New fixed point results in partial quasi-metric spaces

In 1970, D.S. Scott gave applications of Kleene’s fixed point theorem to describe the meaning of recursive denotational specifications in programming languages. Later on, in 1994, S.G. Matthews and, in 1995, M.P. Schellekens gave quantitative counterparts of the Kleene fixed point theorem which allowed to apply partial metric and quasi-metric fixed point techniques to denotational semantics and asymptotic complexity analysis of algorithms in the spirit of Scott. Recently, in 2005, J.J. Nieto and R. Rodríguez-López made an in-depth study of how to reconcile order-theoretic and metric fixed point techniques in the classical metric case with the aim of providing the existence and uniqueness of solutions to first-order differential equations admitting only the existence of a lower solution. Motivated by the aforesaid fixed point results we prove a partial quasi-metric version, when the specialization order is under consideration, of the fixed point results of Nieto and Rodríguez-López in such a way that the results of Matthews and Schellekens can be retrieved as a particular case.