Some Results on Three Asymptotically Pseudocontractive Mappings

Let $K$ be a nonempty closed convex subset of a real Banach space $E$ and $S$, $T,H:K\rightarrow K$ be three asymptotically pseudocontractive mappings having bounded ranges and a sequence $\{k_{n}\}_{n\geq 0}$ $\subset \lbrack 1,\infty ),$ $\underset{n\rightarrow \infty }{\lim }k_{n}=1$ such that $p\in F(S)\cap F(T)\cap F(H)=\{x\in K:Sx=x=Tx=Hx\}$. Further let $S$ be uniformly continuous and $\{\alpha _{n}\}_{n\geq 0},\{\beta _{n}\}_{n\geq 0},\{\gamma _{n}\}_{n\geq 0}\in \lbrack 0,1]$ be sequences such that $\sum_{n\geq 0}\alpha _{n}^{2}=\infty $ and $\underset{n\rightarrow \infty }{\lim }\alpha _{n}=0=\underset{n\rightarrow \infty }{\lim }\beta _{n}.$ For arbitrary $x_{0}\in K$ let $\{x_{n}\}_{n\geq 0}$ be iteratively defined by \begin{eqnarray*} x_{n+1} &=&\left( 1-\alpha _{n}\right) x_{n}+\alpha _{n}S^{n}y_{n},\text{ } \\ y_{n} &=&\left( 1-\beta _{n}\right) x_{n}+\beta _{n}T^{n}z_{n}, \\ z_{n} &=&\left( 1-\gamma _{n}\right) x_{n}+\gamma _{n}H^{n}x_{n},\text{ }% n\geq 0.\; \end{eqnarray*}% Then $\{x_{n}\}_{n\geq 0}$ converges strongly to $p\in F(S)\cap F(T)\cap F(H) $.