Existence and uniqueness of best proximity points in geodesic metric spaces

A mapping T:A∪B→A∪BT:A∪B→A∪B such that T(A)⊆BT(A)⊆B and T(B)⊆AT(B)⊆A is called a cyclic mapping. A best proximity point xx for such a mapping TT is a point such that d(x,Tx)=d(x,Tx)= dist(A,B)(A,B). In this work we provide different existence and uniqueness results of best proximity points in both Banach and geodesic metric spaces. We improve and extend some results on this recent theory and give an affirmative partial answer to a recently posed question by Eldred and Veeramani in [A.A. Eldred, P. Veeramani Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006) 1001–1006].