*(English)*Zbl 1135.47054

Let $E$ be a uniformly convex Banach space, $D$ be a nonempty closed convex subset of $E$ and $T:D\to K\left(E\right)$ be a multimap, where $K\left(E\right)$ is the family of all nonempty compact subsets of $E$. If we denote

then ${P}_{T}:D\to K\left(E\right)$ is nonempty and compact for every $x\in D$.

The first main result of the paper (Theorem 3.1) shows that, if $D$ is a nonexpansive retract of $E$ and if, for each $u\in D$ and $t\in (0,1)$, the multivalued contraction ${S}_{t}$ defined by ${S}_{t}x=t{P}_{T}x+(1-t)u$ has a fixed point ${x}_{t}\in D$, then $T$ has a fixed point if and only if $\left\{{x}_{t}\right\}$ remains bounded as $t\to 1$. Moreover, in this case, $\left\{{x}_{t}\right\}$ converges strongly to a fixed point of $T$ as $t\to 1$.

A similar result (Theorem 3.2) is then obtained for nonself-multimaps satisfying the inwardness condition in the case of reflexive Banach spaces having a uniformly Gâteaux differentiable norm. Several corollaries of these results are also presented.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H04 | Set-valued operators |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

47H09 | Mappings defined by “shrinking” properties |