Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces

We can see from (p) and (p) that p(x, y) =  implies x = y. However, the converse is not necessarily true. A typical example of this situation is provided by the partial metric space (R,pmax), where the function pmax : R+ × R+ → R+ is defined by pmax(x, y) =max{x, y} for all x, y ∈ R+. Other examples of partial metric spaces which are interesting from a computational point of view may be found in [] and []. Following [], each partial metric p on X generates a T topology τ (p) on X, whose base is a family of open p-balls: { Bp(x, ε) : x ∈ X, ε >  } ,