scholarly journals On the range of a coercive maximal monotone operator in a nonreflexive Banach space

1972 ◽  
Vol 35 (1) ◽  
pp. 88-88 ◽  
Author(s):  
Jean-Pierre Gossez
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone

scholarly journals The convex function determined by a multifunction

1996 ◽  
Vol 54 (1) ◽  
pp. 87-97 ◽  
Author(s):  
M. Coodey ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Convex Function ◽  
Monotone Operator ◽  
Open Problem ◽  
Maximal Monotone Operator ◽  
Convex Functions ◽  
Maximal Monotone ◽  
Minimax Theorem ◽  
Local Boundedness

We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

scholarly journals A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-31 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Generalized Mixed Equilibrium Problem ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

scholarly journals Convergence Theorems for a Maximal Monotone Operator and a -Strongly Nonexpansive Mapping in a Banach Space

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping

LetEbe a smooth Banach space with a norm . Let for any , where stands for the duality pair andJis the normalized duality mapping. With respect to this bifunction , a generalized nonexpansive mapping and a -strongly nonexpansive mapping are defined in . In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a -strongly nonexpansive mapping.

scholarly journals Stronger maximal monotonicity properties of linear operators

1999 ◽  
Vol 60 (1) ◽  
pp. 163-174 ◽  
Author(s):  
H.H. Bauschke ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Real Banach Space ◽  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Linear Mapping ◽  
Linear Operators ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function

The subdifferential mapping associated with a proper, convex lower semicontinuous function on a real Banach space is always a special kind of maximal monotone operator. Specifically, it is always “strongly maximal monotone” and of “type (ANA)”. In an attempt to find maximal monotone operators that do not satisfy these properties, we investigate (possibly discontinuous) maximal monotone linear operators from a subspace of a (possibly nonreflexive) real Banach space into its dual. Such a linear mapping is always “strongly maximal monotone”, but we are only able to prove that is of “type (ANA)” when it is continuous or surjective — the situation in general is unclear. In fact, every surjective linear maximal monotone operator is of “type (NA)”, a more restrictive condition than “type (ANA)”, while the zero operator, which is both continuous and linear and also a subdifferential, is never of “type (NA)” if the underlying space is not reflexive. We examine some examples based on the properties of derivatives.

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

scholarly journals NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

2010 ◽  
Vol 83 (1) ◽  
pp. 22-29 ◽  
Author(s):  
FILOMENA CIANCIARUSO ◽  
GIUSEPPE MARINO ◽  
LUIGI MUGLIA ◽  
HONG-KUN XU
Keyword(s):  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Null Sequence ◽  
Graph Convergence ◽  
Common Domain

AbstractWe construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

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