Iterative Process for a Common Fixed Point of a Finite Family of Asymptotically k-Pseudocontractive Maps

: Let $K$ be a closed convex nonempty subset of a Hilbert space $H$ and let $\{T_i\}_{i=1}^N$ be a finite family of Asymptotically k-pseudocontractive maps from $K$ into itself with $F = ∩_{i=1}^NF(T_i) not empty. Sufficient conditions for the strong convergence of the sequence of successive approximations generated by a Picard-like process to a common fixed point of the family are proved.


Introduction
Let $K$ be nonempty subset of a Hilbert space $H$.$K$ is said to be (sequentially) compact if every bounded sequence in $K$ has a subsequence that converges in $K$ and is said to be boundedly compact if every bounded subset of $K$ is compact.In finite dimensional spaces, closed subsets are boundedly compact.
Given a subset $S$ of $K$, we shall denote by $co(S)$ and $ccl(S)$ the convex hull and the closed convex hull of $S$ respectively.If $K$ is boundedly compact convex and $S$ is bounded, $co(S)$ and hence $ccl(S)$ are compact convex subsets of $K$.
A nonlinear map $ T: K \to E\ $ is said to be completely continuous if it maps bounded sets into relatively compact sets, and is said to be Lipschitzian} if $\exists \; L ≥ 0 such that ||Tx -Ty|| ≤ L||x -y|| for all x,y in K

Suggested Citation
If $L = 1$ then $T$ is called nonexpansive and if $ L < 1$ then the mapping $T$ is called a contraction.A self-map $T$ on $K$ is said to be uniformly L-Lipschitzian if there exists L ≥ 0 such that ||T^nx -T^ny|| ≤ L||x -y|| for all x,y in K, for all n in N(set of natural numbers).
The mapping $T$ with domain $D(T)$ and the range $R(T)$ in $H$ is called pseudocontractive if $for all x,y in D(T)$, If (eq:2) holds for all $x in D(T)$ and $y in F(T)$(the fixed point set of $T$), then $T$ is said to be hemicontractive.$T$ is $k$-pseudocontractive or strictly pseudocontractive in the sense of  if there exists k in (0,1) such that for all x,y in D(T), (3)  Osilike, 1993;Weng, 1993;Wong, 1974).
Browder and Petryshyn (1967) introduced the map and proved that the Mann iteration converges strongly to a fixed point of such map defined on a compact convex subset of a Hilbert space.
Our purpose in this paper is to extend the result of Browder-Petryshyn (1976) to the case of a finite family of asymptotically k-pseudontractive self-maps of a closed convex nonempty subset of a Hilbert space.
We need the following lemma in this work.

Main result Proposition 1
Let $E$ be a normed linear space and let $T:E → E$ be asymptotically k-pseudocontractive.