*(English)*Zbl 0970.47039

An operator $T:K\to X$, where $K$ is a subset of a real Banach space $X$, is said to be pseudo-contractive if

for all $u,v\in K$ and $\lambda >1$. This generalises the notion of nonexpansive mapping, and is equivalent to the following: for all $u,v\in K$, there exists $j\in J(u-v)=\{j\in {X}^{*}:\langle u-v,j\rangle =\parallel u-v{\parallel}^{2}=\parallel j{\parallel}^{2}\}$ such that

Furthermore, $T$ is said to satisfy the weakly inward condition if $Tx\in \phantom{\rule{4.pt}{0ex}}\text{cl}\left({I}_{K}\left(x\right)\right)$ for all $x\in K$, where ${I}_{K}\left(x\right):=\{x+\lambda (u-x):u\in K,\lambda \ge 1\}$ is the inward set of $x$. If $T$ satisfies these hypotheses and $K$ is closed and convex, it is shown that for each ${x}_{0}\in K$, there exists a unique path $t\mapsto {x}_{t}$, $t\in [0,1)$, satisfying

Then the main result of the paper asserts that, if $X$ has a uniformly Gâteaux differentiable norm, and every closed bounded convex subset of $K$ has the fixed point property for nonexpansive self-mappings, and the set $E=\{x\in K:Tx=\lambda x+(1-\lambda ){x}_{0}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\lambda >1\}$ is bounded, then the path defined above converges strongly, as $t\to {1}^{-}$, to a fixed point of $T$.

The proof is then adapted to obtain the same conclusion under modified hypotheses.

##### MSC:

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

47H09 | Mappings defined by “shrinking” properties |

46B03 | Isomorphic theory (including renorming) of Banach spaces |