Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings
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RESEARCH
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Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings Koji Aoyama1* and Fumiaki Kohsaka2 * Correspondence: [email protected] 1 Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba, 263-8522, Japan Full list of author information is available at the end of the article
Abstract We show that a strongly relatively nonexpansive sequence of mappings can be constructed from a given sequence of firmly nonexpansive-like mappings in a Banach space. Using this result, we study the problem of approximating common fixed points of such a sequence of mappings. MSC: Primary 47H09; secondary 47H05; 65J15 Keywords: Banach space; firmly nonexpansive-like mapping; firmly nonexpansive mapping; fixed point; mapping of type (P); proximal point algorithm; uniform convexity constant
1 Introduction The aim of the present paper is twofold. Firstly, we construct a strongly relatively nonexpansive sequence from a given sequence of firmly nonexpansive-like mappings with a common fixed point in Banach spaces. Secondly, we obtain two convergence theorems for firmly nonexpansive-like mappings in Banach spaces and discuss their applications. The class of firmly nonexpansive-like mappings (or mappings of type (P)) introduced in [] plays an important role in nonlinear analysis and optimization. In fact, the fixed point theory for such mappings can be applied to several nonlinear problems such as zero point problems for monotone operators, convex feasibility problems, convex minimization problems, equilibrium problems, and so on; see [–] and Section for more details. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space X, J the normalized duality mapping of X into X ∗ , and T : C → X a firmly nonexpansive-like mapping; see (.). The set of all fixed points of T is denoted by F(T). It is known [, Theorem .] that if C is bounded, then F(T) is nonempty. Martinet’s theorem [, Théorème ] ensures that if X is a Hilbert space and C is bounded, then the sequence {T n x} converges weakly to an element of F(T) for each x ∈ C. However, we do not know whether Martinet’s theorem holds for firmly nonexpansive-like mappings in Banach spaces. On the other hand, using the metric projections in Banach spaces, Kimura and Nakajo [, Theorems and ] recently obtained generalizations of the results due to Crombez [, Theorem ] and Brègman [, Theorem ]. ©2014Aoyama and Kohsaka; 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.
Aoyama and Kohsaka Fixed Point Theory and Applications 2014, 2014:95 http://www.fixedpointtheoryandapplications.com/content/2014/1/95
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In this paper, inspired by [], we investigate the asymptotic behavior of the following sequences {xn } and {yn } in a unifo
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