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

  • Allaberen AshyralyevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

In the present paper the well-posedness of the elliptic differential equation
$$\begin{aligned} -u^{\prime \prime }(t)+Au(t)=f(t)(-\infty <t<\infty ) \end{aligned}$$
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.

Keywords

Abstract elliptic equation Banach spaces Exact estimates Fractional spaces Well-posedness 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsFatih UniversityIstanbulTurkey
  2. 2.ITTUAshgabatTurkmenistan

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