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General existence results for third-order nonconvex state-dependent sweeping process with unbounded perturbations. (English) Zbl 1248.34095

Summary: We prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form:
\[ {-A(x^{(3)}(t)) \in N(K(t, \dot{x}(t)); A(\ddot{x}(t))) + F(t, x(t), \dot{x}(t), \ddot{x}(t)) + G(x(t), \dot{x}(t), \ddot{x}(t)) \;\mathrm{a.e.} \;[0, T],} \]
\[ A(\ddot{x}(t)) \in K(t, \dot{x}(t)), \;\mathrm{a.e.} \;t \in [0, T], x(0) = x_0, \dot{x}(0) = u_0, \ddot{x}(0) = v_0, \]
\[ \displaystyle \]
where \(T>0, K\) is a nonconvex Lipschitz set-valued mapping, \(F\) is an unbounded scalarly upper semicontinuous convex set-valued mapping, and \(G\) is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space \(\mathbb H\) .

MSC:

34G25 Evolution inclusions
47N20 Applications of operator theory to differential and integral equations
49J40 Variational inequalities
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[1] J.-J. Moreau, “Evolution problem associated with a moving convex set in a Hilbert space,” Journal of Differential Equations, vol. 26, no. 3, pp. 347-374, 1977. · Zbl 0356.34067 · doi:10.1016/0022-0396(77)90085-7
[2] M. Bounkhel and T. Haddad, “Existence of viable solutions for nonconvex differential inclusions,” Electronic Journal of Differential Equations, no. 50, article 10, 2005. · Zbl 1075.34053
[3] M. Bounkhel, “Existence results for first and second order nonconvex sweeping processes with perturbations and with delay: fixed point approach,” Georgian Mathematical Journal, vol. 13, no. 2, pp. 239-249, 2006. · Zbl 1115.49015
[4] M. Bounkhel and M. Yarou, “Existence results for nonconvex sweeping processes with perturbations and with delay: Lipschitz case,” Arab Journal of Mathematical Sciences, vol. 8, no. 2, pp. 15-26, 2002. · Zbl 1035.49015
[5] C. Castaing, T. X. Dúc H\Ba, and M. Valadier, “Evolution equations governed by the sweeping process,” Set-Valued Analysis, vol. 1, no. 2, pp. 109-139, 1993. · Zbl 0813.34018 · doi:10.1007/BF01027688
[6] C. Castaing, “Quelques problèmes d’évolution du second ordre,” in Seminaire d’Analyse Convexe, Univ. Sci. Tech. Languedoc, Montpellier, France, 1988. · Zbl 0676.47029
[7] A. H. Siddiqi, P. Manchanda, and M. Brokate, “On some recent developments concerning Moreau’s sweeping process,” in Trends in Industrial and Applied Mathematics, vol. 72 of Appl. Optim., pp. 339-354, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
[8] M. Bounkhel and T. Haddad, “An existence result for a new variant of the nonconvex sweeping process,” Portugaliae Mathematica, vol. 65, no. 1, pp. 33-47, 2008. · Zbl 1135.49014 · doi:10.4171/PM/1797
[9] M. Bounkhel, “General existence results for second order nonconvex sweeping process with unbounded perturbations,” Portugaliae Mathematica. Nova Série, vol. 60, no. 3, pp. 269-304, 2003. · Zbl 1055.34116
[10] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998. · Zbl 1047.49500
[11] R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Transactions of the American Mathematical Society, vol. 352, no. 11, pp. 5231-5249, 2000. · Zbl 0960.49018 · doi:10.1090/S0002-9947-00-02550-2
[12] M. Bounkhel and L. Thibault, “Nonconvex sweeping process and prox-regularity in Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 2, pp. 359-374, 2005. · Zbl 1086.49016
[13] M. Bounkhel and L. Thibault, “On various notions of regularity of sets in nonsmooth analysis,” Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 2, pp. 223-246, 2002. · Zbl 1012.49013 · doi:10.1016/S0362-546X(00)00183-8
[14] M. A. Gamal, “Perturbation non convexe d’un probleme d’evolution dans un espace Hilbertien,” Séminaire d’Analyse Convexe Montpellier, no. 16, 1981.
[15] T. X. Duc Ha and M. D. P. Monteiro Marques, “Nonconvex second-order differential inclusions with memory,” Set-Valued Analysis, vol. 3, no. 1, pp. 71-86, 1995. · Zbl 0824.34020 · doi:10.1007/BF01033642
[16] T. Xuan Duc Ha, “Existence of viable solutions of nonconvex differential inclusions,” Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 47, no. 2, pp. 457-471, 1999. · Zbl 0941.34006
[17] L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable, thesis, Université Montpellier, 1976. · Zbl 0343.46030
[18] B. Hopkins, “Existence of solutions for nonconvex third order differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 22, article 11, 2005. · Zbl 1111.34012
[19] M. Bounkhel and B. Al-Senan, “General existence results for nonconvex third order differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 21, article 10, 2010. · Zbl 1202.34108
[20] M. Bounkhel and A. Jofré, “Subdifferential stability of the distance function and its applications to nonconvex economies and equilibrium,” Journal of Nonlinear and Convex Analysis, vol. 5, no. 3, pp. 331-347, 2004. · Zbl 1084.49018
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