Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary

For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Φ≦Mu≦Ψ⇒−ϕ≦u≦ψ and an algorithm to construct these functions ϕ, ψ, Φ, Ψ. This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions Φ=Ψ we require ϕ=ψ∈C ₀[0, 1] ∩C ₂([0, 1]−T) whereT is some finite set), (Mψ) (t)≧Ψ(t), (t∈[0, 1]−T) and certain additional conditions for eacht∈T. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds-ϕ and ψ.