×

Multivalued perturbations for a class of nonlinear evolution equations. (English) Zbl 0752.34013

An evolution problem of the type \(u'(t)\in-\partial^ - f(u(t))+{\mathcal G}(u)(t)\) is considered. Here \({\mathcal G}: {\mathcal C}([0,T];\Omega)\rightsquigarrow L^ 2(0,T;H)\) is a set-valued map, \(H\) denotes a Hilbert space and \(\partial^ -\) denotes subdifferential. An existence theorem for the above differential inclusion is proved for \(\mathcal G\) Hausdorff-upper semicontinuous, locally bounded with closed and convex values and \(f: \Omega\mapsto\mathbb{R}\cup\{+\infty\}\) lower semicontinuous with \(\phi\)-monotone subdifferential of order two (i.e. there exists a continuous map \(\phi: D(f)^ 2\times\mathbb{R}^ 2\mapsto\mathbb{R}^ +\) such that for every \(x,y\in D(\partial^ - f)\), \(\alpha\in\partial^ - f(x)\), \(\beta\in\partial^ - f(x)\), \(\langle\alpha-\beta,x-y\rangle\geq- \phi(x,y,f(x),f(y))(1+\|\alpha\|^ 2+\|\beta\|^ 2)\| x- y\|^ 2\)).
As an application an existence result for \(u'(t)\in-\partial^ - f(u(t))+G_ 1(t,u(t))+G_ 2(t,u(t))\), where \(G_ 1\) (resp. \(G_ 2\)) is a lower semicontinuous (resp. upper semicontinuous, convex valued) are set-valued maps from \([0,T]\times\Omega\) into \(H\), is proved.

MSC:

34A60 Ordinary differential inclusions
49J45 Methods involving semicontinuity and convergence; relaxation
47H20 Semigroups of nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Attouch, H.; Damlamian, A., On multivalued equations in Hilbert spaces, Isr. J. Math., 12, 373-390 (1972) · Zbl 0243.35080
[2] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Berlin: Springer-Verlag, Berlin · Zbl 0538.34007
[3] Bressan, A.; Colombo, G., Estensions and selections of maps with decomposable values, Studia Math., 90, 69-86 (1988) · Zbl 0677.54013
[4] Cellina, A., Approximations of set valued functions and fixed point theorems, Ann. di Mat. Pura Appl., 82, 17-24 (1969) · Zbl 0187.07701
[5] Cellina, A., Notes from the course held at S.I.S.S.A. (198788), Trieste: Academic, Trieste · Zbl 1356.49033
[6] Cellina, A.; Marchi, M. V., Non-convex perturbations of maximal monotone differential inclusions, Isr. J. Math., 46, 1-11 (1983) · Zbl 0542.47036
[7] Chobanov, G.; Marino, A.; Scolozzi, D., Evolution equations for the Laplace operator with respect to an obstacle, Rendiconti Acc. Naz. delle Scienze, Memorie di Matematica, 108‡, XIV, 139-162 (1990) · Zbl 0729.35088
[8] Cohn, D. J., Measure Theory (1980), Boston: BirkhÄuser, Boston · Zbl 0436.28001
[9] Colombo, G.; Fonda, A.; Ornelas, A., Lower semicontinuous perturbations of maximal monotone differential inclusions, Isr. J. Math., 61, 211-218 (1988) · Zbl 0661.47038
[10] De Giorgi, E.; Degiovanni, M.; Marino, A.; Tosques, M., Evolution equations for a class of nonlinear operators, Atti Acc. Naz. Lincei. Cl. Sci. Fis. Mat. Nat. (8), 75, 1-8 (1983) · Zbl 0597.47045
[11] Degiovanni, M.; Marino, A.; Tosques, M., General properties of (p, q)-convex functions and (p, q)-monotone operators, Ricerche di Matematica, 32, 285-319 (1983) · Zbl 0555.49007
[12] M.Degiovanni, A.Marino - M.Tosques,Evolution equations associated with (p, q)-convex functions and (p, q)-monotoneoperators, Ricerche di Matematica (1984), pp- 81-112. · Zbl 0582.49005
[13] Degiovanni, M.; Marino, A.; Tosques, M., Evolution equations with lack of convexity, Nonlin. Anal. Th. Meth. Appl., 9, 1401-1443 (1985) · Zbl 0545.46029
[14] Degiovanni, M.; Tosques, M., Evolution equations for (Φ,f)-monotone operators, Boll. Un. Mat. Ital., 5-B, 537-568 (1986) · Zbl 0615.47047
[15] Frigon, M.; Marino, A.; Saccon, C., Some problems of parabolic type with discontinuous non linearities, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (9), 1, 41-52 (1990) · Zbl 0705.35074
[16] Fryszkowski, A., Continuous selections for a class of non-convex multivalued maps, Studia Math., 78, 163-174 (1983) · Zbl 0534.28003
[17] Himmelberg, C. J., Measurable relations, Fund. Math., 87, 52-72 (1975) · Zbl 0296.28003
[18] S.łojasiewicz jr.,Some theorem of Scorza-Dragoni type for multifunctions with applications to the problem of existence of solutions for differential multivalued equations, Institute of Mathematics, Polish Academy of Sciences, preprint # 5/82/155 (1982).
[19] M.Frigon - G.Saccon,Evolution equations with discontinuous nonlinearities and non convex constraints, to appear in Nonl. Anal. Th. Meth. Appl. · Zbl 0753.35046
[20] Marino, A.; Saccon, C.; Tosques, M., Curves of maximal slope and parabolic variational inequalities on non convex constraints, Ann. Sc. Norm. Sup. Pisa, Serie IV, XVI, 281-330 (1989) · Zbl 0699.49015
[21] E.Mitidieri - M.Tosques,Nonlinear integrodifferential equations in Hilbert spaces: the variational case, Proceedings of Volterra Integrodifferential Equations in Banach Spaces and Applications,Da Prato, M. Iannelli (Editors), Res. Notes in Math., Pitman,190 (1989). · Zbl 0674.45011
[22] M.Tosques,Quasi-autonomous evolution equations associated with (Φ,f)-monotoneoperators, Proceedings of Integral Functionals in Calculus of Variations, Suppl. Rend. Circ. Mat. Palermo, II,15 (1987). · Zbl 0659.47048
[23] Tosques, M., Quasi-autonomous parabolic evolution equations associated with a class of non linear operators, Ricerche di Matematica, XXXVIII, 63-92 (1989) · Zbl 0736.47026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.