Topological Methods in Nonlinear Analysis

A second order differential inclusion with proximal normal cone in Banach spaces

Fatine Aliouane and Dalila Azzam-Laouir

Full-text: Open access

Abstract

In the present paper we mainly consider the second order evolution inclusion with proximal normal cone: \begin{equation} \begin{cases} -\ddot{x}(t)\in N_{K(t)}(\dot{x}(t))+F(t,x(t),\dot{x}(t)), \quad \text{a.e.}\\ \dot x(t)\in K(t),\\ x(0)=x_0,\quad\dot x(0)=u_0, \end{cases} \tag{$*$} \end{equation} where $t\in I=[0,T]$, $E$ is a separable reflexive Banach space, $K(t)$ a ball compact and $r$-prox-regular subset of $E$, $N_{K(t)}(\,\cdot\,)$ the proximal normal cone of $K(t)$ and $F$ an u.s.c. set-valued mapping with nonempty closed convex values. First, we prove the existence of solutions of $(*)$. After, we give an other existence result of $(*)$ when $K(t)$ is replaced by $K(x(t))$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 44, Number 1 (2014), 143-160.

Dates
First available in Project Euclid: 11 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1460381474

Mathematical Reviews number (MathSciNet)
MR3289012

Zentralblatt MATH identifier
1360.34132

Citation

Aliouane, Fatine; Azzam-Laouir, Dalila. A second order differential inclusion with proximal normal cone in Banach spaces. Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 143--160. https://projecteuclid.org/euclid.tmna/1460381474


Export citation

References

  • D. Azzam-Laouir and S. Izza, Existence of solutions for second-order perturbed nonconvex sweeping process, Comp. Math. Appl 62 (2001), 1736–1744.
  • F. Bernard and L. Thibault, Prox-regularity of functions and sets in Banach spaces, Set-Valued Anal. 12 (2004), 25–47.
  • F. Bernard, L. Thibault and N. Zlateva, Characterizations of prox-regular sets in uniformaly convex Banach spaces, J. Convex Anal. 13 (2006), 525–560.
  • ––––, Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties, Trans. Amer. Math. Soc. 363, no. 4 (2010), 2211–2247.
  • F. Bernicot and J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity, J. Convex Anal. 17 (2010), 451–484.
  • M. Bounkhel, Existence results for second order convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces, Electron, J. Qual. Theory Differ. Equ. 27 (2012), 1–10.
  • ––––, General existence results for second order nonconvex sweeping process with unbounded perturbations, Port. Math. (N.S.) 60 (2003), no. 3, 269–304.
  • M. Bounkhel and R. AL-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal. 18 (2010), no. 2, 151–182.
  • M. Bounkhel and D. Laouir-Azzam, Existence results for second order nonconvex sweeping processes, Set-Valued Anal. 12 (2004), no. 3, 291–318.
  • M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2001), 359–374.
  • A. Canino, On $p$-convex sets and geodesics, J. Differential Equations 75 (1988), 118–157.
  • C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math. vol. 580, Springer–Verlag, Berlin, 1977.
  • C. Castaing, T.X. Dúc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal. 1 (1993), 109–139.
  • F.H. Clarke, R.J. Stern and P.R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal. 2 (1995), 117–144.
  • F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer–Verlag, 1998.
  • G. Colombo and V.V. Goncharov, The sweeping processes without convexity, Set-Valued Anal. 7 (1999), 357–374.
  • G. Colombo and M.D.P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Differential Equations 187, no. 1 (2003), 46–62.
  • J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer–Verlag, New York, 1975.
  • J.F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program Ser. B 104 (2005), 347–373.
  • ––––, \romBV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations 226, no. 1 (2006), 135–179.
  • H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.
  • A.G. Ibrahim and F.A. Aladsani, Second order evolution inclusions governed by sweeping process in Banach spaces, Le Matematiche, vol. LXIV (2009), Fasc. II, pp. 17–39.
  • J.J. Moreau, Evolution problem associated with a moving convex set in Hilbert space, J. Differential Equations 26 (1977), no. 3, 347–374.
  • R.A. Poliquin, R.T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000), 5231–5249.
  • A.S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4 (1994), 130–141.
  • L. Thibault, Sweeping process with regular and nonregular sets, J. Differential Equations 193, no. 1 (2003), 1–26.