A GENERAL ITERATIVE ALGORITHM FOR THE SOLUTION OF VARIATIONAL INEQUALITIES FOR A NONEXPANSIVE SEMIGROUP IN BANACH SPACES

Authors

  • P. Sunthrayuth DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING MONGKUT' S UNIVERSITY OF TECHNOLOGYTHONBURI (KMUTT), BANGKOK 10140, THAILAND
  • P. Kumam DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING MONGKUT'S UNIVERSITY OF TECHNOLOGY THONBURI (KMUTT), BANGKOK 10140, THAILAND

Abstract

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 $S = {T(s): 0 _ 0 < \infty}$ be a nonexpansive semigroup on $C$ such that $F(S) 6= \emptyset$ and $f : C \to C$ is a contraction mapping with coefficient $\alpha \in (0, 1)$,  $A$ a strongly positive linear bounded operator with coefficient  $\gamma > 0$.  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 ${t}, {an}$ and ${bn}$ in $(0, 1)$ and ${lt}0<t<1, {tn}$ 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.

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Published

2011-12-20

How to Cite

Sunthrayuth, P., & Kumam, P. (2011). A GENERAL ITERATIVE ALGORITHM FOR THE SOLUTION OF VARIATIONAL INEQUALITIES FOR A NONEXPANSIVE SEMIGROUP IN BANACH SPACES. Journal of Nonlinear Analysis and Optimization: Theory & Applications, 1(1), 139-150. Retrieved from http://www.math.sci.nu.ac.th/ojs302/index.php/jnao/article/view/32

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Research Article

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