Fifty years of maximal monotonicity

Alizadeh, Mohammad Hossein

doi:10.22108/msci.2026.147744.1780

Fifty years of maximal monotonicity

Articles in Press

Document Type : Translation Paper

Author

Mohammad Hossein Alizadeh

Department of Mathematics-Institute for Advanced Studies in Basic Sciences (IASBS) Zanjan ,-Iran

10.22108/msci.2026.147744.1780

Abstract

Abstract:[The author Abstract is here] Maximal monotone operator theory is about to turn (or just has turned)

50. I intend to briefly survey the history of the subject. I shall try to explain why

maximal monotone operators are both interesting and important, -culminating with a

description of the remarkable progress made during the past decade.

Abstract: [Translator’s abstract]: This paper reviews fifty years of research on maximal monotone

operators, from Minty’s foundational work in the 1960s to recent advances driven by Fitzpatrick’s

representation theorem. It explains the central role of monotone operators in convex analysis, optimization,

and partial differential equations, highlighting key results such as Rockafellar’s sum theorem

and Minty’s surjectivity theorem. The article describes how the Fitzpatrick function revolutionized

the theory, leading to simpler proofs and new insights into the structure and classification of monotone

operators. It concludes by outlining major open questions regarding the monotone and maximal monotone operators.

Keywords