Strong convergence theorems for asymptotically k-strictly pseudocontractive maps

Abstract

Let $C$ be a nonempty closed convex subset of a real Hilbert space, $H$ and let $ T: C \rightarrow C $ be an asymptotically $k$-strictly pseudo-contractive mapping with a nonempty fixed-point set, $F(T)=\{x\in C: Tx=x\}$. Let $\{t_n\}$, $\lbrace\alpha_{n}\rbrace$~ and $\lbrace\beta_{n}\rbrace$~ be real ~sequences in $( 0, 1)$. We consider the sequence $\lbrace x_{n}\rbrace$, ge nerated from an arbitrary $ x_{1} \in C $, by either

I. \hskip 3.0cm $x_{n+1} = P_C[\left( 1-\alpha_{n} - \beta_{n}\right) x_{n}+ \beta_{n}T^{n}x_{n}], \; n\geq 1,$ or

II. $\left\{\begin{array}{ll} \nu_n=P_C((1-t_n)x_n)

x_{n+1}=(1-\alpha_n)\nu_n+\alpha_nT^n\nu_n, \; n\geq 1\end{array}\right.$

We prove that under some mild conditions on the real sequences $\lbrace\alpha_{n}\rbrace$ and $\lbrace\beta_{n}\rbrace$, the sequence $ \lbrace x_{n}\rbrace$ generated by I converges strongly to a fixed point of $T$. Furthermore, under some mild conditions on the sequences $\{t_n\}$ and $\{\alpha_n\}$, the sequence generated by II converges strongly to the least norm element of the fixed point set of $T$. Some examples are used to compare the convergence rates of these two iteration schemes. Our results compliment and extend several strong convergence results in the literature to the class of mappings considered in our work.