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 solution of the GSVI, which is also a common solution of these finitely many variational inclusions and the CMP. Our results extend, improve, complement and develop the corresponding ones given by some authors recently in this area.

General iterative methods for systems of variational inequalities with the constraints of generalized mixed equilibria and fixed point problem of pseudocontractions

“…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 ].…”

“…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 ).…”

“…Very recently, Kong et al [ 11 ] established the strong convergence of two hybrid steepest-descent schemes to the same solution of GSVI ( 1.2 ), which is also a common solution of finitely many variational inclusions and a minimization problem.…”