Let ((Omega, {mathcal {U}})) be a measurable space, X be a real separable complete generalized metric space with partial ordering ⪯ and (F: Omegatimes Xto X) be a continuous random operator such that, for each (omegainOmega), the function (F omega,cdot)) is a monotone (either order-preserving or order-reversing) operator.
Let ((Omega, {mathcal {F}})) be a measurable space, ((X,d,preceq)) be a separable complete partially ordered metric space, and (F: Omegatimes Xto X) be a continuous random operator such that, for each (omegainOmega), the function (F omega,cdot )) is a monotone (either order-preserving or order-reversing) operator.
Along this line, Li and Ok [2] explore the order-preserving of generalized metric projection operator and study the existence of maximum (minimum) solutions to (operatorname{GVI}(C,Gamma)) on Banach lattices, where the considered mappings are required to have topped (bottomed) values.
If, for every (omegainOmega), the operator (F_{omega }^{-1}(cdot)=[F omega,cdot)]^{-1}) is monotone (either order-preserving or order-reversing) and continuous, then there exists a random variable (x: Omegato X) that is a random fixed point of F (the unique random fixed point of F in the case of validity of the condition (({mathcal {H} }_{3}))).
Order-preserving encryption.
F is upper order-preserving.
Therefore, Λ is upper order-preserving.
That is, Λ is lower order-preserving.
One-to-many order-preserving encryption.
