• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.377-392
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• 2008

Let X be a real reflexive Banach space with a uniformly $G\hat{a}teaux$ differentiable norm, C a nonempty closed convex subset of X, T : C $\rightarrow$ X a continuous pseudocontractive mapping, and A : C $\rightarrow$ C a continuous strongly pseudocontractive mapping. We show the existence of a path ${x_t}$ satisfying $x_t=tAx_t+(1- t)Tx_t$, t $\in$ (0,1) and prove that ${x_t}$ converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path ${x_t}$ defined by $x_t=tAx_t+(1-t)(2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping T.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.333-337
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• 1994

Many authors[3][4][5] constructed and examined some processes for the fixed point of strictly pseudocontractive mapping in various Banach spaces. In fact the fixed point of strictly pseudocontractive mapping is the zero of strongly accretive operators. So the same processes are used for the both circumstances. Reich[3] proved that Mann-iteration precess can be applied to approximate the zero of strongly accretive operator in uniformly smooth Banach spaces. In the above paper he asked whether the fact can be extended to other Banach spaces the duals of which are not necessarily uniformly convex. Recently Schu[4] proved it for uniformly continuous strictly pseudocontractive mappings in smooth Banach spaces. In this paper we proved that Mann-iteration process can be applied to approximate the fixed point of strictly pseudocontractive mapping in certain Banach spaces.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.465-478
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• 2006

In this paper, we introduce a new class of completely generalized strongly nonlinear quasivariational inequalities and establish its equivalence with a class of fixed point problems by using the resolvent operator technique. Utilizing this equivalence, we develop a three-step iterative scheme with errors, obtain a few existence theorems of solutions for the completely generalized non-linear strongly quasivariational inequality involving relaxed monotone, relaxed Lipschitz, strongly monotone and generalized pseudocontractive mappings and prove some convergence and stability results of the sequence generated by the three-step iterative scheme with errors. Our results include several previously known results as special cases.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.295-305
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• 2010

In this paper, we show that the modified Mann iteration with errors converges strongly to fixed point for uniformly L-Lipschitzian asymptotically $\Phi$-pseudocontractive mappings in real Banach spaces. Meanwhile, it is proved that the convergence of Mann and Ishikawa iterations is equivalent for uniformly L-Lipschitzian asymptotically $\Phi$-pseudocontractive mappings in real Banach spaces. Finally, we obtain the convergence theorems of Ishikawa iterative sequence and the modified Ishikawa iterative process with errors.

• Communications of the Korean Mathematical Society
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• v.18 no.2
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• pp.251-261
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• 2003

The purpose of this paper is to establish the necessary and sufficient conditions which ensure the strong convergence of the Ishikawa iterative schemes with errors to the unique fixed point of a $\Phi$-hemicontractive mapping defined on a nonempty convex subset of a normed linear space. The results of this paper extend substantially most of the recent results.

• East Asian mathematical journal
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• v.26 no.5
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• pp.599-605
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• 2010

In this paper, we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of an asymptotically strictly pseudocontractive mapping in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets.

• Journal of applied mathematics & informatics
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• v.31 no.3_4
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• pp.565-575
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• 2013

Let H be a real Hilbert space and let T : H ${\rightarrow}$ H be a Lipschitz pseudocontractive mapping. We introduce a modified Ishikawa iterative algorithm and prove that if $F(T)=\{x{\in}H:Tx=x\}{\neq}{\emptyset}$, then our proposed iterative algorithm converges strongly to a fixed point of T. No compactness assumption is imposed on T and no further requirement is imposed on F(T).

• 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.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.337-348
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• 2004

In this paper, we suggest and analyze Noor (three-step) iterative scheme for solving nonlinear strongly accretive operator equation Tχ = f. The results obtained in this paper represent an extension as well as refinement of previous known results.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.461-475
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• 2000

In the present paper, we first give some examples of self-mappings which are asymptoticaly nonexpansive in the intermediate, not strictly hemicontractive, but satisfy the property (H). It is then shown that the modified Mann and Ishikawa iteration processes defined by $x_{n+1}=(1-\alpha_n)x_n+\alpha_nT^nx_n\ and\ x_{n+1}=(1-\alpha_n)x_n+\alpha_nT^n[(1-\beta_n)x_n+\beta_nT^nx_n]$,respectively, converges strongly to the unique fixed point of such a self-mapping in general Banach spaces.