Almost Periodic Processes and Almost Periodic Solutions of Evolution Equations,
BROWN UNIV PROVIDENCE R I LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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The problem of existence of almost periodic solutions of nonlinear evolution equations has been studied extensively. A careful study of the literature reveals that in most cases the methodology is in the spirit of the classical treatment of ordinary differential equations by Amerio. The details, however, vary considerably, from case to case, in consequence of the variety of forms of partial differential equations. Since the proofs involve translates of the equation, assumptions are usually made guaranteeing the compactness of the set of translates of derivatives of the solution which seem extraneous to the problem. In an effort to bring forward the essential features of the methodology and, at the same time, avoid the imposition of extraneous assumptions, we attempt here a study of the problem at the level of the evolution operator generated by the equation rather than for the equation itself. Roughly speaking a process is a nonautonomous dynamical system. Almost periodic evolution equations generate the class of almost periodic processes.
- Theoretical Mathematics