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Wataru Takahashi - One of the best experts on this subject based on the ideXlab platform.
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strong convergence of an iterative sequence for Maximal Monotone operators in a banach space
Abstract and Applied Analysis, 2004Co-Authors: Fumiaki Kohsaka, Wataru TakahashiAbstract:We first introduce a modified proximal point algorithm for Maximal Monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of Maximal Monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.
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strong convergence theorems for resolvents of Maximal Monotone operators in banach spaces
Archiv der Mathematik, 2003Co-Authors: Shigeo Ohsawa, Wataru TakahashiAbstract:In this paper, we prove a strong convergence theorem for resolvents of Maximal Monotone operators in Banach spaces by the hybrid method in mathematical programming. Using this, we consider the problem of finding a minimizer of a convex function.
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approximating solutions of Maximal Monotone operators in hilbert spaces
Journal of Approximation Theory, 2000Co-Authors: Shoji Kamimura, Wataru TakahashiAbstract:Let H be a real Hilbert space and let T:H->2^H be a Maximal Monotone operator. In this paper, we first introduce two algorithms of approximating solutions of Maximal Monotone operators. One of them is to generate a strongly convergent sequence with limit v@?T^-^10. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.
Maria Elena Verona - One of the best experts on this subject based on the ideXlab platform.
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On the Regularity of Maximal Monotone Operators and Related Results
arXiv: Functional Analysis, 2012Co-Authors: Maria Elena Verona, Andrei VeronaAbstract:In the first part of the note we prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a Monotone operator is also necessary when the operator has bounded domain and is Maximal. Simons' condition is closely related to the notion of regular Maximal Monotone operator. In the second part of the note we give several characterizations for the regularity of a Maximal Monotone operator, show that a Maximal Monotone operator of type (FPV) is regular and improve a previous sum theorem type result.
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Regular Maximal Monotone Multifunctions and Enlargements
arXiv: Functional Analysis, 2008Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:In this note we use recent results concerning the sum theorem for Maximal Monotone multifunctions in general Banach spaces to find new characterizations and properties of regular Maximal Monotone multifunctions and then use these to describe the domain of certain enlargements.
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Regularity and the Brøndsted–Rockafellar Properties of Maximal Monotone Operators
Set-valued Analysis, 2006Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:The purpose of this paper is to establish connections between the class of Maximal Monotone operators of Brondsted–Rockafellar type and that of regular Maximal Monotone operators.
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Regular Maximal Monotone Operators
Set-valued Analysis, 1998Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:The purpose of this paper is to introduce a class of Maximal Monotone operators on Banach spaces that contains all Maximal Monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all Maximal Monotone operators that verify the simplest possible sum theorem. Dually strongly Maximal Monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then \(\overline {{\text{dom(}}T{\text{)}}} \) (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of \(\overline {{\text{dom(}}T{\text{)}}} \) at which T is locally bounded, and T is Maximal Monotone locally, as well as other results.
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Characterizations of Maximal Monotone operators
Nonlinear Analysis-theory Methods & Applications, 1992Co-Authors: Maria Elena Verona, Andrei VeronaAbstract:LET X BE A Banach space, A be a subset of X, and T: A + 2x* be a Monotone operator. It is known that in the case of an open A, the following assertions are equivalent (see [l]): (1) T is Maximal Monotone; (2) T is convex and w*-compact valued, and w*-upper semicontinuous; (3) T is minimal (with respect to graph inclusion) among all convex and w*-compact valued, w *-upper semicontinuous multivalued maps defined on A. In [2, corollary I] we proved that (1) is equivalent to (2) for Monotone operators defined on much more general sets (e.g. for relatively open or dense subsets of the set of nonsupport points of a convex set whose affine hull is X). When A has “support points”, T may be unbounded and the above statements are no longer equivalent. In [2] we were able to characterize Maximal Monotone operators in terms of their behavior at the support points and a certain upper semicontinuity property. This note contains improvements of some of our results in [2] and also extensions of some results obtained in [3]. Among others, we give, in a very general context, characterizations of the Maximality of Monotone operators, which, for open sets, reduce to those stated at the beginning of this introduction.
Andrei Verona - One of the best experts on this subject based on the ideXlab platform.
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On the Regularity of Maximal Monotone Operators and Related Results
arXiv: Functional Analysis, 2012Co-Authors: Maria Elena Verona, Andrei VeronaAbstract:In the first part of the note we prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a Monotone operator is also necessary when the operator has bounded domain and is Maximal. Simons' condition is closely related to the notion of regular Maximal Monotone operator. In the second part of the note we give several characterizations for the regularity of a Maximal Monotone operator, show that a Maximal Monotone operator of type (FPV) is regular and improve a previous sum theorem type result.
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Regular Maximal Monotone Multifunctions and Enlargements
arXiv: Functional Analysis, 2008Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:In this note we use recent results concerning the sum theorem for Maximal Monotone multifunctions in general Banach spaces to find new characterizations and properties of regular Maximal Monotone multifunctions and then use these to describe the domain of certain enlargements.
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Regularity and the Brøndsted–Rockafellar Properties of Maximal Monotone Operators
Set-valued Analysis, 2006Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:The purpose of this paper is to establish connections between the class of Maximal Monotone operators of Brondsted–Rockafellar type and that of regular Maximal Monotone operators.
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Regular Maximal Monotone Operators
Set-valued Analysis, 1998Co-Authors: Andrei Verona, Maria Elena VeronaAbstract:The purpose of this paper is to introduce a class of Maximal Monotone operators on Banach spaces that contains all Maximal Monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all Maximal Monotone operators that verify the simplest possible sum theorem. Dually strongly Maximal Monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then \(\overline {{\text{dom(}}T{\text{)}}} \) (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of \(\overline {{\text{dom(}}T{\text{)}}} \) at which T is locally bounded, and T is Maximal Monotone locally, as well as other results.
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Characterizations of Maximal Monotone operators
Nonlinear Analysis-theory Methods & Applications, 1992Co-Authors: Maria Elena Verona, Andrei VeronaAbstract:LET X BE A Banach space, A be a subset of X, and T: A + 2x* be a Monotone operator. It is known that in the case of an open A, the following assertions are equivalent (see [l]): (1) T is Maximal Monotone; (2) T is convex and w*-compact valued, and w*-upper semicontinuous; (3) T is minimal (with respect to graph inclusion) among all convex and w*-compact valued, w *-upper semicontinuous multivalued maps defined on A. In [2, corollary I] we proved that (1) is equivalent to (2) for Monotone operators defined on much more general sets (e.g. for relatively open or dense subsets of the set of nonsupport points of a convex set whose affine hull is X). When A has “support points”, T may be unbounded and the above statements are no longer equivalent. In [2] we were able to characterize Maximal Monotone operators in terms of their behavior at the support points and a certain upper semicontinuity property. This note contains improvements of some of our results in [2] and also extensions of some results obtained in [3]. Among others, we give, in a very general context, characterizations of the Maximality of Monotone operators, which, for open sets, reduce to those stated at the beginning of this introduction.
Szilard Laszlo - One of the best experts on this subject based on the ideXlab platform.
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on the generalized parallel sum of two Maximal Monotone operators of gossez type d
Journal of Mathematical Analysis and Applications, 2012Co-Authors: Radu Ioan Boţ, Szilard LaszloAbstract:Abstract 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 interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is Maximal Monotone of Gossez type (D), too. This result will follow as a particular instance of a more general one concerning the Maximal monotonicity of Gossez type (D) of an extended parallel sum defined for the Maximal Monotone extensions of the two operators to the corresponding biduals.
Fumiaki Kohsaka - One of the best experts on this subject based on the ideXlab platform.
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strong convergence of an iterative sequence for Maximal Monotone operators in a banach space
Abstract and Applied Analysis, 2004Co-Authors: Fumiaki Kohsaka, Wataru TakahashiAbstract:We first introduce a modified proximal point algorithm for Maximal Monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of Maximal Monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.
