Approximation of fixed points of a finite family of multivalued Lipschitz pseudo-contractive mappings in Banach spaces

Implicit and Explicit Algorithms

Abstract

Let $E$ be a uniformly convex Banach space and $D\subseteq E$ be

nonempty, open and convex. Let $\mathcal{P}(\overline{D})$ be the

collection of all

nonempty proximinal and bounded subsets of $\overline{D}$ and let

$T_{i}:\overline{D}\rightarrow \mathcal{P}(\overline{D}),~i=1,2,3,...,N$,

be multi-valued Lipschitz pseudo-contractive mappings such that

$F=\bigcap^{N}_{i=1}F(T_{i})\neq \emptyset$. Let $\{y_n\}$ be a

sequence generated by $y_{1}\in \overline{D}$,

$y_{n}=(1-t_n)y_{n-1}+t_{n}{z}_{n},~{z}_n\in T_{n}y_{n},~n\geq2,

T_n= T_{n~mod~N}$, where $\{t_{n}\}\subseteq (0,1)$ such that

$t_{n}\rightarrow1$~as~$n\rightarrow \infty$. Suppose that

$T_{i}(p)=\{p\}$, $\forall~p\in F$. Then $\lim_{n\rightarrow \infty}

{\rm dist}(y_{n},T_{l}y_{n})=0~\forall~l\in\{1,2,3,...,N\}$. Under

the additional condition that $T_{i_{0}}$ is hemicompact for some

$i_{0}\in \{1,2,3,...,N\}$, the sequence $\{y_{n}\}$ is shown to

converge strongly to an element of $F$. Finally, using the implicit

scheme above, an explicit scheme for a multivalued

pseudo-contractive mapping $T$ is shown to strongly converge to

fixed point of $T$.