Abstract and Applied Analysis

A Note on the Parabolic Differential and Difference Equations

Allaberen Ashyralyev, Yasar Sozen, and Pavel E. Sobolevskii

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Abstract

The differential equation u ' ( t ) + A u ( 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 ( , 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 C α ( E ) = C α ( , E ) with norm ϕ C α ( E ) = sup < t < ϕ ( t ) E + sup < t < t + s < (‖ ϕ ( t + s ) ϕ ( t ) E / s α ), 0 < α < 1 , is established. The almost coercivity inequality for solutions of the Rothe difference scheme in C ( τ , E ) spaces is proved. The well-posedness of this difference scheme in C α ( τ , E ) spaces is obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 61659, 16 pages.

Dates
First available in Project Euclid: 5 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1183666871

Digital Object Identifier
doi:10.1155/2007/61659

Mathematical Reviews number (MathSciNet)
MR2302192

Zentralblatt MATH identifier
1153.35353

Citation

Ashyralyev, Allaberen; Sozen, Yasar; Sobolevskii, Pavel E. A Note on the Parabolic Differential and Difference Equations. Abstr. Appl. Anal. 2007 (2007), Article ID 61659, 16 pages. doi:10.1155/2007/61659. https://projecteuclid.org/euclid.aaa/1183666871


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References

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