Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces
Abstract
:1. Introduction
2. Preliminaries
- 1.
- ;
- 2.
- .
- 1.
- Any closed convex set is uniformly generalized prox-regular w.r.t. any ;
- 2.
- The set (with ) is a closed nonconvex set which is uniformly generalized prox-regular w.r.t. some positive number .
- 3.
- Let with and C is a closed convex set in X. Using the same reasoning in Example 4.10 in [8], we can prove that S is not convex but uniformly generalized prox-regular w.r.t. some positive number .
- 1.
- 2.
- 3.
- 4.
3. Main Result
- We present a simple example showing the novelty and importance of our previous results. Assume that with and fix any point with and let be defined as: , where C is a convex compact set in X. Then, obviously, X is a q-uniformly convex and 2-uniformly smooth separable Banach space and K is Lipschitz continuous in the sense of (8) and for any we have , with (the closed convex hull of ), which is a convex compact set in . By Example 1 the set-valued mapping K has uniformly generalized prox-regular values in X. Therefore, all the assumptions of our main result in Theorem 2 are fulfilled and hence there exists a Lipschitz solution of (NSSP) associated with this K. We have to point out that this existence of solutions of (NSSP) cannot be derived from any existing result proved in other works.
- We can consider the cases of set-valued mappings K of the form and with is a bounded Lipschitz single-valued mapping and is a bounded real-valued function, and S is the set used in the above example. These set-valued mappings satisfy the hypothesis of Theorem 2 but their checks are very long and need more tools from nonsmooth analysis.
4. Conclusions
- The main results in the present paper can be summarized as follows: In the framework of separable Banach spaces which are 2-uniformly smooth and q-uniformly convex, we proved: The existence of approximate solutions for generalized prox-regular set-valued mappings which are Lipschitz in the sense of (8).
- If in addition, the image by J of the values of the set-valued mapping are contained in a convex compact set in , then the approximate solutions converge uniformly to a solution of (NSSP).
- The Lipschitz assumption (8) is very easy to check relatively to the Lipschitz conditions used in the previous papers [1,2]. In [2], instead of (8) the authors used the following assumption:Obviously, all the conditions (8), (27), and (28) coincide in Hilbert spaces. However, in Banach spaces the condition (27) is very hard to check even for simple forms of K. The difficulty comes from the definition of the function (see Definition 1) and the fact that the function does not preserve all the nice properties of the usual distance function . To compare (8) and (28), we take for example and , with is a Lipschitz single-valued mapping. Obviously, the condition (8) is satisfied and it can be verified easily. The condition (28) is not satisfied since the expression cannot be bounded from below by a positive number for any and any .
- As future works and perspectives we are investigating the case of p-uniformly smooth with any .
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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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. https://doi.org/10.3390/sym11010028
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(1):28. https://doi.org/10.3390/sym11010028
Chicago/Turabian StyleBounekhel, Djalel, Messaoud Bounkhel, and Mostafa Bachar. 2019. "Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces" Symmetry 11, no. 1: 28. https://doi.org/10.3390/sym11010028