On the density of extremal solutions of differential inclusions

F. S. De Blasi; G. Pianigiani

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 2, page 133-142
  • ISSN: 0066-2216

Abstract

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An existence theorem for the cauchy problem (*) ẋ ∈ ext F(t,x), x(t₀) = x₀, in banach spaces is proved, under assumptions which exclude compactness. Moreover, a type of density of the solution set of (*) in the solution set of ẋ ∈ f(t,x), x(t₀) = x₀, is established. The results are obtained by using an improved version of the baire category method developed in [8]-[10].

How to cite

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F. S. De Blasi, and G. Pianigiani. "On the density of extremal solutions of differential inclusions." Annales Polonici Mathematici 56.2 (1992): 133-142. <http://eudml.org/doc/262527>.

@article{F1992,
abstract = {An existence theorem for the cauchy problem (*) ẋ ∈ ext F(t,x), x(t₀) = x₀, in banach spaces is proved, under assumptions which exclude compactness. Moreover, a type of density of the solution set of (*) in the solution set of ẋ ∈ f(t,x), x(t₀) = x₀, is established. The results are obtained by using an improved version of the baire category method developed in [8]-[10].},
author = {F. S. De Blasi, G. Pianigiani},
journal = {Annales Polonici Mathematici},
keywords = {differential inclusions; extremal solutions; Choquet function; Banach space; Baire-category theorem},
language = {eng},
number = {2},
pages = {133-142},
title = {On the density of extremal solutions of differential inclusions},
url = {http://eudml.org/doc/262527},
volume = {56},
year = {1992},
}

TY - JOUR
AU - F. S. De Blasi
AU - G. Pianigiani
TI - On the density of extremal solutions of differential inclusions
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 2
SP - 133
EP - 142
AB - An existence theorem for the cauchy problem (*) ẋ ∈ ext F(t,x), x(t₀) = x₀, in banach spaces is proved, under assumptions which exclude compactness. Moreover, a type of density of the solution set of (*) in the solution set of ẋ ∈ f(t,x), x(t₀) = x₀, is established. The results are obtained by using an improved version of the baire category method developed in [8]-[10].
LA - eng
KW - differential inclusions; extremal solutions; Choquet function; Banach space; Baire-category theorem
UR - http://eudml.org/doc/262527
ER -

References

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  1. [1] A. Antosiewicz and A. Cellina, Continuous selections and differential relations, J. Differential Equations 19 (1975), 386-398. 
  2. [2] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984. Zbl0538.34007
  3. [3] S. Bahi, Quelques propriétés topologiques de l'ensemble des solutions d'une classe d'équations différentielles multivoques (II), Séminaire d'Analyse Convexe, Montpellier, 1983, exposé No. 4. 
  4. [4] A. Bressan and G. Colombo, Generalized Baire category and differential inclusions in Banach spaces, J. Differential Equations 76 (1988), 135-158. Zbl0655.34013
  5. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin 1977. 
  6. [6] G. Choquet, Lectures on Analysis, Benjamin, Reading 1969. 
  7. [7] P. V. Chuong, Un résultat d'existence de solutions pour des équations différentielles multivoques, C. R. Acad. Sci. Paris 301 (1985), 399-402. Zbl0581.34012
  8. [8] F. S. De Blasi and G. Pianigiani, A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac. (2) 25 (1982), 153-162. Zbl0535.34009
  9. [9] F. S. De Blasi and G. Pianigiani, The Baire category method in existence problems for a class of multivalued differential equations with nonconvex right hand side, Funkcial. Ekvac. 28 (1985), 139-156. Zbl0584.34007
  10. [10] F. S. De Blasi and G. Pianigiani, Differential inclusions in Banach spaces, J. Differential Equations 66 (1987), 208-229. Zbl0609.34013
  11. [11] A. F. Filippov, The existence of solutions of generalized differential equations, Math. Notes 10 (1971), 608-611. Zbl0265.34074
  12. [12] A. N. Godunov, Peano's theorem in Banach spaces, Funktsional. Anal. i Prilozhen. 9 (1) (1974), 59-60 (in Russian). 
  13. [13] H. Kaczyński and C. Olech, Existence of solutions of orientor fields with nonconvex right hand side, Ann. Polon. Math. 29 (1974), 61-66. Zbl0285.34008
  14. [14] N. S. Papageorgiou, On the solution set of differential inclusions in Banach spaces, Appl. Anal. 25 (1987), 319-329. Zbl0623.34062
  15. [15] N. S. Papageorgiou, On measurable multifunctions with applications to random multivalued equations, Math. Japon. 32 (1987), 437-464. Zbl0634.28005
  16. [16] G. Pianigiani, On the fundamental theory of multivalued differential equations, J. Differential Equations 25 (1977), 30-38. Zbl0398.34017
  17. [17] A. A. Tolstonogov, On differential inclusions in Banach spaces, Soviet Math. Dokl. 20 (1979), 186-190. Zbl0439.34052

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