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Convergence of paths for pseudo-contractive mappings in Banach spaces. (English) Zbl 0970.47039
An operator $T\colon K\to X$, where $K$ is a subset of a real Banach space $X$, is said to be pseudo-contractive if $$(\lambda-1)\|u-v\|\le \|(\lambda I-T)u-(\lambda I-T)v\|$$ 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 $j\in J(u-v)=\{j\in X^*:\langle u-v,j\rangle=\|u-v\|^2=\|j\|^2\}$ such that $$\langle Tu-Tv,j\rangle\le\|u-v\|^2.$$ Furthermore, $T$ is said to satisfy the weakly inward condition if $Tx\in\text{ cl}(I_K(x))$ for all $x\in K$, where $I_K(x):=\{x+\lambda(u-x):u\in K,\lambda\ge 1\}$ 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 $$x_t=tTx_t+(1-t)x_0.$$ 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 $E=\{x\in K:Tx=\lambda x+(1-\lambda)x_0\text{ for some }\lambda>1\}$ 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.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
46B03Isomorphic theory (including renorming) of Banach spaces
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