Geodesic
Continuous Approximations of Multivalued Mappings and Fixed Points
Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 212-222

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In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset $X$ of the Hilbert space $H$ which satisfies the tangential condition $F(x)\cap(x+T_X(x))\ne\varnothing$, where $T_X(x)$ is the cone tangent to the set $X$ at a point $x$. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper. 
@article{MZM_2005_78_2_a5,
     author = {B. D. Gel'man},
     title = {Continuous {Approximations} of {Multivalued} {Mappings} and {Fixed} {Points}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {212--222},
     publisher = {mathdoc},
     volume = {78},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a5/}
}
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B. D. Gel'man. Continuous Approximations of Multivalued Mappings and Fixed Points. Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 212-222. http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a5/