Fixed points for extrema of contractions
It is assumed that X is a complete metric space which is partially ordered by \(\leq\). The required compatibility between the metric and the partial order assumes that (X,\(\leq)\) is a countably Dedekind complete lattice with \(\leq\) closed in X \((R=\{(x,y)\in X\times X:\) \(x\leq y\}\) is closed in \(X\times X)\), and d is assumed to be order preserving (x\(\leq y\) and \(y\leq z\) implies d(x,y),d(y,z)\(\leq d(x,z))\). The author consider families of order preserving contraction mappings. It is shown that if an extremum (infimum or supremum) of the contractions exists, then the fixed point of the extremum contraction is less than or equal to the extremum of fixed points of the individual contractions, and sufficient conditions are given for equality to hold. It is also shown that given suitable starting points in the space, the fixed point of the extremum can be monotonically approximated from above and below, thus yielding an algorithm for approximating this fixed point with a prescribed tolerance. The authors conclude by showing how these results may be applied to Markov decision models and families of integral equations.
families of order preserving contraction mappings
scientific article; zbMATH DE number 4180161