Set-valued random differential equations in Banach space

• Autorzy: Mariusz Michta
• Języki publikacji: EN

Abstrakty

EN

We consider the problem of the existence of solutions of the random set-valued equation:
(I) $D_HX_t = F(t,X_t)P.1$, t ∈ [0,T] -a.e.; X₀ = U p.1
where F and U are given random set-valued mappings with values in the space $K_c(E)$, of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

Słowa kluczowe

EN

set-valued mappings; Hukuchara's derivative; Aumann's integral; measurability of multifunctions; measurable selectors

Kategorie tematyczne

• 26E25: Set-valued functions
• 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 1995
• Tom: 15
• Numer: 2
• Strony: 191-200

Daty

wydano

1995

Twórcy

autor

Mariusz Michta

• Institute of Mathematics, Technical University, Podgórna 50, 65-246 Zielona Góra, Poland

Bibliografia

• [1] J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, M. Dakkera 1980.
• [2] F.S. De Blasi, F. Iervolino, Euler method for differential equation with compact, convex valued solutions, Boll. U. M. I. 4 (4) (1971), 941-949.
• [3] F.S. De Blasi, F. Iervolino, Equazioni differenziali con soluzini a valore compatto convesso, Boll. U. M. I. 2 (4) (1969), 501-591.
• [4] F. Hiai, H. Umegaki, Integrals, conditional expectation and martingales of multivalued functions, J. Multiv. Anal. 7 (1977), 149-182.
• [5] C.J. Himmelberg, F.S. Van Vleck, The Hausdorff metric and measurable selections, Topology and its Appl. North-Holland 20 (1985), 121-133.
• [6] D.A. Kandilakis, N.S. Papageorgiou, On the existence of solutions of random differential inclusions in Banach space, J. Math. Anal. Appl. 126 (1987),11-23.
• [7] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluver 1991.
• [8] M. Kisielewicz, Method of averaging for differential equation with compact convex valued solutions, Rend. di. Matem. serie VI, 9 (3) (1976), 1-12.
• [9] M. Kisielewicz, B. Serafin, W. Sosulski, Existence theorem for functional-differential equation with compact convex valued solutions, Demmonstratio Math. IX (2) (1976), 229-237.
• [10] K. Kuratowski, Cz. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci, Ser. Sci Math. Astronom. Phys. 13 (1965), 397-403.
• [11] P. Lopes Pinto, F.S. De Blasi, F. Iervolino, Uniqueness and existence theorem for differential equations with compact convex valued solutions, Boll. U. M. I. 4 (1970), 45-54.
• [12] M. Michta, Istnienie i jednoznaczność rozwiązań losowych równań różniczkowych o wielowartościowych, zwartych i wypukłych prawych stronach, Praca dokt. UAM Poznań, WSI Zielona Góra (1993).
• [13] A. Nowak, Random differential inclusions:measurable selection approach, Ann. Polon. Math. XLIX (1989), 291-296.
• [14] A. Tołstonogow, Differencjalnyje wkluczenija w Banachowych prostranstwach, Nauka 1986.