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General almost automorphy.

S Bochner1

  • 1Department of Mathematics, Rice University, Houston, Texas 77001.

Proceedings of the National Academy of Sciences of the United States of America
|October 1, 1975
PubMed
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This study introduces A-invariance, a generalized concept of almost automorphy for linear operators. It proves that if the operator and homogeneous solutions are A-invariant, then inhomogeneous solutions are also A-invariant.

Area of Science:

  • Differential Equations
  • Functional Analysis
  • Dynamical Systems

Background:

  • Almost automorphy is a key concept in analyzing the behavior of solutions to differential equations.
  • Existing theories often require strong conditions on operators and solutions.

Purpose of the Study:

  • To generalize the concept of almost automorphy to "A-invariance" for linear operators.
  • To establish conditions under which bounded solutions of inhomogeneous linear equations inherit A-invariance.
  • To explore the relationship between A-invariance, minimality, and a new form of almost periodicity.

Main Methods:

  • Development of a generalized framework for "A-invariance".
  • Analysis of strong and weak solutions within this new framework.
  • Comparative study of A-invariance with existing concepts like minimality.

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Main Results:

  • A general proposition is established: if a linear operator L is A-invariant and all bounded solutions to Lg=0 are A-invariant, then for an A-invariant function phi, all bounded solutions to Lf=phi are also A-invariant.
  • The study introduces a "thinned-out" almost periodicity linked to A-invariance.

Conclusions:

  • The proposed A-invariance offers a more flexible approach to studying differential equations.
  • The findings extend the understanding of solution properties for linear operators.
  • The work bridges concepts in dynamical systems and functional analysis.