Transactions of the American Mathematical Society

Necessary and sufficient conditions for viability for semilinear differential inclusions

Author(s): Ovidiu Cârja; Mihai Necula; Ioan I. Vrabie
Journal: Trans. Amer. Math. Soc. 361 (2009), 343-390.
MSC (2000): Primary 34G20, 47J35; Secondary 35K57, 35K65
Posted: August 21, 2008
Retrieve article in: PDF DVI PostScript

Abstract: Given a set

in a Banach space

, we define: the tangent set, and the quasi-tangent set to at $\xi\in K$ , concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931). Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions, conditions expressed in terms of the tangency concepts introduced.

1.: J. P. Aubin and A. Cellina, Differential inclusions, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. MR 755330 (85j:49010)
2.: W. Bebernes and I. D. Schuur, The Ważewski topological method for contingent equations, Ann. Mat. Pura Appl., 87(1970), 271-278.
3.: H. Brezis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. in Mathematics, 21(1976), 355-364. MR 0425688 (54:13641)
4.: H. Bouligand, Sur les surfaces dépourvues de points hyperlimités, Ann. Soc. Polon. Math., 9(1930), 32-41.
5.: O. Cârjă, On the minimal time function and the minimum energy problem: a nonlinear case, Systems Control Lett., 55 (2006), 543-548. MR 2225363 (2007b:93018)
6.: O. Cârjă and M. D. P. Monteiro Marques, Weak tangency, weak invariance and Carathéodory mappings, J. Dynam. Control Systems, 8(2002), 445-461. MR 1931893 (2003k:34033)
7.: O. Cârjă and C. Ursescu, The characteristics method for a first order partial differential equation, An. Ştiin. Univ. Al. I. Cuza Iaşi, Secţ. I a Mat., 39(1993), 367-396. MR 1328937 (96h:35252)
8.: O. Cârjă and I. I. Vrabie, Some new viability results for semilinear differential inclusions, NoDEA Nonlinear Differential Equations Appl., 4(1997), 401-424. MR 1458535 (98h:34029)
9.: F. H. Clarke, Yu. S. Ledyaev and M. L. Radulescu, Approximate invariance and differential inclusions in Hilbert spaces, J. Dynam. Control Systems, 3 (1997), pp. 493-518. MR 1481624 (98k:49011)
10.: K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985. MR 787404 (86j:47001)
11.: J. Diestel, Remarks on weak compactness in $L_1(\mu;X)$ , Glasg. Math. J., 18(1977), 87-01.
12.: J. Diestel and J. J. Uhl, Jr., Vector measures, Mathematical Surveys, 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
13.: N. Dunford and J. T. Schwartz, Linear operators Part I: General theory, Interscience Publishers, Inc., New York, 1958. MR 1009162 (90g:47001a)
14.: R. E. Edwards, Functional analysis theory and applications, Holt, Rinehart and Winston, New York Chicago San Francisco Toronto London, 1965. MR 0221256 (36:4308)
15.: S. Gautier, Equations differentielles multivoques sur un fermé, Publications de l'Université de Pau, (1973).
16.: E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, 31, Fourth Printing of Revised Edition, 1981. MR 0089373 (19:664d)
17.: H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. T.M.A., 4(1980), 985-999. MR 586861 (82c:34075)
18.: M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan, 24(1942), 551-559. MR 0015180 (7:381e)
19.: N. H. Pavel and I. I. Vrabie, Equations d'évolution multivoques dans des espaces de Banach, C. R. Acad. Sci. Paris Sér. A-B, 287(1978), A315-A317. MR 0513204 (58:23829)
20.: N. H. Pavel and I. I. Vrabie, Semilinear evolution equations with multivalued right-hand side in Banach spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat., 25(1979), 137-157. MR 553132 (81a:47060)
21.: F. Severi, Su alcune questioni di topologia infinitesimale, Annales Soc. Polonaise, 9(1931), 97-108.
22.: S. Z. Shi, Viability theorems for a class of differential operator inclusions, J. Differential Equations, 79(1989), 232-257. MR 1000688 (90e:34025)
23.: I. I. Vrabie, Compactness methods for nonlinear evolutions, Second Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics 75, Longman, 1995. MR 1375237 (96k:47116)
24.: I. I. Vrabie, -semigroups and applications, North-Holland Publishing Co. Amsterdam, 2003. MR 1972224 (2004c:47088)
25.: I. I. Vrabie, Differential equations. An introduction to basic concepts, results and applications, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2092912 (2005d:34001)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 34G20, 47J35, 35K57, 35K65

Ovidiu Cârja
Affiliation: Faculty of Mathematics, ``Al. I. Cuza'' University, Iasi 700506, Romania - and - ``Octav Mayer'' Mathematics Institute, Romanian Academy, Iasi 700506, Romania
Email: ocarja@uaic.ro

Mihai Necula
Affiliation: Faculty of Mathematics, ``Al. I. Cuza'' University Iasi 700506, Romania
Email: necula@uaic.ro

Ioan I. Vrabie
Affiliation: Faculty of Mathematics, ``Al. I. Cuza'' University, Iasi 700506, Romania - and - ``Octav Mayer'' Mathematics Institute, Romanian Academy, Iasi 700506, Romania
Email: ivrabie@uaic.ro

DOI: 10.1090/S0002-9947-08-04668-0
PII: S 0002-9947(08)04668-0
Keywords: Viability, tangency condition, reaction-diffusion systems, compact semigroup.
Received by editor(s): February 15, 2007
Posted: August 21, 2008
Additional Notes: The first and third authors were supported by the Project CEx05-DE11-36/05.10.2005. The second author was supported by CNCSIS Grant A 1159/2006.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS