On Some Metric Spaces, Their Properties And Applications

In 1956 Aronszajn and Panitchpakdi found that the necessary and sufficient condition which guarantees the extension of a uniformly continuous mapping T (with a subadditive modulus of continuity) between two metric spaces is that the range of T has to be hyperconvex. Since then these spaces have played an important role among metric spaces. Also the Kuratowski measure of noncompactness introduced in 1930 has important applications in various areas of mathematics, for instance, in nonlinear differential equations in Banach spaces and fixed point theory. In this thesis, we will define some new metrics on the plane R2 and we will study the properties of the space R2 with these metrics from the point of view of hyperconvexity, measure of noncompactness and completeness. We will also give the solution of an open problem concerning hyperconvexity of a space via the measure of noncompactness of bounded sets in such a metric space

On Some Properties of Hyperconvex Spaces

<p/> <p>We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete <inline-formula> <graphic file="1687-1812-2010-213812-i1.gif"/></inline-formula>-trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for <inline-formula> <graphic file="1687-1812-2010-213812-i2.gif"/></inline-formula>-trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.</p