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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 theorems for Maximal and inverse strongly Monotone mappings in hilbert spaces and applications
Journal of Optimization Theory and Applications, 2013Co-Authors: Wataru TakahashiAbstract:In this paper, we prove two strong convergence theorems for finding a common point of the set of zero points of the addition of an inverse-strongly Monotone mapping and a Maximal Monotone Operator and the set of zero points of a Maximal Monotone Operator, which is related to an equilibrium problem in a Hilbert space. Such theorems improve and extend the results announced by Y. Liu (Nonlinear Anal. 71:4852–4861, 2009). As applications of the results, we present well-known and new strong convergence theorems which are connected with the variational inequality, the equilibrium problem and the fixed point problem in a Hilbert space.
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a general iterative method for hierarchical variational inequality problems in hilbert spaces and applications
Positivity, 2012Co-Authors: Laijiu Lin, Wataru TakahashiAbstract:Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let α > 0 and let A be an α-inverse-strongly Monotone mapping of C into H and let B be a Maximal Monotone Operator on H. Let F be a Maximal Monotone Operator on H such that the domain of F is included in C. Let 0 0 }\) and L > 0. Take \({\mu, \gamma \in \mathbb R}\) as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption \({(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}\), we prove a strong convergence theorem for finding a point \({z_0\in (A+B)^{-1}0\cap F^{-1}0}\) which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.
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Generalized Projection Algorithms for Maximal Monotone Operators and Relatively Nonexpansive Mappings in Banach Spaces
Taiwanese Journal of Mathematics, 2011Co-Authors: Chakkrid Klin-eam, Suthep Suantai, Wataru TakahashiAbstract:In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a Maximal Monotone Operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of Maximal Monotone Operators and relatively nonexpansive mappings in Banach spaces.
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Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators
Abstract and Applied Analysis, 2009Co-Authors: Chakkrid Klin-eam, Suthep Suantai, Wataru TakahashiAbstract:We establish strong convergence theorems for finding a common element of the zero point set of a Maximal Monotone Operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a Maximal Monotone Operator and two nonexpansive mappings in a Hilbert space.
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weak and strong convergence theorems for new resolvents of Maximal Monotone Operators in banach spaces
2007Co-Authors: Takanori Ibaraki, Wataru TakahashiAbstract:In this paper, we prove weak and strong convergence theorems for new resolvents of a Maximal Monotone Operator in a Banach space which are connected with the proximal point algorithm of Rockafellar (SIAM J. Control. Optim. 14:877–898, 1976). Using these results, we consider the problem of finding minimizers of convex functions defined on Banach spaces.
Binchao Deng - One of the best experts on this subject based on the ideXlab platform.
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strong convergence theorem by hybrid method for equilibrium problems variational inequality problems and Maximal Monotone Operators
Nonlinear Analysis: Hybrid Systems, 2010Co-Authors: Qiaoli Dong, Binchao DengAbstract:We introduce an iterative scheme for finding a common element of the solution set of the equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-Monotone Operators and the solution set of a Maximal Monotone Operator in a 2-uniformly convex and uniformly smooth Banach space, and then we present strong convergence theorems which generalize the results of many others.
Wanna Sriprad - One of the best experts on this subject based on the ideXlab platform.
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an extragradient method and proximal point algorithm for inverse strongly Monotone Operators and Maximal Monotone Operators in banach spaces
Fixed Point Theory and Applications, 2009Co-Authors: Somyot Plubtieng, Wanna SripradAbstract:We introduce an iterative scheme for finding a common element of the solution set of a Maximal Monotone Operator and the solution set of the variational inequality problem for an inverse strongly-Monotone Operator in a uniformly smooth and uniformly convex Banach space, and then we prove weak and strong convergence theorems by using the notion of generalized projection. The result presented in this paper extend and improve the corresponding results of Kamimura et al. (2004), and Iiduka and Takahashi (2008). Finally, we apply our convergence theorem to the convex minimization problem, the problem of finding a zero point of a Maximal Monotone Operator and the complementary problem.
Qiaoli Dong - One of the best experts on this subject based on the ideXlab platform.
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strong convergence theorem by hybrid method for equilibrium problems variational inequality problems and Maximal Monotone Operators
Nonlinear Analysis: Hybrid Systems, 2010Co-Authors: Qiaoli Dong, Binchao DengAbstract:We introduce an iterative scheme for finding a common element of the solution set of the equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-Monotone Operators and the solution set of a Maximal Monotone Operator in a 2-uniformly convex and uniformly smooth Banach space, and then we present strong convergence theorems which generalize the results of many others.
Benar Fux Svaiter - One of the best experts on this subject based on the ideXlab platform.
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Geometric properties of Maximal Monotone Operators and convex functions which may represent them
arXiv: Functional Analysis, 2012Co-Authors: Benar Fux SvaiterAbstract:We study the relations between some geometric properties of Maximal Monotone Operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We also investigate under which conditions a convex function represents a Maximal Monotone Operator with bounded range and provide an example of a non type (D) Operator on this class.
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Maximal Monotone Operators with a unique extension to the bidual
arXiv: Functional Analysis, 2009Co-Authors: Marques M Alves, Estrada Dona Castorina, Benar Fux SvaiterAbstract:We present a new sufficient condition under which a Maximal Monotone Operator $T:X\tos X^*$ admits a unique Maximal Monotone extension to the bidual $\widetilde T:X^{**} \rightrightarrows X^*$. For non-linear Operators this condition is equivalent to uniqueness of the extension. The class of Maximal Monotone Operators which satisfy this new condition includes class of Gossez type D Maximal Monotone Operators, previously defined and studied by J.-P. Gossez, and all Maximal Monotone Operators of this new class satisfies a restricted version of Brondsted-Rockafellar condition. The central tool in our approach is the $\mathcal{S}$-function defined and studied by Burachik and Svaiter in 2000 \cite{BuSvSet02}(submission date, July 2000). For a generic Operator, this function is the supremum of all convex lower semicontinuous functions which are majorized by the duality product in the graph of the Operator. We also prove in this work that if the graph of a Maximal Monotone Operator is convex, then this graph is an affine linear subspace.
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Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces
arXiv: Functional Analysis, 2008Co-Authors: M. Marques Alves, Benar Fux SvaiterAbstract:Maximal Monotone Operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a Maximal Monotone Operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces was obtained. In this work we study some others conditions which guarantee that a convex function represents a Maximal Monotone Operator in non-reflexive Banach spaces.
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e enlargements of Maximal Monotone Operators in banach spaces
Set-valued Analysis, 1999Co-Authors: Regina Sandra Burachik, Benar Fux SvaiterAbstract:Given a Maximal Monotone Operator T in a Banach space, we consider an enlargement Te, in which monotonicity is lost up to e, in a very similar way to the e-subdifferential of a convex function. We establish in this general framework some theoretical properties of Te, like a transportation formula, local Lipschitz continuity, local boundedness, and a Brondsted–Rockafellar property.
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ε-Enlargements of Maximal Monotone Operators in Banach Spaces
Set-Valued Analysis, 1999Co-Authors: Regina Sandra Burachik, Benar Fux SvaiterAbstract:Given a Maximal Monotone Operator T in a Banach space, we consider an enlargement Te, in which monotonicity is lost up to e, in a very similar way to the e-subdifferential of a convex function. We establish in this general framework some theoretical properties of Te, like a transportation formula, local Lipschitz continuity, local boundedness, and a Brondsted–Rockafellar property.