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Keywords:
pseudo-contractive mappings
pseudo-contractive mappings
Summary:
Let $X$ be a real Banach space. A multivalued operator $T$ from $K$ into $2^X$ is said to be pseudo-contractive if for every $x,y$ in $K$, $u\in T(x)$, $v\in T(y)$ and all $r>0$, $\|x-y\|\leq \|(1+r)(x-y)-r(u-v)\|$. Denote by $G(z,w)$ the set $\{u\in K :\|u-w\|\leq \|u-z\|\}$. Suppose every bounded closed and convex subset of $X$ has the fixed point property with respect to nonexpansive selfmappings. Now if $T$ is a Lipschitzian and pseudo-contractive mapping from $K$ into the family of closed and bounded subsets of $K$ so that the set $G(z,w)$ is bounded for some $z\in K$ and some $w\in T(z)$, then $T$ has a fixed point in $K$.
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