Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

Kuo-Shou Chiu ()
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Kuo-Shou Chiu: Departamento de Matemática, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación, José Pedro Alessandri 774, Santiago 7760197, Chile

Mathematics, 2024, vol. 12, issue 22, 1-32

Abstract: In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the ( μ 1 , μ 2 ) -exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the ( μ 1 , μ 2 ) -exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Keywords: almost periodic solutions; almost periodic sequences; almost periodic functions; piecewise constant argument of generalized type; exponential dichotomy; stability of solutions; Gronwall integral inequality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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