×

zbMATH — the first resource for mathematics

Existence of solutions for operator inclusions: A unified approach. (English) Zbl 0936.47037
Summary: For a class of operator inclusions with multivalued terms fulfilling mixed semicontinuity hypotheses, the existence of solutions is established by chiefly using Bressan and Colombo’s directionally continuous selection theorem as well as Ky Fan’s fixed point principle. An application to the Cauchy problem is then performed.

MSC:
47J05 Equations involving nonlinear operators (general)
46H05 General theory of topological algebras
34A60 Ordinary differential inclusions
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] J.-P. Aubin - A. CELLINA, Differential Inclusions , Springer-Verlag , Berlin ( 1984 ). MR 755330 | Zbl 0538.34007 · Zbl 0538.34007
[2] D. Averna , Lusin type theorems for multifunctions, Scorza Dragoni’s property and CarathĂ©odory selections , Boll. Un. Mat. Ital. A ( 7 ), 8 ( 1994 ), pp. 193 - 202 . MR 1288374 | Zbl 0817.28007 · Zbl 0817.28007
[3] D. Averna - G. Bonanno , Existence of solutions for a multivalued boundary value problem with non-convex and unbounded right-hand side , Ann. Polon. Mat. , 71 ( 1999 ), pp. 253 - 271 . MR 1704302 | Zbl 0935.34009 · Zbl 0935.34009
[4] D. Averna - S.A. Marano , Existence theorems for inclusions of the type \Psi (u)(t)\in F(t, \Phi (u)(t)) , Appl. Anal. , to appear. Zbl 1027.47063 · Zbl 1027.47063
[5] G. Bonanno - S.A. Marano , Higher order ordinary differential equations , Differential Integral Equations , 6 ( 1993 ), pp. 1119 - 1123 . MR 1230485 | Zbl 0785.34021 · Zbl 0785.34021
[6] G. Bonanno - S.A. Marano , Positive solutions of elliptic equations with discontinuous nonlinearities , Topol. Methods Nonlinear Anal. , 8 ( 1996 ), pp. 263 - 273 . MR 1483627 | Zbl 0891.35038 · Zbl 0891.35038
[7] A. Bressan , Upper and lower semicontinuous differential inclusions: A unified approach , in: H. J. Sussman (ed.), Nonlinear Controllability and Optimal Control , Pure Appl. Math. , 133 , Dekker , New York ( 1990 ), pp. 21 - 31 . MR 1061382 | Zbl 0704.49011 · Zbl 0704.49011
[8] 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
[9] M. Day , Normed Linear Spaces , Springer-Verlag , Berlin ( 1962 ). Zbl 0082.10603 · Zbl 0082.10603
[10] K. Deimling , Multivalued Differential Equations , de Gruyter Ser. Nonlinear Anal. Appl. 1 , de Gruyter , Berlin ( 1992 ). MR 1189795 | Zbl 0760.34002 · Zbl 0760.34002
[11] C.J. Himmelberg , Measurable relations , Fund. Math. , 87 ( 1975 ), pp. 53 - 72 . Article | MR 367142 | Zbl 0296.28003 · Zbl 0296.28003
[12] C.J. Himmelberg - F. S. VAN VLECK, Existence of solutions for generalized differential equations with unbounded right-hand side , J. Differential Equations , 61 ( 1986 ), pp. 295 - 320 . MR 829366 | Zbl 0582.34002 · Zbl 0582.34002
[13] S. Lojasiewicz (jr.), Some theorems of Scorza Dragoni type for multifunctions with application to the problem of existence of solutions for differential multivalued equations , in: C. OLECH - B. JAKUBCZYK - J. ZABCZYK (eds), Mathematical Control Theory , Banach Center Publ. , 14 , PWN , Warsaw ( 1985 ), pp. 625 - 643 . MR 851255 | Zbl 0576.49024 · Zbl 0576.49024
[14] A. Majorana - S.A. Marano , Boundary value problems for higher order ordinary differential equations , Comment. Math. Univ. Carolin. , 35 ( 1994 ), pp. 451 - 466 . Article | MR 1307273 | Zbl 0809.34034 · Zbl 0809.34034
[15] S.A. Marano , Existence theorems for a multivalued boundary value problem , Bull. Austral. Math. Soc. , 45 ( 1992 ), pp. 249 - 260 . MR 1155483 | Zbl 0741.34008 · Zbl 0741.34008
[16] O. Naselli RICCERI - B. RICCERI, An existence theorem for inclusions of the type W(u)(t) \in F(t, \Phi (u)(t)) and application to a multivalued boundary value problem , Appl. Anal. , 38 ( 1990 ), pp. 259 - 270 . Zbl 0687.47044 · Zbl 0687.47044
[17] C. Olech , Existence of solutions of non-convex orientor fields , Boll. Un. Mat. Ital. ( 4 ), 11 ( 1975 ), pp. 189 - 197 . MR 394372 | Zbl 0322.34002 · Zbl 0322.34002
[18] E. Schechter , A survey of local existence theories for abstract nonlinear initial value problems , in: T. L. GILL - W. W. ZACHARY (eds), Nonlinear Semigroups, Partial Differential Equations and Attractors , Lecture Notes in Math. 1394 , Springer-Verlag , Berlin ( 1987 ), pp. 136 - 184 . MR 1021021 | Zbl 0678.34005 · Zbl 0678.34005
[19] A.A. Tolstonogov , Solutions of a differential inclusion with unbounded right-hand side , Siberian Math. J. , 29 ( 1988 ), pp. 857 - 868 . MR 971244 | Zbl 0702.34019 · Zbl 0702.34019
[20] R.L. Wheeden - A. Zygmund , Measure and Integral , Monographs Textbooks Pure Appl. Math. , 43 , Dekker , New York ( 1977 ). MR 492146 | Zbl 0362.26004 · Zbl 0362.26004
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.