {\it 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”.

MSC:
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.
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Full Text: DOI
References:
[1] Agarwal, R. P.; O’regan, D.: A note on the existence of multiple fixed points for multivalued maps with applications. J. differential equations 160, 389-403 (2000) · Zbl 1008.47055
[2] Agarwal, R. P.; O’regan, D.: Fixed points in Fréchet spaces and variational inequalities. Nonlinear anal. 42, 1091-1099 (2000)
[3] R. P. Agarwal, and, D. O’Regan, A fixed point theorem of Leggett Williams type with applications to single and multivalued equations, to appear.
[4] Cecchi, M.; Marini, M.; Zecca, P.: Existence of bounded solutions for multivalued differential systems. Nonlinear anal. 9, 775-786 (1985) · Zbl 0534.34020
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[6] Frigon, M.: Théorèmes d’existence de solutions d’inclusions différentielles. NATO ASI series C 472, 51-87 (1995)
[7] Lasota, A.; Opial, Z.: An application of the kututani--Ky Fan theorem in the theory of ordinary differential equations. Bull. acad. Polon. sci. Ser. sci. Math. astron. Phys. 13, 781-786 (1965) · Zbl 0151.10703
[8] O’regan, D.: Integral inclusions of upper semi-continuous or lower semi-continuous type. Proc. amer. Math. soc. 124, 2391-2399 (1996) · Zbl 0860.45007
[9] Pruszko, T.: Topological degree methods in multivalued boundary value problems. Nonlinear analysis 5, 953-973 (1981) · Zbl 0478.34017
[10] Tan, K. K.; Yuan, X. Z.: Random fixed point theorems and approximation in cones. J. math. Anal. appl. 185, 378-390 (1994) · Zbl 0856.47036