A convergence theorem of multi-step iterative scheme for nonlinear maps

Adesanmi Alao Mogbademu

Abstract: Let $K$ be a nonempty closed convex subset of a real Banach space $X$, $T:K\to K$ a nearly uniformly $L$-Lipschitzian (with sequence $\{r_n\}$) asymptotically generalized $\Phi$-hemicontractive mapping (with sequence ${k_n}\subset [1,\infty)$, $\lim_{n\to\infty} k_n=1$) such that $F(T)=\{\rho\in K:T\rho=\rho\}$. Let $\{\alpha_n\}_{n\geq 0}$, $\{\beta^k_n\}_{n\geq 0}$ be real sequences in $[0,1]$ satisfying the conditions: (i) $\sum_{n\geq 0}\alpha_n=\infty$ (ii) $\lim_{n\to\infty}\alpha_n,\beta^k_n=0,\quad k=1, 2,\ldots,p-1$. For arbitrary $x_0\in K$, let $\{x_n\}_{n\geq 0}$ be a multi-step sequence iteratively defined by \begin{align} x_{n+1}&=(1-\alpha_n)x_n+\alpha_nT^ny^1_n,\quad n\geq 0,\notag
y^k_n&=(1-\beta^k_n)x_n+\beta^k_nT^ny^{k+1}_n,\quad k=1, 2,\ldots, p-2,\notag
y^{p-1}_n&=(1-\beta^{p-1}_n)x_n+\beta^{p-1}_nT^nx_n,\quad n\geq 0, p\geq 2. \end{align} Then, $\{x_n\}_{n\geq 0}$ converges strongly to $\rho\in F(T)$. The result proved in this note significantly improve the results of Kim et al. \cite{k1}.

Keywords: Mann iteration; multi-step iteration; asymptotically generalized $\Phi$-hemi contractive mappings; nearly Lipschitzian mapping; uniformly $L$-Lipschitzian; Banach space

Classification (MSC2000): 47H10; 46A03

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