Ishikawa iterative process with errors for generalized Lipschitzian and Phi-accretive mappings in uniformly smooth Banach spaces

Xue Zhiqun

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.