Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

Abstract

A first order of accuracy difference scheme for the approximate solution of abstract nonlocal boundary value problem −𝑑2𝑢(𝑡)/𝑑𝑡2+sign(𝑡)𝐴𝑢(𝑡)=𝑔(𝑡), (0≤𝑡≤1), 𝑑𝑢(𝑡)/𝑑𝑡+sign(𝑡)𝐴𝑢(𝑡)=𝑓(𝑡), (−1≤𝑡≤0), 𝑢(0+)=𝑢(0−),𝑢(0+)=𝑢(0−),and𝑢(1)=𝑢(−1)+𝜇 for differential equations in a Hilbert space 𝐻 with a self-adjoint positive definite operator A is considered. The well-posedness of this difference scheme in Hölder spaces without a weight is established. Moreover, as applications, coercivity estimates in Hölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.