Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces
For a set-valued mapping M defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x,y)∈E×F, we study the minimal time function associated with the images of M and a bounded set Ω⊂F defined by 𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and...
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/707603 |
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| author | Messaoud Bounkhel |
| author_facet | Messaoud Bounkhel |
| author_sort | Messaoud Bounkhel |
| collection | DOAJ |
| description | For a set-valued mapping M defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x,y)∈E×F, we study the minimal time function associated with the images of M and a bounded set Ω⊂F defined by 𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of 𝒯M,Ω at those points of (x,y)∈E×F (both cases: points in the graph gph M and points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph of M at points inside gph M and to the graph of the enlargement set-valued mapping at points outside gph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function 𝒯M,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper. |
| format | Article |
| id | doaj-art-bfd88b13b2a944978db7f1670fab0ec6 |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-bfd88b13b2a944978db7f1670fab0ec62025-02-03T05:53:32ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/707603707603Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector SpacesMessaoud Bounkhel0Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaFor a set-valued mapping M defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x,y)∈E×F, we study the minimal time function associated with the images of M and a bounded set Ω⊂F defined by 𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of 𝒯M,Ω at those points of (x,y)∈E×F (both cases: points in the graph gph M and points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph of M at points inside gph M and to the graph of the enlargement set-valued mapping at points outside gph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function 𝒯M,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.http://dx.doi.org/10.1155/2013/707603 |
| spellingShingle | Messaoud Bounkhel Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces Journal of Function Spaces and Applications |
| title | Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces |
| title_full | Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces |
| title_fullStr | Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces |
| title_full_unstemmed | Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces |
| title_short | Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces |
| title_sort | subdifferential properties of minimal time functions associated with set valued mappings with closed convex graphs in hausdorff topological vector spaces |
| url | http://dx.doi.org/10.1155/2013/707603 |
| work_keys_str_mv | AT messaoudbounkhel subdifferentialpropertiesofminimaltimefunctionsassociatedwithsetvaluedmappingswithclosedconvexgraphsinhausdorfftopologicalvectorspaces |