This paper is concerned with periodic solutions for perturbations of the sweeping process introduced by J.J. Moreau in 1971. The perturbed equation has the form where C is a T-periodic multifunction from [0,T] into the set of nonempty convex weakly compact subsets of a separable Hilbert space H, is the normal cone of C(t) at u(t), f:[0,T] × H∪H is a Carathéodory function and Du is the differential measure of the periodic BV solution u. Several existence results of periodic solutions for this differential inclusion are stated under various assumptions on the moving convex set C(t) and the perturbation f.
@article{bwmeta1.element.bwnjournal-article-div15i2n1bwm,
author = {Charles Castaing and Manuel D.P. Monteiro Marques},
title = {Periodic solutions of evolution problem associated with moving convex sets},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {15},
year = {1995},
pages = {99-127},
zbl = {0854.35134},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-div15i2n1bwm}
}
Charles Castaing; Manuel D.P. Monteiro Marques. Periodic solutions of evolution problem associated with moving convex sets. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 15 (1995) pp. 99-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-div15i2n1bwm/
[000] [1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, 1976. | Zbl 0328.47035
[001] [2] R.I. Becker, Periodic solutions of semilinear equations of evolutions of compact type, J. Math. Anal. Appl. 82 (1981), 33-48. | Zbl 0465.34014
[002] [3] C. Benassi and A. Gavioli, Approximation from the exterior of a multifunction with connected values defined on an interval, Atti Sem. Mat. Fis. Modena XLII (1994), 237-252. | Zbl 0873.54021
[003] [4] H. Brezis, Opérateurs maximaux monotones, North-Holland, Amsterdam 1973.
[004] [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin 1977.
[005] [6] C. Castaing, Truong Xuan Duc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Analysis 1 (1993), 109-119. | Zbl 0813.34018
[006] [7] C. Castaing and V. Jalby, Integral functionals on the space of vector measures. Applications to the sweeping process, Preprint, Université Montpellier II (1993), 29 pages. | Zbl 0837.46019
[007] [8] C. Castaing and V. Jalby, Epiconvergence of integral functionals on the space of vector measures, C.R. Acad. Sci. Paris 319 (1994), 669-674. | Zbl 0821.49014
[008] [9] A. Gamal, Perturbations semicontinues supérieurement de certaines équations d'évolution, Sém. Anal. Convexe, Montpellier, (1981), Exposé 14 (15 pages).
[009] [10] A. Gavioli, Approximation from the exterior of a multifunction and its application to the sweeping process, J. Diff. Equations 92 (1991), 373-383. | Zbl 0744.41018
[010] [11] A. Haraux, Opérateurs maximaux monotones et oscillations forcées non linéaires, These, Université Pierre et Marie Curie, Paris 6 Juin 1978.
[011] [12] N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), 185-192. | Zbl 0795.34051
[012] [13] M.D.P.M. Marques, Perturbations convexes semicontinues supérieurement des problemes d'évolution dans les espaces de Hilbert, Sém. Anal. Convexe, Montpellier (1984), Exposé 2 (23 pages).
[013] [14] M.D.P.M. Marques, Differential inclusions in nonsmooth mechanical problems shocks and dry friction, Birkhäuser Verlag, 1993. | Zbl 0802.73003
[014] [15] S. Maury, Un probleme de frottement équivalent a un probleme de poursuite; étude asymptotique, Sém. Anal. Convexe, Montpellier (1973), Exposé 8 (18 pages). | Zbl 0362.70005
[015] [16] J.J. Moreau, Sur les mesures différentielles de fonctions vectorielles et certains problemes d'évolution, C.R. Acad. Sci. Sci. Paris 282 (1976), 837-840. | Zbl 0329.34050
[016] [17] J.J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equations 26 (1977), 347-374. | Zbl 0356.34067
[017] [18] J.C. Péralba, Equations d'évolution dans un espace de Hilbert associées a des opérateurs sous-différentiels, Thése, Université Montpellier II (1973), (96 pages).
[018] [19] J.C. Péralba, Equations d'évolution dans un espace de Hilbert associées a des opérateurs sous-différentiels, C.R. Acad. Sci. Paris 275 (1972), 93-96. | Zbl 0238.35018
[019] [20] M. Valadier, Applications des mesures de Young aux suites uniformément intégrables dans un espace de Banach, Sém. Anal. Convexe, Montpellier (1990), Exposé 3 (14 pages).
[020] [21] M. Valadier, Lipschitz approximation of the sweeping process (Moreau process), J. Diff. Equations 88 (1990), 248-264. | Zbl 0716.34059
[021] [22] I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653-661. | Zbl 0701.34074