A Note on the Parabolic Differential and Difference Equations

Ashyralyev, Allaberen; Sozen, Yasar; Sobolevskii, Pavel E.

doi:https://doi.org/10.1155/2007/61659

Abstract and Applied Analysis

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Volume 2007 | Article ID 061659 | https://doi.org/10.1155/2007/61659

A Note on the Parabolic Differential and Difference Equations

Allaberen Ashyralyev,¹Yasar Sozen,¹and Pavel E. Sobolevskii^2,3

Academic Editor: Martin J. Bohner

Received13 Aug 2006

Revised12 Dec 2006

Accepted06 Mar 2007

Published24 Apr 2007

Abstract

The differential equation u'(t)+Au(t)=f(t)(−∞<t<∞) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach space C(E)=C(ℝ,E) with norm ‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder space Cα(E)=Cα(ℝ,E) with norm ‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα), 0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in C(ℝτ,E) spaces is proved. The well-posedness of this difference scheme in Cα(ℝτ,E) spaces is obtained.

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Copyright © 2007 Allaberen Ashyralyev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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