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Transactions of the American Mathematical Society

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Necessary and sufficient conditions for viability for semilinear differential inclusions

Authors: Ovidiu Cârja, Mihai Necula and Ioan I. Vrabie
Journal: Trans. Amer. Math. Soc. 361 (2009), 343-390
MSC (2000): Primary 34G20, 47J35; Secondary 35K57, 35K65
Published electronically: August 21, 2008
MathSciNet review: 2439410
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a set $ K$ in a Banach space $ X$, we define: the tangent set, and the quasi-tangent set to $ K$ 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.

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  • 1. Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330
  • 2. W. Bebernes and I. D. Schuur, The Ważewski topological method for contingent equations, Ann. Mat. Pura Appl., 87(1970), 271-278.
  • 3. H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances in Math. 21 (1976), no. 3, 355–364. MR 0425688
  • 4. H. Bouligand, Sur les surfaces dépourvues de points hyperlimités, Ann. Soc. Polon. Math., 9(1930), 32-41.
  • 5. Ovidiu Cârjă, On the minimum time function and the minimum energy problem; a nonlinear case, Systems Control Lett. 55 (2006), no. 7, 543–548. MR 2225363, 10.1016/j.sysconle.2005.11.005
  • 6. Ovidiu Cârjă and Manuel D. P. Monteiro Marques, Weak tangency, weak invariance, and Carathéodory mappings, J. Dynam. Control Systems 8 (2002), no. 4, 445–461. MR 1931893, 10.1023/A:1020765401015
  • 7. Ovidiu Cârjă and Corneliu Ursescu, The characteristics method for a first order partial differential equation, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 39 (1993), no. 4, 367–396. MR 1328937
  • 8. Ovidiu Cârjă and Ioan I. Vrabie, Some new viability results for semilinear differential inclusions, NoDEA Nonlinear Differential Equations Appl. 4 (1997), no. 3, 401–424. MR 1458535, 10.1007/s000300050022
  • 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), no. 4, 493–518. MR 1481624, 10.1023/A:1021873607769
  • 10. Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404
  • 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, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • 13. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • 14. R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
  • 15. S. Gautier, Equations differentielles multivoques sur un fermé, Publications de l'Université de Pau, (1973).
  • 16. Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • 17. Harald Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), no. 5, 985–999. MR 586861, 10.1016/0362-546X(80)90010-3
  • 18. Mitio Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan (3) 24 (1942), 551–559 (German). MR 0015180
  • 19. Nicolae H. Pavel and Ioan I. Vrabie, Équations d’évolution multivoques dans des espaces de Banach, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 5, A315–A317 (French, with English summary). MR 0513204
  • 20. Nicolae H. Pavel and Ioan I. Vrabie, Semilinear evolution equations with multivalued right-hand side in Banach spaces, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 25 (1979), no. 1, 137–157. MR 553132
  • 21. F. Severi, Su alcune questioni di topologia infinitesimale, Annales Soc. Polonaise, 9(1931), 97-108.
  • 22. Shu Zhong Shi, Viability theorems for a class of differential-operator inclusions, J. Differential Equations 79 (1989), no. 2, 232–257. MR 1000688, 10.1016/0022-0396(89)90101-0
  • 23. I. I. Vrabie, Compactness methods for nonlinear evolutions, 2nd ed., Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. With a foreword by A. Pazy. MR 1375237
  • 24. Ioan I. Vrabie, 𝐶₀-semigroups and applications, North-Holland Mathematics Studies, vol. 191, North-Holland Publishing Co., Amsterdam, 2003. MR 1972224
  • 25. Ioan I. Vrabie, Differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. An introduction to basic concepts, results and applications. MR 2092912

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Additional Information

Ovidiu Cârja
Affiliation: Faculty of Mathematics, “Al. I. Cuza” University, Iaşi 700506, Romania – and – “Octav Mayer” Mathematics Institute, Romanian Academy, Iaşi 700506, Romania

Mihai Necula
Affiliation: Faculty of Mathematics, “Al. I. Cuza” University Iaşi 700506, Romania

Ioan I. Vrabie
Affiliation: Faculty of Mathematics, “Al. I. Cuza” University, Iaşi 700506, Romania – and – “Octav Mayer” Mathematics Institute, Romanian Academy, Iaşi 700506, Romania

Keywords: Viability, tangency condition, reaction-diffusion systems, compact semigroup.
Received by editor(s): February 15, 2007
Published electronically: 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.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.