[Paper Review] Discontinuous Almost Automorphic Functions and Almost Automorphic Solutions of Differential Equations with Piecewise Constant Argument

Alan Chávez, Samuel Castillo+1 more|arXiv (Cornell University)|Jun 4, 2013

Nonlinear Differential Equations Analysis26 references16 citations

TL;DR

This paper introduces Z-almost automorphic functions—discontinuous functions arising naturally in differential equations with piecewise constant argument (DEPCA)—to study almost automorphic solutions. By leveraging exponential dichotomy and discrete Bi-almost automorphic Green's functions, the authors prove that bounded solutions of non-autonomous DEPCAs are almost automorphic if and only if their discrete restrictions to integers are discrete almost automorphic, extending the theory of almost automorphic solutions to hybrid continuous-discrete systems.

ABSTRACT

In this article we introduce a class of discontinuous almost automorphic functions which appears naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument. Their fundamental properties are used to prove the almost automorphicity of bounded solutions of a system of differential equations with piecewise constant argument. Due to the strong discrete character of these equations, the existence of a unique discrete almost automorphic solution of a non-autonomous almost automorphic difference system is obtained, for which conditions of exponential dichotomy and discrete Bi-almost automorphicity are fundamental.

Motivation & Objective

• To address the lack of systematic treatment of discontinuous almost automorphic functions in DEPCA theory.
• To establish a framework for almost automorphic solutions in hybrid differential equations with piecewise constant argument.
• To prove that bounded solutions of non-autonomous DEPCAs are almost automorphic if and only if their integer-valued sequences are discrete almost automorphic.
• To generalize existing results on almost periodic and pseudo almost periodic solutions to the almost automorphic setting in DEPCA.
• To introduce and analyze the properties of Z-almost automorphic functions, which are discontinuous but retain structural regularity at integers.

Proposed method

• Introduce Z-almost automorphic functions as a new class of discontinuous functions that generalize almost automorphic functions to discrete points.
• Use simultaneous triangularization of matrices A and B to decouple the system into a hierarchical chain of scalar equations.
• Apply exponential dichotomy conditions to ensure stability and invertibility of the associated difference operator.
• Employ a discrete Bi-almost automorphic Green's function to construct a unique discrete almost automorphic solution for the difference equation x(n+1) = D(n)x(n) + h(n).
• Prove that if the discrete sequence x(n) is discrete almost automorphic, then the continuous solution x(t) is almost automorphic via backward substitution in the triangularized system.
• Leverage the fact that compositions of almost automorphic and Z-almost automorphic functions (e.g., f([t])) remain Z-almost automorphic, enabling recursive analysis.

Experimental results

Research questions

• RQ1 Can bounded solutions of differential equations with piecewise constant argument be almost automorphic even when the solution's integer restriction is discontinuous?
• RQ2 What conditions ensure the existence and uniqueness of discrete almost automorphic solutions for non-autonomous difference equations?
• RQ3 How can Z-almost automorphic functions be defined and characterized to handle the discontinuous nature of x([t]) in DEPCA?
• RQ4 Is the almost automorphicity of the continuous solution x(t) equivalent to the discrete almost automorphicity of the sequence x(n)?
• RQ5 What role does exponential dichotomy play in ensuring the existence of discrete almost automorphic solutions for difference equations arising from DEPCA?

Key findings

• A bounded solution x(t) of the DEPCA x'(t) = Ax(t) + Bx([t]) + f(t) is almost automorphic if and only if the sequence x(n) is discrete almost automorphic.
• The existence of a unique discrete almost automorphic solution for the difference equation x(n+1) = D(n)x(n) + h(n) is guaranteed under exponential dichotomy and discrete Bi-almost automorphicity of D(n) and h(n).
• Z-almost automorphic functions generalize the discontinuous almost periodic functions introduced by Dads and Lachimi, providing a natural framework for analyzing DEPCA.
• The solution x(t) of the DEPCA is almost automorphic even though x([t]) is discontinuous, due to the Z-almost automorphic nature of x([t]) and the recursive structure of the system.
• The result holds under the condition that matrices A and B admit simultaneous triangularization, which allows for a recursive proof by induction on the system's dimension.
• Corollaries show that if f is Z-almost periodic or Z-periodic with rational period, then bounded solutions are almost periodic or periodic respectively, depending on the discrete behavior of x(n).