Geodesic
VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY
HE, ZHENHUA1 ; CHEN , CAN 1 ; GU, FENG2
1 Department of Mathematics, Honghe university, Mengzi, Yunnan, 661100, China.
2 Department of Mathematics, Hangzhou normal university, Zhejiang, 310036, China.
Journal of nonlinear sciences and its applications, Tome 1 (2008) no. 3, p. 169-178.

Voir la notice de l'article provenant de la source International Scientific Research Publications

Let $E$ be a real reflexive Banach space which has uniformly Gâteaux differentiable norm. Let $K$ be aclosed convex subset of $E$ which is also a sunny nonexpansive retract of $E$, and $T : K \rightarrow E$ be nonexpansive mapping satisfying the weakly inward condition and $F(T) = \{x \in K, Tx = x\} \neq\emptyset$, and $f : K \rightarrow K$ be a contractive mapping. Suppose that $x_0 \in K,\quad \{x_n\}$ is defined by
$ \begin{cases} x_{n+1} = \alpha_nf(x_n) + (1 - \alpha_n)((1 - \delta)x_n + \delta y_n)\\ y_n = P(\beta_nx_n + (1 - \beta_n)Tx_n),\quad n \geq 0, \end{cases} $
where $\delta \in (0; 1), \alpha_n, \beta_n \in [0; 1], P$ is sunny nonexpansive retractive from $E$ into $K$. Under appropriate conditions, it is shown that $\{x_n\}$ converges strongly to a fixed point $T$ and the fixed point solutes some variational inequalities. The results in this paper extend and improve the corresponding results of [2] and some others.
DOI : 10.22436/jnsa.001.03.05
Classification : 47H09, 47H10, 47J05, 54H25
Keywords: Strong convergence, Nonexpansive nonself-mapping, Viscosity approximation method, Uniformly Gâteaux differentiable norm, Variational inequality.

HE, ZHENHUA 1 ; CHEN , CAN  1 ; GU, FENG 2

1 Department of Mathematics, Honghe university, Mengzi, Yunnan, 661100, China.
2 Department of Mathematics, Hangzhou normal university, Zhejiang, 310036, China. 
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HE, ZHENHUA; CHEN , CAN ; GU, FENG. VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY. Journal of nonlinear sciences and its applications, Tome 1 (2008) no. 3, p. 169-178. doi : 10.22436/jnsa.001.03.05. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.001.03.05/

[1] S. S. Chang Some problems and results in the study of nonlinear analysis, Nonlinear Anal., Volume 30 (1997), pp. 4197-4208 | DOI

[2] Song, Y.; Chen, R. Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Apple., Volume 321 (2006), pp. 316-326 | DOI

[3] Tomonari Suzuki Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,, Fixed Point Theory and Applications, Volume 1 (2005), pp. 103-123 | Zbl | DOI

[4] W. Takahashi Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000

[5] Takahashi, W.; Y. Ueda On Reich' s strong convergence for resolvents of accretive operators, J. Math. Anal. Appl. , Volume 104 (1984), pp. 546-553 | DOI | Zbl

[6] Xu, H. K. Viscosity approximation methods for nonexpansive mappings , J. Math. Anal. Apple., Volume 298 (2004), pp. 279-291 | DOI

[7] Xu, H.-K. Approximating curves of nonexpansive nonself-mappings in Banach spaces , in: Mathematical Analysis, C.R.Acad. Sci. Paris, Volume 325 (1997), pp. 151-156 | Zbl | DOI

[8] Hong-Kun Xu Iterative algorithms for nonlinear operators, J. London. Math. Soc., Volume 2 (2002), pp. 240-256 | DOI

[9] Zegeye, Habtu; Naseer Shahzad Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal., Volume 66 (2007), pp. 1161-1169 | Zbl | DOI

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