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Convergence of paths for pseudo-contractive mappings in Banach spaces

Authors: Claudio H. Morales and Jong Soo Jung
Journal: Proc. Amer. Math. Soc. 128 (2000), 3411-3419
MSC (1991): Primary 47H10
Published electronically: May 18, 2000
MathSciNet review: 1707528
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Let $X$ be a real Banach space, let $K$ be a closed convex subset of $X$, and let $T$, from $K$ into $X$, be a pseudo-contractive mapping (i.e. $(\lambda-1)$ $\Vert u-v\Vert\le\Vert(\lambda I-T)(u)-(\lambda I-T)(v)\Vert$ for all $u,v\in K$and $\lambda>1)$. Suppose the space $X$ has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of $K$enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path $t\to x_t\in K$, $t\in[0,1)$, defined by the equation $x_t=tTx_t+(1-t)x_0$ is continuous and strongly converges to a fixed point of $T$ as $t\to 1^-$, provided that $T$ satisfies the weakly inward condition.

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Additional Information

Claudio H. Morales
Affiliation: Department of Mathematics, University of Alabama, Huntsville, Alabama 35899

Jong Soo Jung
Affiliation: Department of Mathematics, Dong-A University, Pusan 604-714, Korea

Keywords: Pseudo-contractive mappings, uniformly Gâteaux differentiable norm.
Received by editor(s): September 18, 1998
Received by editor(s) in revised form: January 22, 1999
Published electronically: May 18, 2000
Additional Notes: This paper was carried out while the second author was visiting the University of Alabama in Huntsville under the financial support of the LG Yonam Foundation, 1998, and he would like to thank Professor Claudio H. Morales for his hospitality in the Department of Mathematics.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society