JournalspmVol. 77 , No. 2pp. 197–218

State dependent nonconvex sweeping processes in smooth Banach spaces

  • Djalel Bounekhel

    Université Frères Mentouri, Constantine, Algeria
  • Messaoud Bounkhel

    King Saud University, Riyadh, Saudi Arabia
  • Mostafa Bachar

    King Saud University, Riyadh, Saudi Arabia
State dependent nonconvex sweeping processes in smooth Banach spaces cover
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Abstract

In the setting of 2-uniformly smooth and qq-uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation:

ddtJ(u(t))NC(C(t,u(t));u(t))\mboxa.e.in[0,T].(SDNSP)-\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \mbox{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)}

This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here NC(C(t,u(t));u(t))N^C(C(t,u(t)); u(t)) stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on VV-uniformly generalized prox-regular sets in Banach spaces.

Cite this article

Djalel Bounekhel, Messaoud Bounkhel, Mostafa Bachar, State dependent nonconvex sweeping processes in smooth Banach spaces. Port. Math. 77 (2020), no. 2 pp. 197–218

DOI 10.4171/PM/2049