• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.465-472
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• 2010

The purpose of this article is to introduce a new generalized class of the Wiener-Hopf equations and a new generalized class of the variational inequalities. Using the projection technique, we show that the generalized Wiener-Hopf equations are equivalent to the generalized variational inequalities. We use this alternative equivalence to suggest and analyze an iterative scheme for finding the solution of the generalized Wiener-Hopf equations and the solution of the generalized variational inequalities. The results presented in this paper may be viewed as significant and improvement of the previously known results. In special, our results improve and extend the resent results of M.A. Noor and Z.Y.Huang[M.A. Noor and Z.Y.Huang, Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings, Appl. Math. Comput.(2007), doi:10.1016/j.amc.2007.02.117].

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.37-51
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• 2002

Let X be a reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty bounded open subset of X, and T a continuous mapping from the closure of C into X which is locally pseudo-contractive mapping on C. We show that if the closed unit ball of X has the fixed point property for nonexpansive self-mappings and T satisfies the following condition: there exists z $\in$ C such that ∥z-T(z)∥<∥x-T(x)∥ for all x on the boundary of C, then the trajectory tlongrightarrowz$_{t}$$\in$C, t$\in$[0, 1) defined by the equation z$_{t}$ = tT(z$_{t}$)+(1-t)z is continuous and strongly converges to a fixed point of T as t longrightarrow 1￣.ow 1￣.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.179-190
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• 2007

In this paper, we consider the general variational inequality GVI(F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type method $u_{n+l}=(1-{\alpha}+{\theta}_{n+1})Tu_n+{\alpha}u_n-{\theta}_{n+1g}(Tu_n)-{\lambda}_{n+1}{\mu}F(Tu_n),\;n{\geq}0$. for solving the general variational inequalities. The sequence $\{x_n}\$ is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.149-160
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• 2018

In this paper, we prove a strong convergence result under an iterative scheme for N finite asymptotically $k_i-strictly$ pseudo-contractive mappings and a firmly nonexpansive mappings $S_r$. Then, we modify this algorithm to obtain a strong convergence result by hybrid methods. Our results extend and unify the corresponding ones in [1, 2, 3, 8]. In particular, some necessary and sufficient conditions for strong convergence under Algorithm 1.1 are obtained.

• East Asian mathematical journal
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• v.26 no.1
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• pp.59-67
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• 2010

In this paper, we suggest and analyze some three step iterative scheme for finding the common elements of the set of the solutions of the extended general variational inequalities involving three operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of suggested iterative schemes under some mild conditions. Since the extended general variational inequalities include general variational inequalities and several other classes of variational inequalities as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.297-327
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• 2019

In this paper, we introduce and study a general iterative algorithm to approximate a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. Further, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Finally, we derive some consequences from our main result. The results presented in this paper extended and unify many of the previously known results in this area.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.59-82
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• 2022

In this paper, we present a modified (improved) generalized M-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized M-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

• Nonlinear Functional Analysis and Applications
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• v.28 no.1
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• pp.135-173
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• 2023

The purpose of this paper is to introduce and study a method for solving the split equality of variational inequality and f, g-fixed point problems in reflexive real Banach spaces, where the variational inequality problems are for uniformly continuous pseudomonotone mappings and the fixed point problems are for Bregman relatively f, g-nonexpansive mappings. A strong convergence theorem is proved under some mild conditions. Finally, a numerical example is provided to demonstrate the effectiveness of the algorithm.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.813-823
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• 2013

In this paper, a new iterative algorithm involving quasi-nonexpansive mapping in Hilbert space is proposed and proved to be strongly convergent to a point which is simultaneously a fixed point of a quasi-nonexpansive mapping, a solution of an equilibrium problem and the set of solutions of a variational inequality problem. The results of the paper extend previous results, see, for instance, Takahashi and Takahashi (J Math Anal Appl 331:506-515, 2007), P.E.Maing $\acute{e}$ (Computers and Mathematics with Applications, 59: 74-79,2010) and other results in this field.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.991-1002
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• 2011

An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.