Existence of almost periodic solutions to delay differential equations with Lipschitz nonlinearities

• Published in: Journal of Differential Equations Vol. 55; no. 2; pp. 151 - 164
• Main Author: Layton, William
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.01.1984; Academic Press
• Subjects: Exact sciences and technology; Function theory, analysis; Mathematical methods in physics; Physics; Non linear equation; Differential equation; Retarded equation
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/0022-0396(84)90079-2

Differential equations whose nonlinearities depend upon both x( t) and x( t − τ) arise in many settings. In this paper equations of this form subject to periodic and almost periodic forcing are studied: x′( t) + g( x( t), x( t − τ)) = e( t), −∞ < t < ∞. (E) A nonresonance type condition is found under which it is shown that (E) will have a unique Besicovitch almost periodic solution for any Besicovitch almost periodic forcing term e( t). These results are then generalized to systems of equations of the same form as (E). These results hold without any small parameter type restriction upon g.