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Iteration regularized semigroups of set-valued functions

  • Masoud Mosallanezhad und Mohammad Janfada EMAIL logo
Veröffentlicht/Copyright: 10. November 2016

Abstract

In this paper, a set-valued iteration regularized semigroup, i.e. a family {Ft}t0 of set-valued functions for which

Fs+tC=FsFt,F0=C,s,t0,

will be considered, where C is a set-valued function on a closed convex cone in a Banach space. Under some appropriate conditions the generator of a set-valued regularized concave semigroup is introduced and some of its properties are investigated. Also differentiability of the iteration family {CFt}t0 is discussed.

MSC 2010: 26E25; 39B12; 47D03

References

[1] Aubin J.-P. and Frankowska H., Set-Valued Analysis in Control Theory, Kluwer Academic Publishers, Dordrecht, 2000. Suche in Google Scholar

[2] Berge C., Topological Spaces, Oliver and Boyd, Edinburgh, 1963. Suche in Google Scholar

[3] Clarke F., A proximal characterization of the reachable set, Systems Control Lett. 27 (1996), 195–197. 10.1016/0167-6911(95)00056-9Suche in Google Scholar

[4] Clarke F. H., Ledyaev Y. S., Stern R. J. and Wolenski P. R., Nonsmooth Analysis and Control Theory, Grad. Texts in Math. 178, Springer, New York, 1998. Suche in Google Scholar

[5] Davies E. B. and Pang M. M. H., The Cauchy problem and a generalization of the Hille–Yosida theorem, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 181–208. 10.1112/plms/s3-55.1.181Suche in Google Scholar

[6] deLaubenfels R., C-semigroups and the Cauchy problem, J. Funct. Anal. 111 (1993), no. 1, 44–61. 10.1006/jfan.1993.1003Suche in Google Scholar

[7] Edgar G. A., Measure, Topology and Fractal Geometry, Springer, New York, 1990. 10.1007/978-1-4757-4134-6Suche in Google Scholar

[8] Geletu A., Introduction to topological spaces and set-valued maps, Lecture notes, 2006. Suche in Google Scholar

[9] Hukuhara M., Integration des application mesurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223. Suche in Google Scholar

[10] Kwiecinska G., On the intermediate value property of multivalued functions, Real Anal. Exchange 26 (2000/2001), no. 1, 245–260. 10.2307/44153161Suche in Google Scholar

[11] Li Y.-C. and Shaw S.-Y., N-times integrated C-semigroups and the abstract Cauchy problem, Taiwanese J. Math. 1 (1997), no. 1, 75–102. 10.11650/twjm/1500404927Suche in Google Scholar

[12] Li Y.-C. and Shaw S.-Y., On characterization and perturbation of local C-semigroups, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1097–1106. 10.1090/S0002-9939-06-08549-2Suche in Google Scholar

[13] Nadler, Jr. S. B., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. 10.2140/pjm.1969.30.475Suche in Google Scholar

[14] Olko J., Concave iteration semigroups of linear set-valued functions, Ann. Polon. Math. 71 (1999), no. 1, 31–38. 10.4064/ap-71-1-31-38Suche in Google Scholar

[15] Radstrom H., An embedding theorem for space of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165–169. 10.1090/S0002-9939-1952-0045938-2Suche in Google Scholar

[16] Roxin E. O., Reachable zones in autonomous differential systems, Bol. Soc. Mat. Mex. (2) 5 (1960), 125–135. Suche in Google Scholar

[17] Smajdor A., On increasing iteration semigroups of multivalued functions, Iteration Theory and Its Functional Equations (Schloss Hofen 1984), Lecture Notes in Math. 1163, Springer, Berlin (1990), 177–182. 10.1007/BFb0076431Suche in Google Scholar

[18] Smajdor A., Increasing iteration semigroups of Jensen set-valued functions, Aequationes Math. 56 (1998), 131–142. 10.1007/s000100050049Suche in Google Scholar

[19] Smajdor A., On regular multivalued cosine functions, Ann. Math. Sil. 13 (1999), 271–280. Suche in Google Scholar

[20] Smajdor A., Hukuhara’s derivative and concave iteration semigroups of linear set-valued functions, J. Appl. Anal. 8 (2002), no. 2, 297–305. 10.1515/JAA.2002.297Suche in Google Scholar

[21] Smajdor A., On concave iteration semigroups of linear set-valued functions, Aequationes Math. 75 (2008), no. 1–2, 149–162. 10.1007/s00010-007-2876-8Suche in Google Scholar

[22] Smajdor A. and Smajdor W., Concave iteration semigroups of linear continuous set-valued functions, Cent. Eur. J. Math. 10 (2012), no. 6, 2272–2282. 10.2478/s11533-012-0095-6Suche in Google Scholar

[23] Smajdor W., Superadditive set-valued functions and Banach–Steinhaus theorem, Rad. Mat. 3 (1987), 203–214. Suche in Google Scholar

[24] Tanaka N. and Miyadera I., Exponentially bounded C-semigroups and integrated semigroups, Tokyo J. Math. 12 (1989), no. 1, 99–115. 10.3836/tjm/1270133551Suche in Google Scholar

[25] Tanaka N. and Miyadera I., C-semigroups and the abstract Cauchy problem, J. Math. Anal. Appl. 170 (1992), no. 1, 196–206. 10.1016/0022-247X(92)90013-4Suche in Google Scholar

[26] Wolenski P. R., The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J. Control Optim. 28 (1990), no. 5, 1148–1161. 10.1137/0328062Suche in Google Scholar

Received: 2016-1-8
Revised: 2016-9-18
Accepted: 2016-10-12
Published Online: 2016-11-10
Published in Print: 2016-12-1

© 2016 by De Gruyter

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