Parametrized relaxation for evolution inclusions of the subdifferential type
Papageorgiou, Nikolaos S.
Archivum Mathematicum, Tome 031 (1995), p. 9-28 / Harvested from Czech Digital Mathematics Library

In this paper we consider parametric nonlinear evolution inclusions driven by time-dependent subdifferentials. First we prove some continuous dependence results for the solution set (of both the convex and nonconvex problems) and for the set of solution-selector pairs (of the convex problem). Then we derive a continuous version of the “Filippov-Gronwall” inequality and using it, we prove the parametric relaxation theorem. An example of a parabolic distributed parameter system is also worked out in detail.

Publié le : 1995-01-01
Classification:  34A60,  34G20,  46N20,  49J52,  93C20
@article{107520,
     author = {Nikolaos S. Papageorgiou},
     title = {Parametrized relaxation for evolution inclusions of the subdifferential type},
     journal = {Archivum Mathematicum},
     volume = {031},
     year = {1995},
     pages = {9-28},
     zbl = {0839.34075},
     mrnumber = {1342371},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107520}
}
Papageorgiou, Nikolaos S. Parametrized relaxation for evolution inclusions of the subdifferential type. Archivum Mathematicum, Tome 031 (1995) pp. 9-28. http://gdmltest.u-ga.fr/item/107520/

Variational Convergence for Functionals and Operators, Pitman, London (1984). | MR 0773850

Differential Inclusions, Springer-Verlag, Berlin (1984). | MR 0755330

Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands (1976). | MR 0390843 | Zbl 0328.47035

Extensions and selections of maps with decomposable values, Studia Math.90, (1988), pp. 69-86. | MR 0947921

Operateurs Maximaux Monotones, North Holland, Amsterdam (1973). | Zbl 0252.47055

Continuous selections of solution sets of Lipschitzean differential inclusions, Funkcial. Ekvac. - in press.

Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96, (1983), pp. 130-147. | MR 0717499 | Zbl 0558.34011

Continuous approximations for multifunctions, Pacific J. Math.123, (1986), pp. 9-31. | MR 0834135

Vector Measures, Math Surveys,15, AMS, Providence, RI (1977). | MR 0453964

Classical solutions of differential equations with multivalued right-hand side, SIAM J. Control 5, (1967), pp. 609-621. | MR 0220995

On the existence of optimal controls for a class of nonlinear infinite dimensional systems, Math. Nachrichten 150, (1991), pp. 203-217. | MR 1109656

A priori estimates for operational differential inclusions, J. Differential Equations 84, (1990), pp. 100-128. | MR 1042661 | Zbl 0715.49010

Continuous version of Filippov-Wazewski relaxation theorem, J. Differential Equations, 94, (1991), pp. 254-265. | MR 1137615

Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal.7, (1977), pp. 149-182. | MR 0507504

Lipschitzean generalized differential equations, Rend. Sem. Mat. Padova, 48, (1972), pp. 159-169. | MR 0340692

Theory of Correspondences, Wiley Interscience, New York (1984). | MR 0752692

Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26, (1977), pp. 347-374. | MR 0508661 | Zbl 0351.34038

On the theory of Banach space valued multifunctions. Part 1: Integration and conditional expectation, J. Multiv. Anal.17, (1985), pp. 185-207.

On measurable multifunctions with applications to random multivalued equations, Math. Japonica 32, (1987), pp. 437-464. | MR 0914749 | Zbl 0634.28005

Convergence theorems for Banach space valued integrable multifunctions, Intern. J. Math. and Math. Sci. 10, (1987), pp. 433-442. | MR 0896595 | Zbl 0619.28009

A relaxation theorem for differential inclusions in Banach spaces, Tohoku Math. Journ. 39, (1987), pp. 505-517. | MR 0917464 | Zbl 0647.34011

On the relation between relaxability and performance stability for optimal control problems governed by nonlinear evolution equations, Intern. Journ. of Systems Sci. 22, (1991), pp. 237-259. | MR 1087659

On the solution set of evolution inclusions driven by time dependent subdifferentials, Math. Japonica 37, (1992), pp. 1-13. | MR 1196384 | Zbl 0810.34059

Extremal solutions of evolution inclusions associated with time dependent convex subdifferentials, Math. Nacrichten 158, (1992), pp. 22-36. | MR 1235307 | Zbl 0776.34014

Continuous dependence results for subdifferential inclusions, Zeitshrift für Analysis und ihre Anwendungen, 12 (1), (1993), pp. 137-152. | MR 1239434 | Zbl 0777.34042

Convexity of the orientor field and the solution set of a class of evolution inclusions, Math. Slovaca 42 (1993), pp. 593-615. | MR 1273713 | Zbl 0799.34018

Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Math, 1459, Springer-Verlag, New York (1990). | MR 1090951 | Zbl 0732.49002

Survey of measurable selection theorems, SIAM J. Control Optim. 15, (1977), pp. 859-903. | MR 0486391 | Zbl 0427.28009

On certain nonlinear evolution equations, J. Math. Soc. Japan 25, (1973), pp. 446-463. | MR 0326522 | Zbl 0253.35053

On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo 23, (1976), pp. 491-515. | MR 0425701 | Zbl 0343.34053

Evolution equations associated with subdifferentials, J. Math. Soc. Japan 31, (1978), pp. 623-646. | MR 0544681

On the solution set of differential inclusions in Banach spaces, J. Differential Equations 93, (1991), pp. 213-237. | MR 1125218