Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces
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RESEARCH
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Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces Qianglian Huang1,2 , Lanping Zhu1,2* and Gang Li1 *
Correspondence: [email protected] 1 College of Mathematics, Yangzhou University, Yangzhou 225002, China Full list of author information is available at the end of the article
Abstract In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area. MSC: 47H09; 47H10; 47H20 Keywords: asymptotically nonexpansive type mapping; left amenable semigroup; reversible semigroup; fixed point
1 Introduction Let E be a Banach space and C a nonempty bounded closed convex subset of E. A mapping T on C is said to be nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C. A well-known result of Browder [] asserts that if E is uniformly convex, then every nonexpansive mapping on C has a fixed point. Kirk [], Belluce and Kirk [] extended this result to the case that X has a normal structure or Opial’s property. Goebel and Kirk [] proved that if E is a uniformly convex Banach space, then every asymptotically nonexpansive mapping on C has a fixed point. As is well known, not every semigroup of nonexpansive mappings on a subset of a Banach space has a fixed point []. The existence and convergence of fixed points for semigroups of various mappings have been studied extensively [–]. Recently, Suzuki and Takahashi [], Takahashi and Zembayashi [], Zhu and Li [] proved the existence theorems of fixed points for semigroups = {T(t) : t ≥ } of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings, respectively. For instance, in [], Takahashi and Zembayashi proved the following theorem: Theorem . [] Let C be a nonempty compact convex subset of a Banach space E and = {T(t) : t ≥ } be a semigroup of asymptotically nonexpansive mappings on C, then the set of common fixed points F() of is nonempty. Many results are known in the case that the semigroup G is commutative, amenable or reversible [–]. In the case of an amenable semigroup, the first result was established by Takahashi [] where he proved: © 2012 Huang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Huang et al. Fixed Point Theory and Applications 2012, 2012:116 http://www.fixedpointtheoryandapplications.com/content/2012/1/116
Theorem . [] Let C be a nonempty compact convex subset of a Banach space E. Let = {T(t) : t ∈ G} be an amenable semigroup of nonexpansive mappings on C. Then C contains a common fixed point for . Theorem . was proved for a commutative semigroup by DeMarr []. Later in [], Lau showed that the fixed point property
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