VISCOSITY APPROXIMATION METHOD FOR NONEXPANSIVE NONSELF-MAPPING AND VARIATIONAL INEQUALITY
Zhenhua He, Can Chen, Feng Gu · The Journal of Nonlinear Sciences and Applications · 2008
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 \(F(T) = \{x \in K, Tx = x\} eq\emptyset\), 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.