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 theapproximate solution of abstract nonlocal boundary value problem $-d^{2}u(t)/d{t}^2+\text{sign}(t)Au(t)=g(t)$, $(0\le t\le 1)$, $du(t)/dt+\text{sign}(t)Au(t)=f(t)$, $(-1\le t\le 0)$, $u(0+)=u(0-),u^{\prime }(0+)=u^{\prime }(0-),$ $\text{and\hspace{0.17em}}u(1)=u(-1)+\mu$ 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 normsfor the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.