Convergence of paths for pseudo-contractive mappings in Banach spaces
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- by Claudio H. Morales and Jong Soo Jung PDF
- Proc. Amer. Math. Soc. 128 (2000), 3411-3419 Request permission
Abstract:
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. $(\lambda -1)$ $\|u-v\|\le \|(\lambda I-T)(u)-(\lambda I-T)(v)\|$ 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.References
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Additional Information
- Claudio H. Morales
- Affiliation: Department of Mathematics, University of Alabama, Huntsville, Alabama 35899
- Email: morales@math.uah.edu
- Jong Soo Jung
- Affiliation: Department of Mathematics, Dong-A University, Pusan 604-714, Korea
- Email: jungjs@mail.donga.ac.kr
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3411-3419
- MSC (1991): Primary 47H10
- DOI: https://doi.org/10.1090/S0002-9939-00-05573-8
- MathSciNet review: 1707528