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Relationships between Global and Local Monotonicity of Operators

Pham Duy Khanh, Vu Vinh Huy Khoa, Juan Enrique Martínez-Legaz, Boris S. Mordukhovich

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.

Relationships between Global and Local Monotonicity of Operators

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.
Paper Structure (7 sections, 9 theorems, 40 equations)

This paper contains 7 sections, 9 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Global and Local Monotonicity of Multifunctions
  3. Relationships between Global and Local Monotonicity of Single-Valued Operators
  4. Monotonicity of Univariate Set-Valued Operators
  5. Local Monotonicity Is Not Global on the Plane
  6. Relationships between Local and Global Maximal Monotonicity in Hilbert Spaces
  7. Conclusions and Future Research

Key Result

Theorem 3.1

Let $X$ be a locally convex topological vector space. Assume that $T\colon\hbox{\rm dom}\, T\rightarrow X^*$ has a convex domain and that $T$ is continuous relative to any segment in $\hbox{\rm dom}\, T$. Then the local monotonicity of $T$ around any graph point is equivalent to the global monotonic

Theorems & Definitions (15)

  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.4
  • Remark 4.5
  • Remark 4.6
  • Example 5.1
  • ...and 5 more