On a Class of Inclusions in Ordered Spaces

Let W be a non-empty set, X an ordered topological space, L : W → X a single-valued operator and N : W → 2 \ {∅} a set-valued operator. Under approximate assumptions on monotonicity of L and N we prove existence results for inclusions Lx ∈ Nx. An application of the obtained results to implicit elliptic equations of the form Lu = f(x, u, Lu) is given.