Well-Posedness in Holder Spaces of Elliptic Differential and Difference Equations

摘要

In the present paper the well-posedness of the elliptic differential equation -u"(t) + Au(t) = f(t)(-∞ < t < ∞) in an arbitrary Banach space E with the general positive operator in Ho lder spaces C~β (R, E_α) is established. The exact estimates in Holder norms for the solution of the problem for elliptic equations are obtained. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation is studied. The well-posedness of the these difference schemes in the difference analogy of Holder spaces C~β (R_τ, E_α) are obtained. The almost coercive inequality for solutions in C(R_τ, E) of these difference schemes is established.