Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions ξ : A → R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,ξ) : R × E → E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to ξ. Then there exists a solution u: [a,b] × [0,T] → E of the problem ut(x,t) − uxx(x,t) = F(x,t,u(x,t)) ((x,t) ∈ R), u(x,t) = 0 ((x,t) ∈ R ∖ R).