# VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY

Volume 1, Issue 3, pp 169-178
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### Authors

ZHENHUA HE - Department of Mathematics, Honghe university, Mengzi, Yunnan, 661100, China.. CAN CHEN - Department of Mathematics, Honghe university, Mengzi, Yunnan, 661100, China.. FENG GU - Department of Mathematics, Hangzhou normal university, Zhejiang, 310036, China..

### Abstract

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.

### Share and Cite

##### ISRP Style

ZHENHUA HE, CAN CHEN, FENG GU, VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY, Journal of Nonlinear Sciences and Applications, 1 (2008), no. 3, 169-178

##### AMA Style

HE ZHENHUA, CHEN CAN, GU FENG, VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY. J. Nonlinear Sci. Appl. (2008); 1(3):169-178

##### Chicago/Turabian Style

HE, ZHENHUA, CHEN , CAN, GU, FENG. "VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY." Journal of Nonlinear Sciences and Applications, 1, no. 3 (2008): 169-178

### Keywords

• Strong convergence
• Nonexpansive nonself-mapping
• Viscosity approximation method
• Uniformly Gâteaux differentiable norm
• Variational inequality.

•  47H09
•  47H10
•  47J05
•  54H25

### References

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