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Symmetry 2019, 11(1), 28; https://doi.org/10.3390/sym11010028

Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces

1
Faculté des Sciences Exactes, Départment des Mathématiques, Université Frères Mentouri, Constantine 25000, Algérie
2
College of Science, Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Received: 23 November 2018 / Revised: 21 December 2018 / Accepted: 25 December 2018 / Published: 30 December 2018
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
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Abstract

We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, we prove in the nonconvex case an existence result of this type of differential inclusions when the separable Banach space is assumed to be q-uniformly convex and 2-uniformly smooth. In our proofs we use recent results on uniformly generalized prox-regular sets. Part of the novelty of the paper is the use of the usual Lipschitz continuity of the set-valued mapping which is very easy to verify contrarily to the ones used in the previous works. An example is stated at the end of the paper, showing the application of our existence result. View Full-Text
Keywords: uniformly convex spaces; uniformly smooth spaces; generalized proximal normal cone; generalized uniformly prox-regularity; Nonconvex sweeping process uniformly convex spaces; uniformly smooth spaces; generalized proximal normal cone; generalized uniformly prox-regularity; Nonconvex sweeping process
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Bounekhel, D.; Bounkhel, M.; Bachar, M. Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces. Symmetry 2019, 11, 28.

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