In the context of a general q-uniformly smooth Banach space, we will prove the convergence of a scheme which covers Yamada's hybrid steepest-descent method and Moudafi's viscosity approximation in their comple...
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In the context of a general q-uniformly smooth Banach space, we will prove the convergence of a scheme which covers Yamada's hybrid steepest-descent method and Moudafi's viscosity approximation in their complete forms. Moreover, we present an application to the minimization over a set of fixed points.
With the help of the generalized viscosity implicit method and hybrid steepest-descent method, we introduce an iterative scheme for approximating the solution of a variational inequality over the set of fixed points o...
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With the help of the generalized viscosity implicit method and hybrid steepest-descent method, we introduce an iterative scheme for approximating the solution of a variational inequality over the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense. Some strong convergence results for our proposed iterative scheme are established in the framework of Banach spaces. Applicability of our proposed method is shown in variational inclusion problem and convex minimization problem. We discuss some examples to demonstrate the numerical implementation and efficiency of our main results in comparison of other related results. Our results improve, extend and unify previously known results given in literature. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
Based on a viscosity hybrid steepest-descent method, in this paper, we introduce an iterative scheme for finding a common element of a system of equilibrium and fixed point problems of an infinite family of strictly p...
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Based on a viscosity hybrid steepest-descent method, in this paper, we introduce an iterative scheme for finding a common element of a system of equilibrium and fixed point problems of an infinite family of strictly pseudo-contractive mappings which solves the variational inequality <(gamma f - mu F)q, p - q > <= 0 for p epsilon boolean AND F-infinity(i=1)(T-i). Furthermore, we also prove the strong convergence theorems for the proposed iterative scheme and give a numerical example to support and illustrate our main theorem.
To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a...
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To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybridsteepest-descent (MRHSD) method. However, choosing an efficient and implementable nonexpansive mapping is still a difficult problem. We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies. Second, we design an efficient implementation of the MRHSD method for a type of variational inequality problem based on the approximate projection contraction method. Finally, we design a set of practical numerical experiments. The results demonstrate that this is an efficient implementation of the MRHSD method.
In the present paper, we propose a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert sp...
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In the present paper, we propose a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces. A strong convergence theorem is proved under fewer restrictions imposed on the mappings and parameters. An extension and numerical result are also given to illustrate the effectiveness and superiority of the proposed algorithm.
In this paper, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich's extragradient method, hybrid steepest-descent method and Mann's iteration method. It is proven that under approp...
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In this paper, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich's extragradient method, hybrid steepest-descent method and Mann's iteration method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem ((MEP), the solution set of finitely many variational inclusions and the solution set of a system of generalized equilibrium problems (SGEP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the SGEP and Finitely many variational inclusions.
Assume that C is a nonempty closed convex subset of a Hilbert space H and B : C -> H is a strongly monotone mapping. Assume also that F is the intersection of the common fixed points of an infinite family of nonexp...
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Assume that C is a nonempty closed convex subset of a Hilbert space H and B : C -> H is a strongly monotone mapping. Assume also that F is the intersection of the common fixed points of an infinite family of nonexpansive mappings on C and the set of solutions of a system of equilibrium problems. We devise a modified hybrid steepest-descent method which generates a sequence (x(n)) from an arbitrary initial point x(0) is an element of H. The sequence (x(n)) is shown to converge in norm to the unique solution of the variational inequality VI(B, F) under suitable conditions. (C) 2010 Elsevier Ltd. All rights reserved.
In order to reduce the difficulty and complexity on computing the projection from a real Hilbert space onto a nonempty closed convex subset, Yamada has provided the hybrid steepest-descent method for solving varia...
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In order to reduce the difficulty and complexity on computing the projection from a real Hilbert space onto a nonempty closed convex subset, Yamada has provided the hybrid steepest-descent method for solving variational inequalities. Recently Xu has provided the modified and relaxed hybridsteepestdescentmethod for variational inequalities based on the minds of the Gauss-seidel method, and given out the convergence theorem under some suitable conditions(Condition 3.1). In this paper, we give out other different conditions(Condition 3.2) about the modified and relaxed hybrid steepest-descent method for variational inequalities, such the conditions can simplify proof and it is to be noted that the proof of strong convergence is different from the previous results. Furthermore we design a set of practical numerical experiments and numerical results demonstrated that the modified and relaxed hybridsteepest-descentmethod under the Condition 3.2 is more efficient than under the Condition 3.1.
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