# Second-order viability result in Banach spaces

Myelkebir Aitalioubrahim; Said Sajid

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2010)

- Volume: 30, Issue: 1, page 5-21
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topMyelkebir Aitalioubrahim, and Said Sajid. "Second-order viability result in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.1 (2010): 5-21. <http://eudml.org/doc/271139>.

@article{MyelkebirAitalioubrahim2010,

abstract = {
We show the existence result of viable solutions to the second-order differential inclusion
ẍ(t) ∈ F(t,x(t),ẋ(t)),
x(0) = x₀, ẋ(0) = y₀, x(t) ∈ K on [0,T],
where K is a closed subset of a separable Banach space E and F(·,·,·) is a closed multifunction, integrably bounded, measurable with respect to the first argument and Lipschitz continuous with respect to the third argument.
},

author = {Myelkebir Aitalioubrahim, Said Sajid},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {differential inclusion,viability; measurability; selection},

language = {eng},

number = {1},

pages = {5-21},

title = {Second-order viability result in Banach spaces},

url = {http://eudml.org/doc/271139},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Myelkebir Aitalioubrahim

AU - Said Sajid

TI - Second-order viability result in Banach spaces

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2010

VL - 30

IS - 1

SP - 5

EP - 21

AB -
We show the existence result of viable solutions to the second-order differential inclusion
ẍ(t) ∈ F(t,x(t),ẋ(t)),
x(0) = x₀, ẋ(0) = y₀, x(t) ∈ K on [0,T],
where K is a closed subset of a separable Banach space E and F(·,·,·) is a closed multifunction, integrably bounded, measurable with respect to the first argument and Lipschitz continuous with respect to the third argument.

LA - eng

KW - differential inclusion,viability; measurability; selection

UR - http://eudml.org/doc/271139

ER -

## References

top- [1] B. Aghezzaf and S. Sajid, On the second-order contingent set and differential inclusions, J. Conv. Anal. 7 (1) (2000), 183-195. Zbl0958.34010
- [2] K.S. Alkulaibi and A.G. Ibrahim, On existence of monotone solutions for second-order non-convex differential inclusions in infinite dimensional spaces, Portugaliae Mathematica 61 (2) (2004), 231-143. Zbl1062.34062
- [3] S. Amine, R. Morchadi and S. Sajid, Carathéodory perturbation of a second-order differential inclusions with constraints, Electronic J. Diff. Eq. (2005), 1-11. Zbl1096.34008
- [4] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977. doi:10.1007/BFb0087685
- [5] B. Cornet and G. Haddad, Théorème de viabilité pour les inclusions différentielles du seconde ordre, Isr. J. Math. 57 (2) (1987), 225-238. Zbl0659.34012
- [6] A. Dubovitskij and A.A. Miljutin, Extremums problems with constraints, Soviet Math. 4 (1963), 452-455. Zbl0133.05501
- [7] T.X. Duc Ha, Existence of viable solutions for nonconvex-valued differential inclusions in Banach spaces, Portugaliae Mathematica 52 Fasc. 2, 1995. Zbl0824.34021
- [8] M. Larrieu, Invariance d'un fermé pour un champ de vecteurs de Carathéodory, Pub. Math. de Pau, 1981.
- [9] V. Lupulescu, A viability result for second order differential inclusions, Electron J. Diff. Eq. 76 (2003), 1-12. Zbl1023.34010
- [10] V. Lupulescu, Existence of solutions for nonconvex second order differential inclusions, Applied Math. E-notes 3 (2003), 115-123. Zbl1034.34018
- [11] R. Morchadi and S. Sajid, Noncovex second-order differential inclusion, Bulletin of the Polish Academy of Sciences 47 (3) (1999). Zbl0943.34054
- [12] Q. Zhu, On the solution set of differential inclusions in Banach spaces, J. Differ. Eq. 41 (2001), 1-8.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.