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Keywords:
periodic solutions; translation operator along trajectories; set-valued maps; $C_0$-semigroup; $R_\delta$-sets
periodic solutions; translation operator along trajectories; set-valued maps; $C_0$-semigroup; $R_\delta$-sets
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.
References:
[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
[2] Bader R.: Fixpunktindextheorie mengenwertiger Abbildungen und einige Anwendungen. Ph.D. Dissertation, Universität München, 1995. Zbl 0865.47049
[3] Bader R.: Fixed point theorems for compositions of set-valued maps with single-valued maps. submitted. Zbl 1012.47043
[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. MR 1304049 | Zbl 0846.55001
[5] Bothe D.: Multivalued perturbations of $m$-accretive differential inclusions. to appear in Israel J. Math. MR 1669396 | Zbl 0922.47048
[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
[7] Deimling K.: Multivalued differential equations. de Gruyter, Berlin-New York, 1992. MR 1189795 | Zbl 0820.34009
[8] Diestel J.: Geometry of Banach Spaces - Selected Topics. LNM 485, Springer-Verlag, Berlin-Heidelberg-New York, 1975. MR 0461094 | Zbl 0466.46021
[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
[10] Hyman D.M.: On decreasing sequences of compact absolute retracts. Fund. Math. 64 (1969), 91-97. MR 0253303 | Zbl 0174.25804
[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
[12] Kamenskii M., Obukhovskii V., Zecca P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces. book in preparation. Zbl 0988.34001
[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
[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
[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. MR 0248580 | Zbl 0165.43304
[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
[17] Muresan M.: On a boundary value problem for quasi-linear differential inclusions of evolution. Collect. Math. 45 2 (1994), 165-175. MR 1316934 | Zbl 0824.34017
[18] Papageorgiou N.S.: Boundary value problems for evolution inclusions. Comment. Math. Univ. Carolinae 29 (1988), 355-363. MR 0957404 | Zbl 0696.35074
[19] Pazy A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. MR 0710486 | Zbl 0516.47023
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