Abstract
In the present paper the well-posedness of the elliptic differential equation
in an arbitrary Banach space E with the general positive operator in Hö lder spaces \(C^{\beta }(\mathbb {R},E_{\alpha })\) is established. The exact estimates in Hölder 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 are studied. The well-posedness of the these difference schemes in the difference analogy of Hölder spaces \(C^{\beta }(\mathbb {R}_{\tau }, E_{\alpha })\) are obtained. The almost coercive inequality for solutions in \(C(\mathbb {R}_{\tau },E)\) of these difference schemes is established.
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Ashyralyev, A. (2015). Well-Posedness in Hölder Spaces of Elliptic Differential and Difference Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science(), vol 9045. Springer, Cham. https://doi.org/10.1007/978-3-319-20239-6_3
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DOI: https://doi.org/10.1007/978-3-319-20239-6_3
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