The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems
نویسنده
چکیده
and Applied Analysis 3 With respect to the pointwise order u ≤ v ⇐⇒ ( ∀x ∈ Ω : u (x) ≤ v (x) ) (11) the setNL(Ω) is a Dedekind complete lattice. In particular, the supremum and infimum of a setA ⊂ NL(Ω) is given by inf A = (I ∘ S) (φ) , (12) supA = (I ∘ S) (ψ) , (13) respectively, where φ : Ω ∋ x → inf{u(x) : u ∈ A} and ψ : Ω ∋ x → sup{u(x) : u ∈ A}. Furthermore, the lattice NL(Ω) is fully distributive. That is, ∀ v ∈ NL (Ω) : ∀ A ⊂ NL (Ω) : u0 = supA ⇒ sup {inf {u, v} : u ∈ A} = inf {u0, v} . (14) A useful characterization of order bounded sets in terms of pointwise bounded sets is given as follows. If a set A ⊂ NL(Ω) satisfies ∃ B ⊂ Ω of first Baire category :
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