Multivalued linear operators and differential inclusions in Banach spaces

Anatolii Baskakov; Valeri Obukhovskii; Pietro Zecca

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2003)

  • Volume: 23, Issue: 1, page 53-74
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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Anatolii Baskakov, Valeri Obukhovskii, and Pietro Zecca. "Multivalued linear operators and differential inclusions in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 23.1 (2003): 53-74. <http://eudml.org/doc/271435>.

@article{AnatoliiBaskakov2003,
abstract = {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.},
author = {Anatolii Baskakov, Valeri Obukhovskii, Pietro Zecca},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {multivalued linear operator; generalized semigroup; minimal growth at infinity; Hille-Yosida condition; degenerate differential inclusion; Cauchy problem; bounded solution; differential inclusion},
language = {eng},
number = {1},
pages = {53-74},
title = {Multivalued linear operators and differential inclusions in Banach spaces},
url = {http://eudml.org/doc/271435},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Anatolii Baskakov
AU - Valeri Obukhovskii
AU - Pietro Zecca
TI - Multivalued linear operators and differential inclusions in Banach spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2003
VL - 23
IS - 1
SP - 53
EP - 74
AB - 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.
LA - eng
KW - multivalued linear operator; generalized semigroup; minimal growth at infinity; Hille-Yosida condition; degenerate differential inclusion; Cauchy problem; bounded solution; differential inclusion
UR - http://eudml.org/doc/271435
ER -

References

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  1. [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. [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. [3] R. Cross, Multivalued Linear Operators. Monographs and Textbooks in Pure and Applied Mathematics, 213. Marcel Dekker, Inc., New York, 1998. 
  4. [4] A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl. 163 (4) (1993), 353-384. Zbl0786.47037
  5. [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. Zbl0913.34001
  6. [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. Zbl0988.34001
  7. [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. [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. Zbl0982.34001
  9. [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. [10] V. Obukhovskii and P. Zecca, On boundary value problems for degenerate differential inclusions in Banach spaces, Abstr. Appl. Anal. 13 (2003), 769-784. Zbl1076.34070
  11. [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. [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. Zbl1145.47304
  13. [13] K. Yosida, Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin, 1965. Zbl0126.11504
  14. [14] A. Yagi, Generation theorem of semigroup for multivalued linear operators, Osaka J. Math. 28 (2) (1991), 385-410. Zbl0812.47045

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