| MSC: | 47H09 | Nonexpansive mappings, and their generalizations (ultimately compact mappings, measures of noncompactness and condensing mappings, $A$-proper mappings, $K$-set contractions, etc.) |
| 47H10 | Fixed-point theorems [See also 54H25, 55M20, 58C30] |
Abstract:
Let $E$ be a uniformly smooth real
Banach space and $T:E\rightarrow E$ be generalized Lipschitz
$\Phi$-accretive mapping with $\Phi(r)\rightarrow +\infty$ as
$r\rightarrow +\infty$. Let$\left\{{a_n}\right\}$,
$\left\{{b_n}\right\}$, $\left\{{c_n}\right\}$,
$\left\{{a_n^\prime}\right\}$, $\left\{{b_n^\prime}\right\}$,
$\left\{{c_n^\prime}\right\}$ be six real sequences in $[0,1]$
satisfying the following conditions:
(i)$a_n+b_n+c_n=a_n^\prime+b_n^\prime+c_n^\prime=1$,
(ii)$\lim\limits_{n\rightarrow \infty}b_n=\lim\limits_{n\rightarrow
\infty}b_n^\prime =\lim\limits_{n\rightarrow \infty}c_n^\prime=0$,
(iii)$\sum\limits_{n=0}^{\infty}b_n=\infty$, (iv)$c_n=o(b_n)$. For
arbitrary $x_0\in E$, define the Ishikawa iterative process with
errors $\left\{{x_n}\right\}_{n=0}^\infty$ by (ISE): $y_n=a_n^\prime
x_n+b_n^\prime Sx_n+c_n^\prime v_n,
x_{n+1}=a_n x_n+b_n Sy_n+c_n u_n, n\geq 0$. where $S:E\rightarrow E$ is defined by $Sx=f+x-Tx, f\in E, \forall x\in
E$. Assume that the equation $Tx=f$ has solution and $\left\{{u_n}\right\}_{n=0}^\infty,
\left\{{v_n}\right\}_{n=0}^\infty$ are arbitrary
two bounded sequences in $E$. Then the sequence $\left\{{x_n}\right\}_{n=0}^\infty$
converges strongly to the unique solution of the equation $Tx=f$. A related
result deals with approximation of fixed point of generalized
Lipschitz $\Phi$-pseudocontractive mapping.
Notes: Project supported by the National Science Foundation of China and Shijiazhuang Railway College Sciences Foundation.