Abstract
We give a representation of the solution for the best approximation of the harmonic oscillator equation formulated in a general Banach space setting, and a characterization of -maximal regularity—or well posedness—solely in terms of -boundedness properties of the resolvent operator involved in the equation.
Highlights
We give a representation of the solution for the best approximation of the harmonic oscillator equation formulated in a general Banach space setting, and a characterization of lp-maximal regularity—or well posedness—solely in terms of R-boundedness properties of the resolvent operator involved in the equation
In numerical integration of a differential equation, a standard approach is to replace it by a suitable difference equation whose solution can be obtained in a stable manner and without troubles from round off errors
Often the qualitative properties of the solutions of the difference equation are quite different from the solutions of the corresponding differential equations
Summary
In numerical integration of a differential equation, a standard approach is to replace it by a suitable difference equation whose solution can be obtained in a stable manner and without troubles from round off errors. In the article 3 , a characterization of lp-maximal regularity for a discrete second-order equation in Banach spaces was studied, but without taking into account the best approximation character of the equation. This would lead to very interesting problems related to difference equations arising in numerical analysis, for instance From this perspective the results obtained in this work are, to the best of our knowledge, new. We observe that this approach cannot be done by a direct translation of the proofs from the continuous time setting to the discrete time setting The former only allows to construct a solution on a possibly very short time interval, the global solution being obtained by extension results. We obtain a characterization about maximal regularity for 1.2
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