James type constants and fixed points for multivalued nonexpansive mappings
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James type constants and fixed points for multivalued nonexpansive mappings Zhan-fei Zuo1 Accepted: 13 October 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract In this paper, we give some sufficient conditions for the Domínguez–Lorenzo condition in terms of the James type constant J X ,t (τ ), the coefficient of weak orthogonality μ(X ) and the Domínguez Benavides coefficient R(1, X ), which imply the existence of fixed points for multivalued nonexpansive mappings. Our results extend some well known results in the recent literature. Keywords James type constant · Coefficient of weak orthogonality · Domínguez Benavides coefficient · Multivalued nonexpansive mapping · Fixed point Mathematics Subject Classification Primary 47H10 · Secondary 46B20
1 Introduction The theory of fixed points is one of the most powerful tools in modern mathematics. In particular, the fixed point theory for multivalued mappings has many useful applications in applied sciences, such as game theory and mathematical economics. In 1969, Nadler [13] extended the Banach Contraction Principle to multivalued contractive mappings in complete metric spaces. Since then, many researchers studied the possibility of extending some classical fixed point theorems for singlevalued nonexpansive mappings to the setting of multivalued nonexpansive mappings ([14,16]). One of the most celebrated results about multivalued mappings was given by Domínguez Benavides and Lorenzo in [4]. They obtained a relationship between the Chebyshev radius of the asymptotic center of a bounded sequence and the modulus of noncompactness, which gives an affirmative answer to the problem in [19] by proving that every nonempty compact and convex valued nonexpansive self-mapping has a fixed point, whenever Banach space is nearly uniformly convex. However, the fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of singlevalued nonexpansive
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Zhan-fei Zuo [email protected] Department of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, People’s Republic of China
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mappings, many questions remain open, for instance, the possibility of extending the wellknown Kirk’s theorem [10], that is, do Banach spaces with normal structure have the fixed point property (FPP) for multivalued nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for singlevalued mappings ([7,15,17]). Thus, it is natural to study if those properties imply the fixed point property for multivalued mappings. In 2006, Dhompongsa et al. [2] introduced the (DL)condition which imply the FPP for multivalued nonexpansive mappings. Since then, a possible approach to the above problem is to look for some geometric conditions in term of some geometric constants which imply the (DL)-condition. Both the James type constant and the von Neumann–Jordan type constant play an important role in the description of various geometric structures
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