Convergence of paths for pseudo-contractive mappings in Banach spaces. (English) Zbl 0970.47039

Morales, Claudio H.; Jung, Jong Soo

Proc. Am. Math. Soc. 128, No. 11, 3411-3419 (2000).

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.

Reviewer: Line Baribeau (Quebec)

MSC:

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

Keywords:

pseudo-contractive mapping; uniformly Gâteaux differentiable norm; inward set; fixed point property for nonexpansive self-mappings

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