Geodesic
Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 625-630.

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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 $K$.
Classification : 47H04, 47H09, 47H10
Keywords: pseudo-contractive mappings 
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     author = {Morales, Claudio H.},
     title = {Multivalued pseudo-contractive mappings defined on unbounded sets in {Banach} spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {625--630},
     publisher = {mathdoc},
     volume = {33},
     number = {4},
     year = {1992},
     mrnumber = {1240184},
     zbl = {0794.47038},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1992__33_4_a7/}
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Morales, Claudio H. Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 625-630. http://geodesic.mathdoc.fr/item/CMUC_1992__33_4_a7/