Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces
In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in <sup>[<xref ref-type="bibr" r...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231513?viewType=HTML |
| _version_ | 1797544360752971776 |
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| author | Messaoud Bounkhel |
| author_facet | Messaoud Bounkhel |
| author_sort | Messaoud Bounkhel |
| collection | DOAJ |
| description | In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup>. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup> from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper. |
| first_indexed | 2024-03-10T13:59:22Z |
| format | Article |
| id | doaj.art-b1d09dbd5df54e4e8321ac357abe22f8 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-10T13:59:22Z |
| publishDate | 2023-11-01 |
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| spelling | doaj.art-b1d09dbd5df54e4e8321ac357abe22f82023-11-21T01:17:55ZengAIMS PressAIMS Mathematics2473-69882023-11-01812295552956810.3934/math.20231513Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spacesMessaoud Bounkhel 0Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh11451, Saudi ArabiaIn this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup>. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in <sup>[<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b20">20</xref>]</sup> from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper.https://www.aimspress.com/article/doi/10.3934/math.20231513?viewType=HTML$ f $-generalized projection$ (f， \lambda) $-generalized projection$ p $-uniformly convex banach spaces$ q $-uniformly smooth banach spacesnonconvex variational problemuniformly generalized prox-regular |
| spellingShingle | Messaoud Bounkhel Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces AIMS Mathematics $ f $-generalized projection $ (f， \lambda) $-generalized projection $ p $-uniformly convex banach spaces $ q $-uniformly smooth banach spaces nonconvex variational problem uniformly generalized prox-regular |
| title | Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces |
| title_full | Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces |
| title_fullStr | Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces |
| title_full_unstemmed | Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces |
| title_short | Generalized (f,λ)-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces |
| title_sort | generalized f λ projection operator on closed nonconvex sets and its applications in reflexive smooth banach spaces |
| topic | $ f $-generalized projection $ (f， \lambda) $-generalized projection $ p $-uniformly convex banach spaces $ q $-uniformly smooth banach spaces nonconvex variational problem uniformly generalized prox-regular |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231513?viewType=HTML |
| work_keys_str_mv | AT messaoudbounkhel generalizedflprojectionoperatoronclosednonconvexsetsanditsapplicationsinreflexivesmoothbanachspaces |