# Parametrized relaxation for evolution inclusions of the subdifferential type

Archivum Mathematicum (1995)

- Volume: 031, Issue: 1, page 9-28
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topPapageorgiou, Nikolaos S.. "Parametrized relaxation for evolution inclusions of the subdifferential type." Archivum Mathematicum 031.1 (1995): 9-28. <http://eudml.org/doc/247698>.

@article{Papageorgiou1995,

abstract = {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.},

author = {Papageorgiou, Nikolaos S.},

journal = {Archivum Mathematicum},

keywords = {subdifferential; relaxation theorem; Filippov-Gronwall inequality; lower semicontinuous multifunction; continuous selector; weak norm; Filippov-Gronwall estimate; nonlinear evolution inclusions of the subdifferential type; continuous dependence; parametric relaxation theorem; parabolic distributed parameter system},

language = {eng},

number = {1},

pages = {9-28},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Parametrized relaxation for evolution inclusions of the subdifferential type},

url = {http://eudml.org/doc/247698},

volume = {031},

year = {1995},

}

TY - JOUR

AU - Papageorgiou, Nikolaos S.

TI - Parametrized relaxation for evolution inclusions of the subdifferential type

JO - Archivum Mathematicum

PY - 1995

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 031

IS - 1

SP - 9

EP - 28

AB - 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.

LA - eng

KW - subdifferential; relaxation theorem; Filippov-Gronwall inequality; lower semicontinuous multifunction; continuous selector; weak norm; Filippov-Gronwall estimate; nonlinear evolution inclusions of the subdifferential type; continuous dependence; parametric relaxation theorem; parabolic distributed parameter system

UR - http://eudml.org/doc/247698

ER -

## References

top- Variational Convergence for Functionals and Operators, Pitman, London (1984). MR0773850
- Differential Inclusions, Springer-Verlag, Berlin (1984). MR0755330
- Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands (1976). Zbl0328.47035MR0390843
- Extensions and selections of maps with decomposable values, Studia Math.90, (1988), pp. 69-86. MR0947921
- Operateurs Maximaux Monotones, North Holland, Amsterdam (1973). Zbl0252.47055
- 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. Zbl0558.34011MR0717499
- Continuous approximations for multifunctions, Pacific J. Math.123, (1986), pp. 9-31. MR0834135
- Vector Measures, Math Surveys,15, AMS, Providence, RI (1977). MR0453964
- Classical solutions of differential equations with multivalued right-hand side, SIAM J. Control 5, (1967), pp. 609-621. MR0220995
- On the existence of optimal controls for a class of nonlinear infinite dimensional systems, Math. Nachrichten 150, (1991), pp. 203-217. MR1109656
- A priori estimates for operational differential inclusions, J. Differential Equations 84, (1990), pp. 100-128. Zbl0715.49010MR1042661
- Continuous version of Filippov-Wazewski relaxation theorem, J. Differential Equations, 94, (1991), pp. 254-265. MR1137615
- Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal.7, (1977), pp. 149-182. MR0507504
- Lipschitzean generalized differential equations, Rend. Sem. Mat. Padova, 48, (1972), pp. 159-169. MR0340692
- Theory of Correspondences, Wiley Interscience, New York (1984). MR0752692
- Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26, (1977), pp. 347-374. Zbl0351.34038MR0508661
- 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. Zbl0634.28005MR0914749
- Convergence theorems for Banach space valued integrable multifunctions, Intern. J. Math. and Math. Sci. 10, (1987), pp. 433-442. Zbl0619.28009MR0896595
- A relaxation theorem for differential inclusions in Banach spaces, Tohoku Math. Journ. 39, (1987), pp. 505-517. Zbl0647.34011MR0917464
- 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. MR1087659
- On the solution set of evolution inclusions driven by time dependent subdifferentials, Math. Japonica 37, (1992), pp. 1-13. Zbl0810.34059MR1196384
- Extremal solutions of evolution inclusions associated with time dependent convex subdifferentials, Math. Nacrichten 158, (1992), pp. 22-36. Zbl0776.34014MR1235307
- Continuous dependence results for subdifferential inclusions, Zeitshrift für Analysis und ihre Anwendungen, 12 (1), (1993), pp. 137-152. Zbl0777.34042MR1239434
- Convexity of the orientor field and the solution set of a class of evolution inclusions, Math. Slovaca 42 (1993), pp. 593-615. Zbl0799.34018MR1273713
- Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Math, 1459, Springer-Verlag, New York (1990). Zbl0732.49002MR1090951
- Survey of measurable selection theorems, SIAM J. Control Optim. 15, (1977), pp. 859-903. Zbl0427.28009MR0486391
- On certain nonlinear evolution equations, J. Math. Soc. Japan 25, (1973), pp. 446-463. Zbl0253.35053MR0326522
- On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo 23, (1976), pp. 491-515. Zbl0343.34053MR0425701
- Evolution equations associated with subdifferentials, J. Math. Soc. Japan 31, (1978), pp. 623-646. MR0544681
- On the solution set of differential inclusions in Banach spaces, J. Differential Equations 93, (1991), pp. 213-237. MR1125218

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.