2017
DOI: 10.22436/jnsa.010.03.03
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Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems

Abstract: Two hybrid steepest-descent schemes (implicit and explicit) for finding a solution of the general system of variational inequalities (in short, GSVI) with the constraints of finitely many variational inclusions for maximal monotone and inversestrongly monotone mappings and a minimization problem for a convex and continuously Fréchet differentiable functional (in short, CMP) have been presented in a real Hilbert space. We establish the strong convergence of these two hybrid steepestdescent schemes to the same s… Show more

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Cited by 2 publications
(3 citation statements)
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“…Moreover, we established strong convergence of the proposed implicit and explicit iterative schemes to a solution of the GSVI, which is the unique solution of a certain variational inequality. Our Theorems 3.1 – 3.3 not only improve and develop the main results of [ 1 ] and [ 12 ] but also improve and develop Theorems 3.1 and 3.2 of [ 9 ], Theorems 3.1 and 3.2 of [ 10 ], and Proposition 3.1, Theorems 3.2 and 3.5 of [ 11 ].…”
Section: Discussionsupporting
confidence: 59%
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“…Moreover, we established strong convergence of the proposed implicit and explicit iterative schemes to a solution of the GSVI, which is the unique solution of a certain variational inequality. Our Theorems 3.1 – 3.3 not only improve and develop the main results of [ 1 ] and [ 12 ] but also improve and develop Theorems 3.1 and 3.2 of [ 9 ], Theorems 3.1 and 3.2 of [ 10 ], and Proposition 3.1, Theorems 3.2 and 3.5 of [ 11 ].…”
Section: Discussionsupporting
confidence: 59%
“…Compared with Proposition 3.3, Theorem 3.4, and Theorem 3.7 in [ 11 ], respectively, our Theorems 3.1 , 3.2 , and 3.3 improve and develop them in the following aspects: GSVI ( 1.3 ) with solutions being also fixed points of a continuous pseudocontinuous mapping in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7] is extended to GSVI ( 1.3 ) with solutions being also common solutions of a finite family of generalized mixed equilibrium problems (GMEPs) and fixed points of a continuous pseudocontinuous mapping in our Theorems 3.1 , 3.2 , and 3.3 ; in the argument process of our Theorems 3.1 , 3.2 , and 3.3 , we use the variable parameters and (resp., and ) in place of the fixed parameters λ and ν in the proof of [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], and additionally deal with a pool of variable parameters (resp., ) involving a finite family of GMEPs; the iterative schemes in our Theorems 3.1 , 3.2 , and 3.3 are more advantageous and more flexible than the iterative schemes in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], because they can be applied to solving three problems (i.e., GSVI ( 1.3 ), a finite family of GMEPs, and the fixed point problem of a continuous pseudocontractive mapping) and involve much more parameter sequences; it is worth emphasizing that our general implicit iterative scheme ( 3.1 ) is very different from Jung’s composite implicit iterative scheme in [ 12 ], because the term “ ” in Jung’s implicit scheme is replaced by the term “ ” in our implicit scheme ( 3.1 ). Moreover, the term “ ” in Jung’s explicit scheme is replaced by the term “ ” in our explicit scheme ( 3.3 ).…”
Section: Resultsmentioning
confidence: 99%
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