A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces
Pitipong Sunthrayuth, Poom Kumam · 2010
Let $X$ be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, $C$ a nonempty bounded closed convex subset of $X$. Let $S = {T(s): 0 _ 0 0$. ????????????????????????????????We prove that the sequences ${x_t}$ and ${x_n}$ are generated by the following iterative algorithms, respectively $$x_t ??= t \gamma f(x_t) + (I-tA) \frac{1}{\lambda_t}\int_0^{\lambda_t}T(s)x_t ds$$ And $$x_{n+1} = \alpha_n \gamma f(x_n) + \beta_n x_n + ((1-\beta_n)I - \alpha_n A ) \frac{1}{t_n} \int^{t_n}_0 T(s)x_n ds$$ where ${t}, { a n}$ and ${ b n}$ in $(0, 1)$ and ${ l t}0<t<1, {tn}$ are positive real divergent sequences, converging strongly to a common fixed point $x^* \in F(S)$, which solves variational inequality $$\langle (\gamma f-A)x^*, J(x-x^*) \rangle \leq 0$$ for $x \in F(S).$ Our results presented in this paper extend and improve the corresponding results announced by many others.