## 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 $(\lambda-1)\|u-v\|\leq \|(\lambda I-T)u-(\lambda I-T)v\|$ 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 $\langle Tu-Tv,j\rangle\leq\|u-v\|^2.$ Furthermore, $$T$$ is said to satisfy the weakly inward condition if $$Tx\in\text{ cl}(I_K(x))$$ for all $$x\in K$$, where $$I_K(x):=\{x+\lambda(u-x):u\in K,\lambda\geq 1\}$$ 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.

### MSC:

 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46B03 Isomorphic theory (including renorming) of Banach spaces
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### References:

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