maximal monotone operator theory has recently turned fifty years old. In the first part of the talk, I shall try to explain why maximal(ly) monotone operators are both interesting and important objects while briefly survey the history of the subject — culminating with a description of the remarkable progress made during the past decade. In the second part of the talk, I shall describe in more d...

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A+B provided that A,B are maximally monotone and A is a linear relation, as soon as Rockafellar’s constraint qualification holds: domA ∩ int domB 6...

We introduce and study a new system of nonlinear variational-like inclusions involving sG, η maximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, λ, ξ -relaxed cocoercive operators, ζ, φ, g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique ...

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are sum maximal operator monotone, Lipschitzian operator. The requires only unbiased estimations obtain rate $${\mathcal {O}}(log(n)/n)$$ in expectation strongly case, as well almost sure general case. Furthermore, context application to convex–concave saddle point problems, we derive pri...

The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this pa...

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uni...

The concept of a monotone operator — which covers both linear positive semi-definite operators and subdifferentials of convex functions — is fundamental in various branches of mathematics. Over the last few decades, several stronger notions of monotonicity have been introduced: Gossez’s maximal monotonicity of dense type, Fitzpatrick and Phelps’s local maximal monotonicity, and Simons’s monoton...

An important and perhaps interesting topic in nonlinear analysis and convex optimization concerns solving inclusions of the form 0 ∈ A(x), where A is a maximal monotone operator on a Hilbert space H. Its importance in convex optimization is evidenced from the fact that many problems that involve convexity can be formulated as finding zeros of maximal monotone operators. For example, convex mini...