On functional differential inclusions in Hilbert spaces

Myelkebir Aitalioubrahim

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2012)

  • Volume: 32, Issue: 1, page 63-85
  • ISSN: 1509-9407

Abstract

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We prove the existence of monotone solutions, of the functional differential inclusion ẋ(t) ∈ f(t,T(t)x) +F(T(t)x) in a Hilbert space, where f is a Carathéodory single-valued mapping and F is an upper semicontinuous set-valued mapping with compact values contained in the Clarke subdifferential ∂ c V ( x ) of a uniformly regular function V.

How to cite

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Myelkebir Aitalioubrahim. "On functional differential inclusions in Hilbert spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 32.1 (2012): 63-85. <http://eudml.org/doc/270498>.

@article{MyelkebirAitalioubrahim2012,
abstract = {We prove the existence of monotone solutions, of the functional differential inclusion ẋ(t) ∈ f(t,T(t)x) +F(T(t)x) in a Hilbert space, where f is a Carathéodory single-valued mapping and F is an upper semicontinuous set-valued mapping with compact values contained in the Clarke subdifferential $∂_\{c\} V(x)$ of a uniformly regular function V.},
author = {Myelkebir Aitalioubrahim},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {functional differential inclusion; regularity; Clarke subdifferential},
language = {eng},
number = {1},
pages = {63-85},
title = {On functional differential inclusions in Hilbert spaces},
url = {http://eudml.org/doc/270498},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Myelkebir Aitalioubrahim
TI - On functional differential inclusions in Hilbert spaces
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2012
VL - 32
IS - 1
SP - 63
EP - 85
AB - We prove the existence of monotone solutions, of the functional differential inclusion ẋ(t) ∈ f(t,T(t)x) +F(T(t)x) in a Hilbert space, where f is a Carathéodory single-valued mapping and F is an upper semicontinuous set-valued mapping with compact values contained in the Clarke subdifferential $∂_{c} V(x)$ of a uniformly regular function V.
LA - eng
KW - functional differential inclusion; regularity; Clarke subdifferential
UR - http://eudml.org/doc/270498
ER -

References

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  2. [2] J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4 
  3. [3] M. Bounkhel, Existence results of nonconvex differential inclusions, Portugal. Math. 59 (3) (2002) 283-310. Zbl1022.34007
  4. [4] A. Bressan, A. Cellina and G. Colombo, Upper semicontinuous differential inclusions without convexity, Proc. Amer. Math. Soc. 106 (1989) 771-775. doi: 10.1090/S0002-9939-1989-0969314-6 Zbl0698.34014
  5. [5] A. Cernea and V. Lupulescu, Viable solutions for a class of nonconvex functional differential inclusions, Math. Reports 7(57) (2) (2005) 91-103. Zbl1100.34049
  6. [6] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley and Sons, 1983. Zbl0582.49001
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  8. [8] K. Deimling, Multivalued Defferential Equations. De Gruyter Series in Non linear Analysis and Applications, Walter de Gruyter, Berlin, New York, 1992. 
  9. [9] A. Gavioli and L. Malaguti, Viable solutions of differential inclusions with memory in Banach spaces, Portugal. Math. 57 Fasc. 2 (2000). Zbl0963.34059
  10. [10] G. Haddad, Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. Math. 39 (1981) 83-100. doi: 10.1007/BF02762855 Zbl0462.34048
  11. [11]G. Haddad, Monotone trajectories for functional differential inclusions, J. Differential Equations 42 (1981) 1-24. doi: 10.1016/0022-0396(81)90031-0 
  12. [12] R.T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 39 (1980) 257-280. doi: 10.4153/CJM-1980-020-7 Zbl0447.49009
  13. [13] A. Syam, Contributions aux Inclusions Différentielles, Ph. thesis, Université Montpellier II, 1993. 

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