Almost Periodic Solutions to Difference Equations
NEW YORK UNIV NY COURANT INST OF MATHEMATICAL SCIENCES
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The theory of Massera and Schaffer relating the existence of unique almost periodic solutions of an inhomogeneous linear equation to an exponential dichotomy for the homogeneous equation has been completely extended to discretizations by a strongly stable difference scheme. In addition it has been shown that the almost periodic sequence solution will converge to the differential equation solution at a rate Ok sup p where p is the accuracy of the scheme, uniformly in t, if the coefficients are sufficiently smooth. The preceding theory has also been applied to a class of exponentially stable partial differential equations to which one can apply the Hille-Yoshida Theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous equation which can be approximated by almost periodic sequences which are the solutions to appropriate discretizations. Two methods of discretizations are discussed the strongly stable scheme described above and the Lax-Wendroff scheme.
- Theoretical Mathematics