# Article

 Title: The periodic problem for semilinear differential inclusions in Banach spaces (English) Author: Bader, Ralf Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 39 Issue: 4 Year: 1998 Pages: 671-684 . Category: math . Summary: Sufficient conditions on the existence of periodic solutions for semilinear differential inclusions are given in general Banach space. In our approach we apply the technique of the translation operator along trajectories. Due to recent results it is possible to show that this operator is a so-called decomposable map and thus admissible for certain fixed point index theories for set-valued maps. Compactness conditions are formulated in terms of the Hausdorff measure of noncompactness. (English) Keyword: periodic solutions Keyword: translation operator along trajectories Keyword: set-valued maps Keyword: $C_0$-semigroup Keyword: $R_\delta$-sets MSC: 34A60 MSC: 34C25 MSC: 34G25 MSC: 47H11 MSC: 47N20 idZBL: Zbl 1060.34508 idMR: MR1715457 . Date available: 2009-01-08T18:47:24Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119043 . Reference: [1] Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N.: Measures of Noncompactness and Condensing Operators.Birkhäuser Verlag, Basel-Boston-Berlin, 1992. MR 1153247 Reference: [2] Bader R.: Fixpunktindextheorie mengenwertiger Abbildungen und einige Anwendungen.Ph.D. Dissertation, Universität München, 1995. Zbl 0865.47049 Reference: [3] Bader R.: Fixed point theorems for compositions of set-valued maps with single-valued maps.submitted. Zbl 1012.47043 Reference: [4] Bader R., Kryszewski W.: Fixed-point index for compositions of set-valued maps with proximally $\infty$-connected values on arbitrary ANR's.Set-Valued Analysis 2 (1994), 459-480. Zbl 0846.55001, MR 1304049 Reference: [5] Bothe D.: Multivalued perturbations of $m$-accretive differential inclusions.to appear in Israel J. Math. Zbl 0922.47048, MR 1669396 Reference: [6] Conti G., Obukhovskii V., Zecca P.: On the topological structure of the solution set for a semilinear functional-differential inclusion in a Banach space.in: Topology in Nonlinear Analysis, K. Geba and L. Górniewicz (eds.), Polish Academy of Sciences, Institute of Mathematics, Banach Center Publications 35, Warszawa, 1996, pp.159-169. MR 1448435 Reference: [7] Deimling K.: Multivalued differential equations.de Gruyter, Berlin-New York, 1992. Zbl 0820.34009, MR 1189795 Reference: [8] Diestel J.: Geometry of Banach Spaces - Selected Topics.LNM 485, Springer-Verlag, Berlin-Heidelberg-New York, 1975. Zbl 0466.46021, MR 0461094 Reference: [9] Górniewicz L.: Topological approach to differential inclusions in: Topological methods in differential equations and inclusions, A. Granas and M. Frigon (eds.), NATO ASI Series C 472, Kluwer Academic Publishers, 1995, pp.129-190.. MR 1368672 Reference: [10] Hyman D.M.: On decreasing sequences of compact absolute retracts.Fund. Math. 64 (1969), 91-97. Zbl 0174.25804, MR 0253303 Reference: [11] Kamenskii M.I., Obukhovskii V.V.: Condensing multioperators and periodic solutions of parabolic functional-differential inclusions in Banach spaces.Nonlinear Anal. 20 (1993), 781-792. MR 1214743 Reference: [12] Kamenskii M., Obukhovskii V., Zecca P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces.book in preparation. Zbl 0988.34001 Reference: [13] Kamenskii M., Obukhovskii V., Zecca P.: On the translation multioperator along the solutions of semilinear differential inclusions in Banach spaces.to appear in Rocky Mountain J. Math. MR 1661823 Reference: [14] Krasnoselskii M.A.: The operator of translation along trajectories of differential equations.American Math. Soc., Translation of Math. Monographs, vol. 19, Providence, 1968. MR 0223640 Reference: [15] Lasota A., Opial Z.: Fixed-point theorems for multi-valued mappings and optimal control problems.Bull. Polish Acad. Sci. Math. 16 (1968), 645-649. Zbl 0165.43304, MR 0248580 Reference: [16] Mönch H., von Harten G.-F.: On the Cauchy problem for ordinary differential equations in Banach spaces.Archiv Math. 39 (1982), 153-160. MR 0675655 Reference: [17] Muresan M.: On a boundary value problem for quasi-linear differential inclusions of evolution.Collect. Math. 45 2 (1994), 165-175. Zbl 0824.34017, MR 1316934 Reference: [18] Papageorgiou N.S.: Boundary value problems for evolution inclusions.Comment. Math. Univ. Carolinae 29 (1988), 355-363. Zbl 0696.35074, MR 0957404 Reference: [19] Pazy A.: Semigroups of linear operators and applications to partial differential equations.Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. Zbl 0516.47023, MR 0710486 .

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