Abstract
In this paper we prove the existence of strong solutions for evolution inclusions of the form −\(\dot x\)(t) ∈ ∂ϕ(x(t))+F(t,x)) defined in a separable Hilbert space, where ∂ϕ(·) denotes the subdifferential of a proper, closed, convex function ϕ(·) andF(t,x) is a multivalued nonconvex, nonmonotone perturbation satisfying a general growth condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.Antosiewicz, A.Cellina, Continuous selections and differential relations,J. Diff. Equations 19 (1975), pp. 386–398.MR 55: 3373
V.Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands (1976).MR 52: 11666
Ph.Benilan, Solutions integrales d'equations d'evolution dans un espace de Banach,C. R. Acad. Sci. Paris, t.274 (1972), pp. 47–50.MR 45: 9212
A.Bressan, On differential relations with lower semicontinuous right hand side J. Diff. Equations37 (1980), pp. 89–97.MR 81j: 34022
H.Brezis,Operateurs Maximaux Monotones et Semi-groupes Contractions dans les Espases de Hilbert North Holland, Amsterdam (1973).Zbl 252: 47055
H. Brezis, New results concerning monotone operators and nonlinear semigroups Proc. of RIMS on Analysis of Nonlinear Problems, Kyoto University (1975), pp. 2–27.MR 58: 12532
A.Cellina, M.Marchi, Nonconvex perturbations of maximal monotone differential inclusions,Israel J. Math. 46 (1983), pp. 1–11.MR 85b: 34013
G.Colombo, A.Fonda, A.Ornelas, Lower semicontinuous perturbations of maximal monotone differential inclusions,Israel J. Math 60 (1988), pp. 211–218.
J.-P.Delahaye, J.Denel, The continuities of point to set maps, definitions and equivalences,Math. Programming Study 10 (1979), pp. 8–12.
A.Fryszkowski, Continuous selections for a class of nonconvex multivalued mapsStudia Math. 76 (1983), pp. 163–174.Zbl 534: 28003
S.Gutman, Evolutions governed by m-accretive plus compact operators,Nonlinear Anal. T. M. A. 7 (1983), pp. 707–717.Zbl 518: 34055
Y.Konishi, Compacité des résolvantes des opérateurs maximaux cycliquement monotones,Proc. Japan Acad. 49 (1973), 303–305.MR 49: 11325
S.Lojasiewicz, The existence of solutions for lower semicontinuous orientor fields,Bull. Polish. Acad. Sci. 28 (1980), pp. 483–387.Zbl 483: 49028
M.Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems,J. Diff. Equations 46 (1982), pp. 268–299.Zbl 495: 35042
N. S.Papageorgiou, On multivalued evolution equations and differential inclusions in Banach spaces,Comm. Math. Univ. Sancti Pauli 36 (1987), pp. 21–39.Zbl 641: 47052
N. S.Papageorgiou, Convergence theorems for banach space valued integrable multifunctions,Intern, J. Math and Math. Sci. 10 (1987), pp. 433–442.Zbl 619: 28009
N. S.Papageorgiou, On multivalued evolution equations in Hilbert spaces involving time varying subdifferentials,Math. Japonica 34 (1989), pp. 287–296
R. T.Rockafellar,Conjugate Duality and Optimization, Conference Board of Math Sciences, No. 16, SIAM Publications, Philadelphia (1974).MR 51: 9811
D.Wagner, Survey of measurable selection theo=ems,SIAM J. Control Optim. 1 (1977), pp. 859–903.MR 58: 6137
Author information
Authors and Affiliations
Additional information
Research supported by N.S.F. Grant D.M.S.-8802688.
Rights and permissions
About this article
Cite this article
Papageorgiou, N.S. Nonconvex and nonmonotone perturbations of evolution inclusions of subdifferential type. Period Math Hung 21, 167–177 (1990). https://doi.org/10.1007/BF01946854
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01946854