scholarly journals Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Teffera M. Asfaw
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuous ◽  
Sufficient Conditions ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Uniformly Convex ◽  
Evolution Type

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎andA:X⊇D(A)→2X⁎be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved forT+Aunder weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used conditionD(T)∘∩D(A)≠∅and Browder and Hess who used the quasiboundedness ofTand condition0∈D(T)∩D(A). In particular, the maximality ofT+∂ϕis proved provided thatD(T)∘∩D(ϕ)≠∅, whereϕ:X→(-∞,∞]is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.

scholarly journals Subdifferentials are locally maximal monotone

1993 ◽  
Vol 47 (3) ◽  
pp. 465-471 ◽  
Author(s):  
S. Simons
Keyword(s):  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Affirmative Answer ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
New Class ◽  
Locally Maximal ◽  
Convex Lower Semicontinuous Function

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.

scholarly journals Topological degree and application to a parabolic variational inequality problem

2001 ◽  
Vol 25 (4) ◽  
pp. 273-287 ◽  
Author(s):  
A. Addou ◽  
B. Mermri
Keyword(s):  
Variational Inequality ◽  
Topological Degree ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parabolic Variational Inequality ◽  
Inequality Problem ◽  
Monotone Map ◽  
New Formulation

We are interested in constructing a topological degree for operators of the formF=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolic variational inequality problem.

scholarly journals Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1836
Author(s):  
Yaqian Jiang ◽  
Rudong Chen ◽  
Luoyi Shi
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Linear Convergence ◽  
Monotone Operators ◽  
Null Point ◽  
Point Problem ◽  
Bounded Linear ◽  
Zero Points

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

scholarly journals Split variational inclusions for Bregman multivalued maximal monotone operators

2020 ◽  
Author(s):  
Mujahid Abbas ◽  
Faik Gürsoy ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan
Keyword(s):  
Strong Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Strong Convergence Theorem ◽  
Variational Inclusions ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

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.

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
In-Sook Kim ◽  
Jung-Hyun Bae
Keyword(s):  
Reflexive Banach Space ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Dense Subspace ◽  
Uniformly Convex ◽  
Infinite Dimensional ◽  
Surjectivity Result ◽  
Densely Defined

AbstractLet X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.

scholarly journals A New Hybrid Method for Equilibrium Problems, Variational Inequality Problems, Fixed Point Problems, and Zero of Maximal Monotone Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yaqin Wang
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Fixed Points ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Strong Convergence Theorem

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, the set of solutions of the generalized mixed equilibrium problem, and zeros of maximal monotone operators in a 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 obtained in this paper improve and extend the result of Zeng et al. (2010) and many others.

scholarly journals Iterative Schemes for Finite Families of Maximal Monotone Operators Based on Resolvents

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Li Wei ◽  
Ruilin Tan
Keyword(s):  
Banach Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Generalized Resolvent ◽  
Iterative Schemes

The purpose of this paper is to present two iterative schemes based on the relative resolvent and the generalized resolvent, respectively. And, it is shown that the iterative schemes converge weakly to common solutions for two finite families of maximal monotone operators in a real smooth and uniformly convex Banach space and one example is demonstrated to explain that some assumptions in the main results are meaningful, which extend the corresponding works by some authors.

scholarly journals Hybrid Projection Algorithm for Two Countable Families of Hemirelatively Nonexpansive Mappings and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zi-Ming Wang ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Projection Algorithm ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Convex Feasibility ◽  
Hybrid Projection Algorithm

Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

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