Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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
and: BGSMath, Barcelona, Spain

JuanEnrique.Martinez.Legaz@uab.cat

Boris S. Mordukhovich

Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.

aa1086@wayne.edu

Keywords

  • Globally and locally monotone operators
  • maximal global and local monotonicity
  • variational analysis and generalized differentiation
  • coderivatives

Mathematics Subject Classification

26A15, 47H05, 49J52, 49J53, 54D05

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Suggested citation

P. D. Khanh, V. V. H. Khoa, J. E. Martínez-Legaz, B. S. Mordukhovich. “Relationships between Global and Local Monotonicity of Operators.” Journal of Convex Analysis 32 (2025), No. 2, 431–446. https://doi.org/10.68381/jca32021

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press.