We establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a ‘remainder’ monotone mapping, where the remainder is either skew linear, or more broadly ‘acyclic’, in the sense that it contains no nontrivial subdifferential component. Examples are given of indecomposable and acyclic operators. In particular, we present an explicit nonlinear acyclic opera...

In this paper, we introduce two iterative algorithms for finding the solution of the sum of two monotone operators by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we prove strong convergence theorems of such sequences to the solution of the sum of an inverse-strongly monotone and a maximal monotone operator. Finally, we present a numerical re...

We consider whether the “inequality-splitting” property established in the Brøndsted– Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions...

We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a root of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in the PDE and optimization literatures can be retrieved by this generalized PPA scheme. We establish the convergence rate of this generalized PPA scheme under different conditions, in...

In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick’s family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brøndsted-Rockafellar property. In this ...

This paper is concerned with the study of a class of prox-penalization methods for solving variational inequalities of the form Ax + NC(x) 3 0 where H is a real Hilbert space, A : H ⇒ H is a maximal monotone operator and NC is the outward normal cone to a closed convex set C ⊂ H. Given Ψ : H → R ∪ {+∞} which acts as a penalization function with respect to the constraint x ∈ C, and a penalizatio...

The notion of monotone operator in a Hilbert space is extended, and a considerably broader class of possibly multivalued operators is introduced. It is shown that well-known Minty's theorems on maximal monotone operators in Hilbert spaces can be extended to the cases of operators belonging to the class. Typical examples of partial differential operators are given to illustrate that the results ...

The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiorityand...