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2003 | 23 | 1 | 53-74
Tytuł artykułu

Multivalued linear operators and differential inclusions in Banach spaces

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EN
Abstrakty
EN
In this paper, we study multivalued linear operators (MLO's) and their resolvents in non reflexive Banach spaces, introducing a new condition of a minimal growth at infinity, more general than the Hille-Yosida condition. Then we describe the generalized semigroups induced by MLO's. We present a criterion for an MLO to be a generator of a generalized semigroup in an arbitrary Banach space. Finally, we obtain some existence results for differential inclusions with MLO's and various types of multivalued nonlinearities. As a consequence, we give theorems on the existence of local, global and bounded solutions of the Cauchy problem for degenerate differential inclusions.
Twórcy
  • Faculty of Applied Mathematics and Mechanics, University of Voronezh, Universitetskaya pl., 1, 394 006, Voronezh, Russia
  • Faculty of Mathematics, University of Voronezh, Universitetskaya pl., 1, 394 006, Voronezh, Russia
autor
  • Dipartimento di Energetica "S. Stecco", Universita di Firenze, Via S. Marta, 3-1, 50139 Firenze, Italia
Bibliografia
  • [1] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Topological methods in the theory of fixed points of multivalued mappings, (in Russian) Uspekhi Mat. Nauk 35 (1980), no.1(211), 59-126, English transl. in Russian Math. Surveys 35 (1980), 65-143.
  • [2] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Multivalued mappings, (Russian) Mathematical Analysis 19 pp.127-230, 232, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Techn. Informatsii, Moscow, 1982. English transl. in J. Soviet Math. 24 (1984), 719-791.
  • [3] R. Cross, Multivalued Linear Operators. Monographs and Textbooks in Pure and Applied Mathematics, 213. Marcel Dekker, Inc., New York, 1998.
  • [4] A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl. 163 (4) (1993), 353-384.
  • [5] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces. Monographs and Textbooks in Pure and Applied Mathematics, 215. Marcel Dekker, Inc., New York, 1999.
  • [6] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin - New York, 2001.
  • [7] I.V. Mel'nikova and M.A. Al'shanskii, Well-posedness of the degenerate Cauchy problem in a Banach space, (in Russian) Dokl. Akad. Nauk 336 (1994), no.1, 17-20; English translation in Russian Acad. Sci. Dokl. Math. 49 (3) (1994), 449-453.
  • [8] I.V. Mel'nikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 120. Chapman & Hall/CRC, Boca Raton, FL, 2001.
  • [9] V.V. Obukhovskii, On some fixed point principles for multivalued condensing operators, (in Russian) Trudy Mat. Fac. Voronezh Univ. 4 (1971), 70-79.
  • [10] V. Obukhovskii and P. Zecca, On boundary value problems for degenerate differential inclusions in Banach spaces, Abstr. Appl. Anal. 13 (2003), 769-784.
  • [11] G.A. Sviridyuk, On the general theory of operator semigroups, (in Russian) Uspekhi Mat. Nauk 49 (1994), no. 4(208), 47-74; English translation in Russian Math. Surveys 49 (4) (1994), 45-74.
  • [12] G.A. Sviridyuk and V.E. Fedorov, Semigroups of operators with kernels, (in Russian) Vestnik Chelyabinsk Univ. Ser. 3 Mat. Mekh. 1 (6) (2002), 42-70.
  • [13] K. Yosida, Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin, 1965.
  • [14] A. Yagi, Generation theorem of semigroup for multivalued linear operators, Osaka J. Math. 28 (2) (1991), 385-410.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1046
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