Fixed Points and Best Approximation in Menger Convex Metric Spaces
نویسنده
چکیده
Nonexpansive mappings have been studied extensively in recent years by many authors.The first fixed point theorem of a general nature for nonlinear nonexpansive mappings in noncompact setting were proved independently by Browder [8] and Gohde [12]. Later on, Kirk [17] proved the same results under slightly weaker assumptions. A fundamental problem in fixed point theory of nonexpansive mappings is to find conditions under which a set has the fixed point property for these mappings. It is intimately connected with differential equations and with the geometry of the Banach spaces.The interplay between the geometry of Banach spaces and fixed point theory has been very strong and fruitful. In particular, geometrical properties play key role in metric fixed point problems, see, for example, [2–5, 7, 19, 20] and references mentioned therein. These results use mainly convexity hypothesis and geometric properties of Banach spaces. These results were starting point for a new mathematical feild : the application of the geometric theory of Banach spaces to fixed point theory. The problem of proximity of subsets in normed spaces has been studied extensily, see, for example, [13, 15, 19, 20]. Aronszajn and Panitchapakdi [2] and Menger [18] defined the convexity structure on metric spaces through closed ball and studied their properties. Khalil [16] further studied existence of fixed points and best approximation in these convex metric spaces. This paper addresses the convexity structure of a metric space, using geometric
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