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Abstract

We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.
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... Iterative techniques for approximating fixed points of nonexpansive mappings have been studied by using famous. Some variations of Mann and Ishikawa iterations for approximation of several problems can be found in [2,9,15,17,18,22,28,29,45,47,50]. ...
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In this chapter, we discuss several iterative algorithms. We present and analyze a new unified hybrid steepest-descent-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem and a common fixed point problem of uncountable family of nonexpansive mappings.
... Any uniformly convex Banach space and a CAT (0) space [3,11] are uniformly convex hyperbolic spaces (see also [5,8,10] ). Every uniformly convex hyperbolic space is strictly convex but the converse is not true in general. ...
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We establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we avoid assumptions such as smoothness of the norm, necessary in the linear case. MSC: 47H09, 46B20, 47H10, 47E10.
... The metric space X together with a convex structure W is denoted by X. For recent investigations in convex metric spaces, we refer the reader to [5][6][7][8]13]. One can consider Hadamard manifolds [4] and geodesic spaces [1,12] as nonlinear examples of a convex metric space (see also [29]). ...
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We establish strong convergence of the modified Moudafi iterative scheme to a common fixed point of an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi-nonexpansive type mapping on a uniformly convex metric space. Our results either improve or generalize the corresponding results of Kim (Arab J Math 2:279–286, 2013), Kim and Kim (Comput Math Appl 42:1565–1570, 2001) and Rhoades (J Math Anal Appl 183:118–120, 1994).
... If we set S = I in Condition (I), it becomes Condition 3 of Khan et. al. ([12], p.3) and an analogue of the Condition (A) of Maiti and Ghosh[16]in normed spaces. In this paper, we define and study an itetation process for two and three quasi-nonexpansive mappings on a nonlinear domain, namely, a uniformly convex metric space and prove its convergence to common fixed point of the mappings under weaker assumptions.The results established in this paper, in particular, hold for uniformly convex Banach spaces and CAT (0) spaces, simultaneously. ...
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The paper establishes some convergence theorems for Ishikawa type iteration processes associated with two and three quasi-nonexpansive mappings in a convex metric space.
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In this paper we prove some fixed point theorems in fuzzy metric spaces for a class of generalized nonexpansive mappings satisfying Bγ,µ condition. We introduce a type of convexity in fuzzy metric spaces with respect to an altering distance function and prove convergence results for some iteration schemes to the fixed point. The results are supported by suitable examples.
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We study Moudafi’s iterative algorithm for an α-nonexpansive mapping and a fundamentally nonexpansive mapping in the framework of a convex metric space. We prove ▵-convergence and strong convergence results for the algorithm to a common fixed point of the mappings. Our results are new and are also valid in CAT0 spaces and Banach spaces, simultaneously.
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Introduction The accurate measurement of behaviour is vitally important to many disciplines and practitioners of various kinds. While different methods have been used (such as observation, diaries, questionnaire), none are able to accurately monitor behaviour over the long term in the natural context of people’s own lives. The aim of this work was therefore to develop and test a reliable system for unobtrusively monitoring various behaviours of multiple individuals within the same household over a period of several months. Methods A commercial Real Time Location System was adapted to meet these requirements and subsequently validated in three households by monitoring various bathroom behaviours. Results The results indicate that the system is robust, can monitor behaviours over the long-term in different households and can reliably distinguish between individuals. Precision rates were high and consistent. Recall rates were less consistent across households and behaviours, although recall rates improved considerably with practice at set-up of the system. The achieved precision and recall rates were comparable to the rates observed in more controlled environments using more valid methods of ground truthing. Conclusion These initial findings indicate that the system is a valuable, flexible and robust system for monitoring behaviour in its natural environment that would allow new research questions to be addressed.
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A fixed point theorem for a generalized nonexpansive mapping is established in a convex metric space introduced by Takahashi [A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142-149]. Our theorem generalizes simultaneously the fixed point theorem of Bose and Laskar [Fixed point theorems for certain class of mappings, Jour. Math. Phy. Sci., 19 (1985), 503-509] and the well-known fixed point theorem of Goebel and Kirk [A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174] on a nonlinear domain. The fixed point obtained is approximated by averaging Krasnosel'skii iterations of the mapping. Our results substantially improve and extend several known results in uniformly convex Banach spaces and CAT (0) spaces.
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