• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.885-894
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• 2009

In this paper, we consider a hybrid projection algorithm for a pair of inverse-strongly monotone mappings and a quasi-$\phi4-nonexpansive mapping. Strong convergence theorems are established in the framework of Banach spaces.

• East Asian mathematical journal
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• v.27 no.1
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• pp.1-9
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• 2011

In this paper, we consider an iterative scheme for finding a common element of the set of fixed points of a asymptotically quasi nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a asymptotically quasi-nonexpansive mapping and strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a asymptotically quasi-nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.

• East Asian mathematical journal
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• v.26 no.5
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• pp.627-639
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• 2010

In this paper, we introduce a new iterative sequence finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a Banach space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly-monotone mapping, the fixed point problem and the classical variational inequality problem. Our results improve and extend the corresponding results announced by many others.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.621-640
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• 2014

In this paper, we introduce a new approximating method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solution of system variational inequalities for finite family of inverse strongly monotone mappings and strictly pseudo-contractive of Browder-Petryshyn type mappings. We show that the sequence converges strongly to a common element the above two sets under some parameter controling conditions. Our results improve and extend the results announced by many others.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.13-30
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• 2010

Let C be a nonempty closed convex subset of a real Hilbert space H. Consider the following iterative algorithm given by $x_0\;{\in}\;C$ arbitrarily chosen, $x_{n+1}\;=\;{\alpha}_n{\gamma}f(W_nx_n)+{\beta}_nx_n+((1-{\beta}_n)I-{\alpha}_nA)W_nP_C(I-s_nB)x_n$, ${\forall}_n\;{\geq}\;0$, where $\gamma$ > 0, B : C $\rightarrow$ H is a $\beta$-inverse-strongly monotone mapping, f is a contraction of H into itself with a coefficient $\alpha$ (0 < $\alpha$ < 1), $P_C$ is a projection of H onto C, A is a strongly positive linear bounded operator on H and $W_n$ is the W-mapping generated by a finite family of nonexpansive mappings $T_1$, $T_2$, ${\ldots}$, $T_N$ and {$\lambda_{n,1}$}, {$\lambda_{n,2}$}, ${\ldots}$, {$\lambda_{n,N}$}. Nonexpansivity of each $T_i$ ensures the nonexpansivity of $W_n$. We prove that the sequence {$x_n$} generated by the above iterative algorithm converges strongly to a common fixed point $q\;{\in}\;F$ := $\bigcap^N_{i=1}F(T_i)\;\bigcap\;VI(C,\;B)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)q,\;p\;-\;q{\rangle}\;{\leq}\;0$ for all $p\;{\in}\;F$. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.51-65
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• 2011

In this paper, a composite iterative process is introduced for a generalized equilibrium problem and a pair of nonexpansive mappings. It is proved that the sequence generated in the purposed composite iterative process converges strongly to a common element of the solution set of a generalized equilibrium problem and of the common xed point of a pair of nonexpansive mappings.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.373-388
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• 2015

In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.

• East Asian mathematical journal
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• v.26 no.1
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• pp.25-38
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• 2010

In this paper, we introduce a hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of common mixed points of finitely many nonexpansive mappings and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. We show that the iterative sequences converge strongly to a common element of the three sets. The results extended and improved the corresponding results of L.-C.Ceng and J.-C.Yao.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.849-875
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• 2018

The purpose of this paper is to present an operator method of regularization for the problem of finding a solution of a system of nonlinear ill-posed equations with a monotone hemicontinuous mapping and N inverse-strongly monotone mappings in Banach spaces. A regularization parameter choice is given and convergence rate of the regularized solutions is estimated. We also give the convergence and convergence rate for regularized solutions in connection with the finite-dimensional approximation. An iterative regularization method of zero order in a real Hilbert space and two examples of numerical expressions are also given to illustrate the effectiveness of the proposed methods.

• Korean Journal of Mathematics
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• v.25 no.4
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• pp.469-481
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• 2017

In this paper, we established a general nonconvex split variational inequality problem, this is, an extension of general convex split variational inequality problems in two different Hilbert spaces. By using the concepts of prox-regularity, we proved the convergence of the iterative schemes for the general nonconvex split variational inequality problems. Further, we also discussed the iterative method for the general convex split variational inequality problems.