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On the structure of the solution set of evolution inclusions with time-dependent subdifferentials. (English) Zbl 0893.34060

The Cauchy problem \[ \begin{gathered} -x'(t)\in \partial \varphi\bigl(t, x(t)\bigr) +F \bigl(t,x(t) \bigr),\;t\in T=[0,b], \\ x(0)=\xi, \tag{P} \end{gathered} \] is considered in a real separable Hilbert space \(H\). Here \(\xi\in H\), \(\varphi: T\times H \to(-\infty, \infty]\) is proper, convex and lower semicontinuous on \(H\) for each fixed \(t\), and satisfies a time-dependence condition of S. Yotsutani type [J. Math. Soc. Japan 31, 623-646 (1978; Zbl 0414.35041)], while the multifunction \(F:T\times H\to 2^H\) is assumed to be closed valued. The authors first discuss the nonemptiness and the topological structure of the solution set of (P) when \(\varphi(t,.)\) is of compact type for every \(t\in T\), and \(F\) has convex values. The study is then extended to the case when \(F\) is no longer convex-valued and the functions \(\varphi(t,.)\) are not necessarily of compact type. Some examples involving parabolic distributed parameter systems are also presented.

MSC:

34G20 Nonlinear differential equations in abstract spaces
35K55 Nonlinear parabolic equations
34A60 Ordinary differential inclusions

Citations:

Zbl 0414.35041
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References:

[1] M.E. Ballotti , Aronszajn’s theorem for a parabolic partial differential equation , Nonl. Anal. T.M.A. , 9 ( 1985 ), pp. 1183 - 1188 . MR 813652 | Zbl 0583.35053 · Zbl 0583.35053 · doi:10.1016/0362-546X(85)90029-X
[2] V. Barbu , Nonlinear Semigroups and Differential Equations in Banach Spaces , Noordhoff International Publishing , Leyden , The Netherlands ( 1976 ). MR 390843 | Zbl 0328.47035 · Zbl 0328.47035
[3] A. Bressan - G. Colombo , Extension and selections of maps with decomposable values , Studia Math. , 90 ( 1988 ), pp. 69 - 86 . Article | MR 947921 | Zbl 0677.54013 · Zbl 0677.54013
[4] H. Brezis , Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert , North-Holland , Amsterdam ( 1973 ). MR 348562 | Zbl 0252.47055 · Zbl 0252.47055
[5] A. Cellina , On the set of solution to Lipschitzian differential inclusions , Diff. Integ. Equations , 1 ( 1988 ), pp. 495 - 500 . MR 945823 | Zbl 0723.34009 · Zbl 0723.34009
[6] F. De Blasi , Characterizations of certain classes of semicontinuous multifunctions by continuous approximations , J. Math. Anal. Appl. , 106 ( 1985 ), pp. 1 - 18 . MR 780314 | Zbl 0574.54012 · Zbl 0574.54012 · doi:10.1016/0022-247X(85)90126-X
[7] F. De Blasi - J. Myjak , On the solution sets for differential inclusions , Bull. Pol. Acad. Sci. , 33 ( 1985 ), pp. 17 - 23 . MR 798723 | Zbl 0571.34008 · Zbl 0571.34008
[8] F. De Blasi - G. Pianigiani , Non-convex valued differential inclusions in Banach spaces , J. Math. Anal. Appl. , 157 ( 1991 ), pp. 469 - 494 . MR 1112329 | Zbl 0728.34013 · Zbl 0728.34013 · doi:10.1016/0022-247X(91)90101-5
[9] F. De Blasi - G. Pianigiani , Topological properties of nonconvex differential inclusions , Nonl. Anal. T.M.A. , 20 ( 1993 ), pp. 871 - 894 . MR 1214750 | Zbl 0774.34010 · Zbl 0774.34010 · doi:10.1016/0362-546X(93)90075-4
[10] F. De Blasi - G. Pianigiani - V. Staicu , Topological properties of nonconvex differential inclusions of evolution type , Nonl. Anal. T.M.A. , 23 ( 1995 ), pp. 711 - 720 . MR 1319080 | Zbl 0828.34010 · Zbl 0828.34010 · doi:10.1016/0362-546X(94)E0043-G
[11] C.J. Himmelberg , Measurable relations , Fundamenta Math. , 87 ( 1975 ), pp. 53 - 72 . Article | MR 367142 | Zbl 0296.28003 · Zbl 0296.28003
[12] C.J. Himmelberg - F. S. VAN VLECK, A note on solution sets of differential inclusions , Rocky Mountain J. Math. , 12 ( 1982 ), pp. 621 - 625 . MR 683856 | Zbl 0531.34007 · Zbl 0531.34007 · doi:10.1216/RMJ-1982-12-4-621
[13] S. Hu - V. Lakshmikantham - N.S. Papageorgiou , On the properties of the solution set of semilinear evolution inclusions , to appear. MR 1330643 | Zbl 0831.34014 · Zbl 0831.34014 · doi:10.1016/0362-546X(94)00213-2
[14] S. Hu - N. S. PAPAGEORGIOU, On the topological regularity of the solution set of differential inclusions with constraints , J. Diff. Equations , 107 ( 1994 ), pp. 280 - 289 . MR 1264523 | Zbl 0796.34017 · Zbl 0796.34017 · doi:10.1006/jdeq.1994.1013
[15] D.M. Hyman , On decreasing sequences of compact absolute retracts , Fundamenta Math. , 64 ( 1969 ), pp. 91 - 97 . Article | MR 253303 | Zbl 0174.25804 · Zbl 0174.25804
[16] N. Kemnochi , Some nonlinear parabolic variational inequalities , Israel J. Math. , 22 ( 1975 ), pp. 305 - 331 . Zbl 0327.49004 · Zbl 0327.49004 · doi:10.1007/BF02761596
[17] N. Kikuchi , Kneser’s property for du/dt = \Delta u + \surd u , Keio Univ. Math. Sem. Rep. , 3 ( 1978 ), pp. 45 - 48 . Zbl 0391.35011 · Zbl 0391.35011
[18] N.S. Papageorgiou , Convergence theorems for Banach space valued integrable multifunctions, Inter. Math . Math. Sci. , 10 ( 1987 ), pp. 433 - 442 . MR 896595 | Zbl 0619.28009 · Zbl 0619.28009 · doi:10.1155/S0161171287000516
[19] N.S. Papageorgiou , On measurable multifunctions with applications to random multivalued equations , Math. Japonica , 32 ( 1987 ), pp. 437 - 464 . MR 914749 | Zbl 0634.28005 · Zbl 0634.28005
[20] N.S. Papageorgiou , On the solution set of evolution inclusions driven by time-dependent subdifferential , Math. Japonica , 37 ( 1992 ), pp. 1087 - 1099 . MR 1196384 | Zbl 0810.34059 · Zbl 0810.34059
[21] N.S. Papageorgiou , On the topological property of the solution set of evolution inclusions involving time-dependent subdifferential operators , Boll. Un. Mat. Ital. , 8-B ( 1994 ). Zbl 0845.34066 · Zbl 0845.34066
[22] J. Rauch , Discontinuous semilinear differential equations and multiple valued maps , Proc. Amer. Math. Soc. , 64 ( 1977 ). MR 442453 | Zbl 0413.35031 · Zbl 0413.35031 · doi:10.2307/2041442
[23] V. Staicu , Sissa report , 42M ( 1990 ).
[24] D.H. Wagner , Survey of measurable selection theorems , SIAM J. Control Optim. , 15 ( 1977 ), pp. 859 - 903 . MR 486391 | Zbl 0407.28006 · Zbl 0407.28006 · doi:10.1137/0315056
[25] Y. Yamada , On evolution equations generated by subdifferential operators , J. Fac. Sci. Univ. Tokyo , 23 ( 1976 ), pp. 491 - 515 . MR 425701 | Zbl 0343.34053 · Zbl 0343.34053
[26] S. Yotsutani , Evolution equations associated with subdifferentials , J. Math. Soc. Japan , 31 ( 1978 ), pp. 623 - 646 . Article | MR 544681 | Zbl 0405.35043 · Zbl 0405.35043 · doi:10.2969/jmsj/03140623
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