ACTA issues

Iterative approximation of fixed points for $\phi $-hemicontractive operators in arbitrary Banach spaces

Zeqing Liu, Shin Min Kang

Acta Sci. Math. (Szeged) 67:3-4(2001), 821-831

Abstract. In this paper, the following result is shown: Let $X$ be an arbitrary real Banach space and $K$ a nonempty closed convex subset of $X$ and $T\colon K\to K$ a Lipschitz $\phi$-hemicontractive operator. Define the sequence $\{x_n\}_{n=0}^\infty$ iteratively by $x_0, u_0, v_0 \in K$, $$\eqalign{ x_{n+1}&= a_nx_n + b_nTy_n+ c_nu_n, \quad n\ge0,\cr y_{n}&= a'_nx_n + b'_nTx_n+ c'_nv_n\quad n\ge0, }$$ where $\{u_n\}_{n=0}^\infty$, $\{v_n\}_{n=0}^\infty$ are arbitrary bounded sequences in $K$; $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a'_n\}$, $\{b'_n\}$ and $\{c'_n\}$ are real sequences in $[0,1]$ satisfying the following conditions: \item{(i)} $a_n+ b_n+ c_n= a'_n+ b'_n + c'_n=1,$ $n\ge0$; \item{(ii)} $\sum_{n=0}^\infty c_n < \infty, \sum_{n=0}^\infty b_nb'_n < \infty$, $\sum_{n=0}^\infty b_nc'_n < \infty, \sum_{n=0}^\infty b_n^2 < \infty,$ \item{(iii)} $\sum_{n=0}^\infty b_n =+\infty.$ \par\noindent Then the sequence $\{x_n\}_{n=0}^\infty$ converges strongly to the unique fixed point of $T$. Our result extends, improves and unifies the corresponding results in [2]-[8], [10]-[15], [20], [21], [24], [27].

AMS Subject Classification (1991): 47H17, 47H15, 47H05

Keyword(s): The Ishikawa iteration sequence with errors, the Mann iteration sequence with errors, strongly pseudocontractive operator, \phi, \phi, -strongly pseudocontractive operator.-henicontractive operator, real Banach space

Received May 3, 2000, and in revised form March 5, 2001. (Registered under 2820/2009.)