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

A first order of accuracy difference scheme for the approximate solution of abstract nonlocal boundary value problem − ð ‘‘ 2 ð ‘¢ ( ð ‘¡ ) / ð ‘‘ ð ‘¡ 2 + s i g n ( ð ‘¡ ) ð ´ ð ‘¢ ( ð ‘¡ ) = ð ‘” ( ð ‘¡ ) , ( 0 ≤ ð ‘¡ ≤ 1 ) , ð ‘‘ ð ‘¢ ( ð ‘¡ ) / ð ‘‘ ð ‘¡ + s i g n ( ð ‘¡ ) ð ´ ð ‘¢ ( ð ‘¡ ) = ð ‘“ ( ð ‘¡ ) , ( − 1 ≤ ð ‘¡ ≤ 0 ) , ð ‘¢ ( 0 + ) = ð ‘¢ ( 0 − ) , ð ‘¢ î…ž ( 0 + ) = ð ‘¢ î…ž ( 0 − ) , a n d ð ‘¢ ( 1 ) = ð ‘¢ ( − 1 ) + 𠜇 for differential equations in a Hilbert space ð » 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.