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1995 | 15 | 2 | 99-127
Tytuł artykułu

Periodic solutions of evolution problem associated with moving convex sets

Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $-Du ∈ N_{C(t)}(u(t)) + f(t,u(t))$ 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, $N_{C(t)}(u(t))$ 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.
Twórcy
  • Département de Mathématiques, Université Montpellier II,, Case 051, Place Eugéne Bataillon, 34095 Montpellier cedex 05, France
  • Centro de Matemática e Aplicaçoes Fundamentais and, Faculdade de Ciencias da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, P-1699 Lisboa Codex, Portugal
Bibliografia
  • [1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, 1976.
  • [2] R.I. Becker, Periodic solutions of semilinear equations of evolutions of compact type, J. Math. Anal. Appl. 82 (1981), 33-48.
  • [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.
  • [4] H. Brezis, Opérateurs maximaux monotones, North-Holland, Amsterdam 1973.
  • [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin 1977.
  • [6] C. Castaing, Truong Xuan Duc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Analysis 1 (1993), 109-119.
  • [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.
  • [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.
  • [9] A. Gamal, Perturbations semicontinues supérieurement de certaines équations d'évolution, Sém. Anal. Convexe, Montpellier, (1981), Exposé 14 (15 pages).
  • [10] A. Gavioli, Approximation from the exterior of a multifunction and its application to the sweeping process, J. Diff. Equations 92 (1991), 373-383.
  • [11] A. Haraux, Opérateurs maximaux monotones et oscillations forcées non linéaires, These, Université Pierre et Marie Curie, Paris 6 Juin 1978.
  • [12] N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), 185-192.
  • [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).
  • [14] M.D.P.M. Marques, Differential inclusions in nonsmooth mechanical problems shocks and dry friction, Birkhäuser Verlag, 1993.
  • [15] S. Maury, Un probleme de frottement équivalent a un probleme de poursuite; étude asymptotique, Sém. Anal. Convexe, Montpellier (1973), Exposé 8 (18 pages).
  • [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.
  • [17] J.J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equations 26 (1977), 347-374.
  • [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).
  • [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.
  • [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).
  • [21] M. Valadier, Lipschitz approximation of the sweeping process (Moreau process), J. Diff. Equations 88 (1990), 248-264.
  • [22] I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653-661.
Typ dokumentu
Bibliografia
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