×

zbMATH — the first resource for mathematics

Semicontinuous differential inclusions. (English) Zbl 0936.34010
The author deals with the Cauchy problem for the differential inclusion \[ x'\in F(t,x)+ G(t,x)\quad \text{a.e. on}\quad x(0)= x_0\tag{\(*\)} \] in a Banach space \(E\) with uniformly convex dual, where \(F\) is an almost upper-semicontinuous multifunction with compact convex values and \(G\) is an almost lower-semicontinuous multifunction with compact values. A relaxed problem, i.e. the problem \((*)\) with the function \(G\) replaced by a convexified upper-semicontinuous regularization of \(G\), is considered, too. Supposing a suitable growth and suitable Lipschitz type conditions, the author proves that: (i) the problem \((*)\) admits at least one solution; (ii) the solution set to problem \((*)\) is dense in the solution set of the relaxed problem; (iii) the solution set of the relaxed problem is an \(R_\delta\) set; and finally, under some additional condition, that: (iv) the solution set to \((*)\) is connected.

MSC:
34A60 Ordinary differential inclusions
34G25 Evolution inclusions
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] M. Arrate , Local existence of solutions for ordinary differential equations involving dissipative and compact functions , J. Math. Anal. Appl. , 145 ( 1990 ), pp. 39 - 44 . MR 1031173 | Zbl 0684.34061 · Zbl 0684.34061
[2] D. Bothe , Multivalued perturbations of m-accretive differential inclusions , submitted. Zbl 0922.47048 · Zbl 0922.47048
[3] D. Bothe , Multivalued Differential Equations on Graphs and Applications , Ph. D. thesis, Paderborn ( 1992 ). Zbl 0789.34013 · Zbl 0789.34013
[4] A. Bressan , Upper and Lower Semicontinuous Differential Inclusions. A Unified Approach , in H. SUSMAN (edt.) Controllability and Optimal Control , Marcel Dekker ( 1989 ). MR 1061382 | Zbl 0704.49011 · Zbl 0704.49011
[5] A. Bressan - G. Colombo , Selections and representations of multifunctions in paracompact spaces , Studia Math. , 102 ( 1992 ), pp. 209 - 216 . Article | MR 1170551 | Zbl 0807.54020 · Zbl 0807.54020
[6] A. Bressan - V. Staicu , On nonconvex perturbation of maximal monotone differential inclusions , Set-Valued Analysis , 2 ( 1994 ), pp. 415 - 437 . MR 1304047 | Zbl 0820.47072 · Zbl 0820.47072
[7] K. Deimling , Multivalued Differential Equations , De Gruyter , Berlin ( 1992 ). MR 1189795 | Zbl 0760.34002 · Zbl 0760.34002
[8] T. Donchev - R. Ivanov , On the existence of solutions of differential inclusions in uniformly convex banach spaces , Matematica Balkanica , 6 ( 1992 ), pp. 13 - 24 . MR 1170727 | Zbl 0831.34013 · Zbl 0831.34013
[9] A. Dontchev - F. Lempio , Difference method for differential inclusions: a survey , SIAM Review , 34 ( 1992 ), pp. 263 - 294 . MR 1166177 | Zbl 0757.34018 · Zbl 0757.34018
[10] L. Gornievicz - A. Granas - W. Kryszewski , Sur la Metode de L’homotopie dans la Theorie des Point Fixes pour les Applications Multivoques, Parte 2: L’indice dans les ANRs Compactness , C.R. Acad. Sci. Paris , 308 ( 1989 ), pp. 440 - 452 . Zbl 0678.54033 · Zbl 0678.54033
[11] A. Plis , Trajectories and quasitrajectories of an orientor field , Bull. Acad. Polon. Sci., Ser. Math. , 11 ( 1963 ), pp. 369 - 370 . MR 155072 | Zbl 0124.29404 · Zbl 0124.29404
[12] V. Veliov , Diferential inclusions with stable subinclusions , Nonlinear Analysis , 23 ( 1994 ), pp. 1027 - 1038 . MR 1304242 | Zbl 0816.34011 · Zbl 0816.34011
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.