Set-valued random differential equations in Banach space

Mariusz Michta

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1995)

  • Volume: 15, Issue: 2, page 191-200
  • ISSN: 1509-9407

Abstract

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We consider the problem of the existence of solutions of the random set-valued equation: (I) D H X 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.

How to cite

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Mariusz Michta. "Set-valued random differential equations in Banach space." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 15.2 (1995): 191-200. <http://eudml.org/doc/275974>.

@article{MariuszMichta1995,
abstract = {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.},
author = {Mariusz Michta},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {set-valued mappings; Hukuchara's derivative; Aumann's integral; measurability of multifunctions; measurable selectors; set-valued mapping; random set-valued differential equation; Hukuchara derivative; set-valued stochastic process; random differential inclusion},
language = {eng},
number = {2},
pages = {191-200},
title = {Set-valued random differential equations in Banach space},
url = {http://eudml.org/doc/275974},
volume = {15},
year = {1995},
}

TY - JOUR
AU - Mariusz Michta
TI - Set-valued random differential equations in Banach space
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1995
VL - 15
IS - 2
SP - 191
EP - 200
AB - 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.
LA - eng
KW - set-valued mappings; Hukuchara's derivative; Aumann's integral; measurability of multifunctions; measurable selectors; set-valued mapping; random set-valued differential equation; Hukuchara derivative; set-valued stochastic process; random differential inclusion
UR - http://eudml.org/doc/275974
ER -

References

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  1. [1] J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, M. Dakkera 1980. Zbl0441.47056
  2. [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. Zbl0282.34007
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  6. [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. Zbl0629.60075
  7. [7] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluver 1991. Zbl0731.49001
  8. [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. [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. Zbl0332.34062
  10. [10] K. Kuratowski, Cz. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci, Ser. Sci Math. Astronom. Phys. 13 (1965), 397-403. Zbl0152.21403
  11. [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. Zbl0211.12103
  12. [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. [13] A. Nowak, Random differential inclusions:measurable selection approach, Ann. Polon. Math. XLIX (1989), 291-296. Zbl0674.60062
  14. [14] A. Tołstonogow, Differencjalnyje wkluczenija w Banachowych prostranstwach, Nauka 1986. 

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