Residuated functions were introduced in the theory of ordered sets as a counterpart tocontinuous functions in mathematical analysis, as functions having the property that the inverseimage of a principal down-set is a principal down-set.

We study inequalities and equations defined by residuated and residual (dually residuated)functions and we show that the problems of computing the extreme solutions to these equationsand inequalities boil down to the problems of computing the greatest post-fixed points or theleast pre-fixed points, using the well known Knaster-Tarski fixed point theorem. Our approachis very general and it includes, as special cases, many relation inequalities and equations alreadyknown in the literature, which will be considered here, too.