# ON SECOND ORDER NONCONVEX SWEEPING PROCESS WITH NONCONVEX PERTURBATION

• Aitalioubrahim, Myelkebir (High school Ibn Khaldoune commune Bouznika)
• Published : 2011.07.31
• 75 7

#### Abstract

This paper deals with the existence result of solutions of a second order functional differential inclusion, governed by a class of nonconvex sweeping process, with a nonconvex perturbation.

#### Keywords

nonconvex sweeping process;functional differential inclusion;uniformly ${\rho}$-prox regular sets

#### References

1. R. P. Agarwal and D. Oregan, A survey of recent fixed point theory in Frechet spaces, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2, 75-88, Kluwer Acad. Publ., Dordrecht, 2003,
2. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin Heidelberg, 1984.
3. D. Azzam-Laouir, Mixed semicontinuous perturbation of a second order nonconvex sweeping process, Electron. J. Qual. Theory Differ. Equ. 2008 (2008), no. 37, 1-9.
4. H. Benabdellah, C. Castaing, A. Salvadori, and A. Syam, Nonconvex sweeping process, J. Appl. Anal. 2 (1996), no. 2, 217-240. https://doi.org/10.1515/JAA.1996.217
5. M. Bounkhel, General existence results for second order nonconvex sweeping process with unbounded perturbations, Port. Math. (N.S.) 60 (2003), no. 3, 269-304.
6. M. Bounkhel and D. Laouir-Azzam, Existence results on the second-order nonconvex sweeping process with perturbations, Set-Valued Var. Anal. 12 (2004), no. 3, 291-318. https://doi.org/10.1023/B:SVAN.0000031356.03559.91
7. M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2005), no. 2, 359-374.
8. M. Bounkhel and M. Yarou, Existence results for first and second order nonconvex sweeping process with delay, Port. Math. (N.S.) 61 (2004), no. 2, 207-230.
9. C. Castaing, Quelques problemes d'evolution du second ordre, Seminaire d'Analyse Convexe, Vol. 18 (Montpellier, 1988), Exp. No. 5, 18 pp., Univ. Sci. Tech. Languedoc, Montpellier, 1988,
10. C. Castaing and M. D. P. Monteiro Marques, Topological properties of solution sets for sweeping process with delay, Port. Math. 54 (1997), no. 4, 485-507.
11. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
12. C. Castaing, M. Valadier, and T. X. Ducha, Evolution equations governed by the sweeping process, Set-Valued Var. Anal. 1 (1993), no. 2, 109-139. https://doi.org/10.1007/BF01027688
13. F. H. Clarke, R. J. Stern, and P. R. Wolenski, Proximal smoothness and the lower $C^2$ property, J. Convex Anal. 2 (1995), no. 1-2, 117-144.
14. K. Deimling, Multivalued Differential Equations, De Gruyter Series in Non linear Analysis and Applications, Walter de Gruyter, Berlin, New York, 1992.
15. T. X. Duc Ha and M. D. P. Monteiro-Marques, Nonconvex second-order differential inclusions with memory, Set-Valued Var. Anal. 3 (1995), no. 1, 71-86. https://doi.org/10.1007/BF01033642
16. J. F. Edmond, Delay perturbed sweeping process, Set-Valued Var. Anal. 14 (2006), no. 3, 295-317. https://doi.org/10.1007/s11228-006-0021-9
17. T. Haddad and L. Thibault, Mixed semicontinuous perturbations of nonconvex sweeping process, Math. Program. 123 (2010), no. 1, Ser. B, 225-240. https://doi.org/10.1007/s10107-009-0315-4
18. M. D. P. Monteiro Marques, Differential inclusions in nonsmooth mechanical problems, Shocks and dry friction. Progress in Nonlinear Differential Equations and their Applications, 9. Birkhauser Verlag, Basel, 1993.
19. J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations 26 (1977), no. 3, 347-374. https://doi.org/10.1016/0022-0396(77)90085-7
20. J. J. Moreau, Application of convex analysis to the treatment of elasto-plastic systems, in "Applications of Methods of Functional Analysis to Problems in Mechanics", (Germain and Nayroles, Eds.), Lecture Notes in Mathematics, 503, Springer-Verlag, Berlin, (1976), 56-89.
21. J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in "Nonsmooth Mechanics", (J.J. Moreau and P.D. Panagiotopoulos, Eds.), CISM Courses and Lectures, 302, Springer-Verlag, Vienna, New York, (1988), 1-82.
22. R. A. Poliquin, R. T. Rockafellar, and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000), no. 11, 523-{5249.
23. Q. Zhu, On the solution set of differential inclusions in Banach space, J. Differential Equations 93 (1991), no. 2, 213-237. https://doi.org/10.1016/0022-0396(91)90011-W