The Range of a Monotone Operator

“…In the context of Banach spaces, in [14] the preceding notion was considered under the name of pre-maximal monotone operator. Previously, the uniqueness notion was used in [1,10,21]; T is unique in their sense iff T −1 (as a subset of X * × X * * ) is unique in X * × X * * in the present sense.…”

Section: Main Notions and Notationsmentioning

confidence: 97%

“…For uniqueness one has the following characterizations (see also Proposition 33 below for double-cones); these characterizations can also be found in [14,Proposition 36]), [21,Theorem 19], and [1, Fact 2.6].…”

Section: Types Of Monotone Operatorsmentioning

confidence: 99%

Linear Monotone Subspaces of Locally Convex Spaces

Voisei

Zălinescu

2009

Set-Valued Anal

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The main focus of this paper is to study multi-valued linear monotone operators in the context of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negativeinfimum, and (dual-)representability are studied and criteria are provided. Mathematics Subject Classifications (2000)Primary 47A06 · 47H05; Secondary 26B25 · 52A41 PreliminariesMotivated by the facts that, besides the subdifferentials, single-valued linear maximal monotone operators enjoy a set of stronger monotonicity properties and belong to most of the special classes of maximal monotone operators introduced in the non-reflexive Banach space settings (see [1,2]), the linear monotone subspaces of a non-reflexive Banach space made the object of extensive studies in [3,18].

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“…A monotone operator T : X → 2 X * is called 3 * -monotone if for all x * ∈ R (T ) and x ∈ D(T ) there is some (x * , x) ∈ such that inf y * ∈T (y) x * − y * , x − y ≥ (x * , x). Definition 4.3 ([20,21]). An operator T : X → 2 X * is called of type (NI ) if for all (x * * , x * ) ∈ X * * × X * one has inf y * ∈T (y) ŷ − x * * , y * − x * ≤ 0.…”

Section: Brézis-haraux -Type Approximation Of the Range Of The Sum Ofmentioning

confidence: 99%

Weaker Constraint Qualifications in Maximal Monotonicity

Boţ

Grad

Wanka

2007

Numerical Functional Analysis and Optimization

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We give a sufficient condition, weaker than the others known so far, that guarantees that the sum of two maximal monotone operators on a reflexive Banach space is maximal monotone. Then we give a weak constraint qualification assuring the Brézis-Haraux-type approximation of the range of the sum of the subdifferentials of two proper convex lower-semicontinuous functions in nonreflexive Banach spaces, extending and correcting an earlier result due to Riahi.

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The Range of a Monotone Operator

Cited by 45 publications

References 0 publications

Recent Progress on Monotone Operator Theory

Recent Progress on Monotone Operator Theory

Linear Monotone Subspaces of Locally Convex Spaces

Weaker Constraint Qualifications in Maximal Monotonicity

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