A Note on the Parabolic Differential and Difference Equations. (24th April 2007)

• Record Type: Journal Article
• Authors: Ashyralyev, Allaberen; Sozen, Yasar; Sobolevskii, Pavel E.
• Other Names: Bohner Martin J. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2007(2007)
• Publication Date: 2007-04-24
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2007/61659
• Languages: English
• ISSNs: 1085-3375

Abstract

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 ( 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 C α ( E ) = C α ( ℝ, 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 ( ℝ τ, E ) spaces is proved. The well-posedness of this difference scheme in C α ( ℝ τ, E ) spaces is obtained.