Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding (a0,x0)∈A×X(a0,x0)∈A×X such that d(a0,x0)=inf{d(a,x):a∈A,x∈X}d(a0,x0)=inf{d(a,x):a∈A,x∈X} (resp. d(a0,x0)=sup{d(a,x):a∈A,x∈X}d(a0,x0)=sup{d(a,x):a∈A,x∈X}). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.Dirección General de Enseñanza SuperiorJunta de AntalucíaThe Sectoral Operational Programme Human Resources Developmen

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We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction wrt orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces.

We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.