[Paper Review] A class of abstract delay differential equations in the light of suns and stars

Sebastiaan G. Janssens|arXiv (Cornell University)|Jan 31, 2019

Nonlinear Differential Equations Analysis34 references4 citations

TL;DR

This paper establishes a correspondence between a class of abstract delay differential equations (DDEs) with unbounded linear operators in a Banach space and weak*-type integral equations via dual perturbation theory in a non-sun-reflexive setting. The key contribution is showing that even when the state space is not sun-reflexive, the convolution integral in the associated abstract integral equation remains well-defined and takes values in the range of the canonical embedding, enabling the application of sun-star calculus to nonlinear DDEs with unbounded generators.

ABSTRACT

Using dual perturbation theory in a non-sun-reflexive context, we establish a correspondence between 1. a class of nonlinear abstract delay differential equations (DDEs) with unbounded linear part and an unknown taking values in an arbitrary Banach space and 2. a class of abstract weak* integral equations of convolution type involving the sun-star adjoint of a translation-like strongly continuous semigroup. For this purpose we also characterize the sun dual of the underlying state space. More generally we consider bounded linear perturbations of an arbitrary strongly continuous semigroup and we comment on some implications for the particular case of abstract DDEs.

Motivation & Objective

• To extend dual perturbation theory to abstract DDEs with unbounded linear parts in arbitrary Banach spaces.
• To address the challenge of non-sun-reflexivity in the state space while preserving the validity of the sun-star framework.
• To characterize the sun dual of the underlying state space for delay equations.
• To establish a correspondence between mild solutions of abstract DDEs and solutions of weak*-type integral equations involving the sun-star adjoint.
• To generalize the applicability of dual perturbation theory beyond classical or finite-dimensional cases to infinite-dimensional, unbounded settings.

Proposed method

• Utilizes dual perturbation theory in a non-sun-reflexive context to relate the abstract DDE to a weak* integral equation.
• Defines the sun-star calculus framework by introducing the sun-star adjoint of a translation-like semigroup on the dual space.
• Characterizes the sun dual of the state space $X = C([-h,0]; Y)$ to handle non-sun-reflexive cases.
• Employs vector-valued weak* Riemann integration to define the convolution integral in $X^{owtieowtie}$, ensuring convergence even without sun-reflexivity.
• Applies the canonical embedding $j: X \to X^{owtie\star}$ and shows that $j^{-1}$ is well-defined on the image of the integral operator.
• Establishes that the solution of the DDE corresponds to a mild solution of the abstract integral equation involving $T_0^\odot^*$ and the nonlinear operator $G$.

Experimental results

Research questions

• RQ1 Can dual perturbation theory be extended to abstract DDEs with unbounded linear operators in non-sun-reflexive state spaces?
• RQ2 How can the sun-star framework be applied when the state space $X = C([-h,0]; Y)$ fails to be sun-reflexive?
• RQ3 What conditions ensure that the weak* convolution integral in the associated integral equation remains well-defined and takes values in the range of the canonical embedding $j$?
• RQ4 How does the solution of the abstract DDE relate to the solution of the corresponding weak* integral equation in the sun-star setting?
• RQ5 What is the role of the sun-star adjoint semigroup $T_0^{\odot\star}$ in characterizing the dynamics of nonlinear DDEs?

Key findings

• The paper establishes a one-to-one correspondence between mild solutions of the abstract DDE and solutions of a weak* integral equation of convolution type involving the sun-star adjoint semigroup.
• Even when the state space $X$ is not sun-reflexive, the convolution integral in the abstract integral equation takes values in the range of the canonical embedding $j$, justifying the use of $j^{-1}$ in the solution formula.
• The sun dual of the state space $X = C([-h,0]; Y)$ is characterized as the subspace $X^\odot$ of the dual space $X^*$ consisting of functionals that are strongly continuous under the adjoint semigroup $T_0^*$.
• The solution of the DDE is shown to be equivalent to a mild solution of the abstract integral equation involving $T_0^\odot^*$ and the nonlinear operator $G$, which maps $X$ into $X^{\bowtie\star}$.
• The framework allows for bounded linear perturbations of an arbitrary strongly continuous semigroup and extends the applicability of dual perturbation theory to abstract DDEs with unbounded generators.
• The results generalize existing theory by unifying the analysis of abstract DDEs, renewal equations, and coupled systems under a single functional analytic framework based on sun-star calculus.