Relationships between Global and Local Monotonicity of Operators
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929296333864960 |
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| author | Khanh, Pham Duy Khoa, Vu Vinh Huy Martínez-Legaz, Juan Enrique Mordukhovich, Boris S. |
| author_facet | Khanh, Pham Duy Khoa, Vu Vinh Huy Martínez-Legaz, Juan Enrique Mordukhovich, Boris S. |
| contents | The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defined on the real line under a certain connectedness condition. The situation is different for set-valued operators in multidimensional spaces as demonstrated by an example of locally monotone operator on the plane that is not globally monotone. Finally, we invoke coderivative criteria from variational analysis to characterize both global and local maximal monotonicity of set-valued operators in Hilbert spaces to verify the equivalence between these monotonicity properties under the closed-graph and global hypomonotonicity assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relationships between Global and Local Monotonicity of Operators Khanh, Pham Duy Khoa, Vu Vinh Huy Martínez-Legaz, Juan Enrique Mordukhovich, Boris S. Functional Analysis Optimization and Control 26A15, 47H05, 49J53, 49J53, 54D05 The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defined on the real line under a certain connectedness condition. The situation is different for set-valued operators in multidimensional spaces as demonstrated by an example of locally monotone operator on the plane that is not globally monotone. Finally, we invoke coderivative criteria from variational analysis to characterize both global and local maximal monotonicity of set-valued operators in Hilbert spaces to verify the equivalence between these monotonicity properties under the closed-graph and global hypomonotonicity assumptions. |
| title | Relationships between Global and Local Monotonicity of Operators |
| topic | Functional Analysis Optimization and Control 26A15, 47H05, 49J53, 49J53, 54D05 |
| url | https://arxiv.org/abs/2403.20227 |