Let \(A\) and \(B\) be nonempty closed subsets of a complete metric space \((X,d)\). A map \(T: A \cup B \rightarrow A \cup B\) is called cyclic if \(T(A) \subset B\) and \(T(B) \subset A\). A point \(x \in A \cup B\) is called a best proximity point for \(T\) if \(d(x,T(x))\) coincides with the infimum of the set \(\{ d(a,b) : a \in A,\, b \in B \}\). Several conditions on \(T\) are considered to ensure that, if \(A\) and \(B\) are convex subsets of a uniformly convex Banach space \(X\), then there is a unique best proximity point \(x\) for \(T\).