Structural Origin of the Hale Bilinear Form: From the Viewpoint of the Green and Lagrange Identities

Abstract

In the study of linear retarded functional differential equations (RFDEs), the classical bilinear form proposed by J. Hale has long served as an indispensable tool for the geometric theory. Nevertheless, its original definition suffers from ambiguities, such as the non-commutativity of matrix products and the precise meaning of the involved double integrals, due to its definition motivated primarily by eigenvalue problems. Here, by establishing the Green and Lagrange identities associated with a given autonomous linear RFDE, we derive a bilinear form $B$, which is none other than the celebrated Hale bilinear form. This not only resolves the aforementioned issues in the original formulation, but also clarifies its structural origin within the theory of differential equations. Furthermore, based on the recent definition of the $M^p$-space via the memory measure, we demonstrate that the bilinear form $B$ uniquely extends to the product space $M^{*q} \times M^p$ by dense extension, and show that the Green and Lagrange identities still hold in the sense of mild solutions. In terms of its potential for extension to non-autonomous systems, neutral functional differential equations, and differential equations with delay and spatial structures, this work opens a new avenue for the adjoint theory of linear functional differential equations.