Journal of Convex Analysis 32 (2025), No. 2, 431--446
Copyright Heldermann Verlag 2025
Relationships between Global and Local Monotonicity of Operators
Pham Duy Khanh
Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam
khanhpd@hcmue.edu.vn
Vu Vinh Huy Khoa
Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.
khoavu@wayne.edu
Juan Enrique Martínez-Legaz
Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, Barcelona, Spain
JuanEnrique.Martinez.Legaz@uab.cat
Boris S. Mordukhovich
Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.
aa1086@wayne.edu
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.
Keywords: Globally and locally monotone operators, maximal global and local monotonicity, variational analysis and generalized differentiation, coderivatives.
MSC: 26A15, 47H05, 49J52, 49J53, 54D05.
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