The most important 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 + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying domA ∩ int dom...

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds, which is called the “sum problem”. In this paper, we establish the maximal monotonicity of A+ B provided that A and B are maximally monotone operators such that domA ∩ int domB 6= ∅,...

In this work, it is presented iterative schemes for achieving to common points of the solutions set of the system of generalized mixed equilibrium problems, solutions set of the variational inequality for an inverse-strongly monotone operator, common fixed points set of two infinite sequences of relatively nonexpansive mappings and common zero points set of two finite sequences of maximal monot...

We consider the primal problem of finding the zeros of the sum of a maximal monotone operator and the composition of another maximal monotone operator with a linear continuous operator. By formulating its Attouch-Théra-type dual inclusion problem, a primal-dual splitting algorithm which simultaneously solves the two problems in finitedimensional spaces is presented. The proposed scheme uses at ...

We present a new sufficient condition under which a maximal monotone operator T : X ⇉ X∗ admits a unique maximal monotone extension to the bidual T̃ : X∗∗ ⇉ X∗. For non-linear operators this condition is equivalent to uniqueness of the extension. The class of maximal monotone operators which satisfy this new condition includes class of Gossez type D maximal monotone operators, previously defined...

We establish variational principles for monotone and maximal bifunctions of Brøndsted-Rockafellar type by using our characterization of bifunction’s maximality in reflexive Banach spaces. As applications, we give an existence result of saddle point for convex-concave function and solve an approximate inclusion governed by a maximal monotone operator.

Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an “irreducible” monotone operator. In 2007, Borwein and Wiersma [...