Levitan/Bohr almost periodic and almost automorphic solutions of scalar differential equations

ABSTRACT

The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the scalar differential equations. The existence of at least one quasi-periodic (respectively, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent, Levitan almost periodic, almost recurrent, Poisson stable) solution of sclalar differential equations is proved under the condition that it admits at least one bounded solution on the positive semi-axis which is uniformly Lyapunov stable.

Keywords:

• Bohr/Levitan almost periodic solution
• almost automorphic solutions
• scalar differential equations
• uniform stability
• non-autonomous dynamical systems
• cocycle

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 37B05
• 37B55
• 34C27
• 34D05

Acknowledgments

This paper was written while the author was visiting the University of Granada (December 2013–January 2014, Granada, Spain) under the Program EMERGE – Erasmus Mundus European Mobility. He would like to thank the people of this university for their very kind hospitality, especially Professors Rafael Ortega and Pedro Torres.

Disclosure statement

No potential conflict of interest was reported by the author.