Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces
In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, in p-uniformly convex and q-uniformly smooth Banach spaces, the density theorem for the new conc...
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Wiley
2016-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2016/1917387 |
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| author | Messaoud Bounkhel |
| author_facet | Messaoud Bounkhel |
| author_sort | Messaoud Bounkhel |
| collection | DOAJ |
| description | In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, in p-uniformly convex and q-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established. |
| format | Article |
| id | doaj-art-a0324b5d348c4a07afa69c61dfc929ad |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-a0324b5d348c4a07afa69c61dfc929ad2025-02-03T01:12:11ZengWileyJournal of Function Spaces2314-88962314-88882016-01-01201610.1155/2016/19173871917387Calculus Rules for V-Proximal Subdifferentials in Smooth Banach SpacesMessaoud Bounkhel0Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaIn 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, in p-uniformly convex and q-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.http://dx.doi.org/10.1155/2016/1917387 |
| spellingShingle | Messaoud Bounkhel Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces Journal of Function Spaces |
| title | Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces |
| title_full | Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces |
| title_fullStr | Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces |
| title_full_unstemmed | Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces |
| title_short | Calculus Rules for V-Proximal Subdifferentials in Smooth Banach Spaces |
| title_sort | calculus rules for v proximal subdifferentials in smooth banach spaces |
| url | http://dx.doi.org/10.1155/2016/1917387 |
| work_keys_str_mv | AT messaoudbounkhel calculusrulesforvproximalsubdifferentialsinsmoothbanachspaces |