arXiv subjects

Junya Nishiguchi

Publications and source records attributed to Junya Nishiguchi.

8 recordsLinked to original sources

arXiv2026

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

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.

math.DS↗

arXiv2024

On regularity of mild solutions for autonomous linear retarded functional differential equations

The notion of mild solutions for autonomous linear retarded functional differential equations (RFDEs) has been introduced in [J. Nishiguchi, Electron.\ J. Qual.\ Theory Differ.\ Equ.\ \textbf{2023}, No.~32, 1--77] for the purpose of defining fundamental matrix solutions and obtaining a variation of constants formula for the RFDEs. This notion gives a straightforward definition of solutions to the RFDEs under discontinuous history functions compared with previous studies in the literature. For a given autonomous linear RFDE, it holds that the fundamental matrix solutions are locally Lipschitz continuous on the interval $[0, \infty)$. However, it is not apparent whether a similar property is true for the mild solutions. Here we obtain a result which shows the regularity of mild solutions on $[0, \infty)$ for autonomous linear RFDEs. The result makes clear a connection between the mild solutions and solution concepts in previous studies.

math.DS↗

arXiv2023

Mild solutions, variation of constants formula, and linearized stability for delay differential equations

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a \textit{mild solution}, which is a solution under an initial condition having a discontinuous history function. Then the \textit{principal fundamental matrix solution} is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré-Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.

math.DS↗

arXiv2022

Stability region and critical delay

The location of roots of the characteristic equation of a linear delay differential equation (DDE) determines the stability of the linear DDE. However, by its transcendency, there is no general criterion on the contained parameters for the stability. Here we mainly concentrate on the study of a simple transcendental equation $(*)$ $z + a - w \mathrm{e}^{-τz} = 0$ with coefficients of real $a$ and complex $w$ and a delay parameter $τ> 0$ to tackle this transcendency brought by delay. The consideration is twofold: (i) to give the stability region in the parameter space for Eq.~$(*)$ by using the critical delay and (ii) to compare this with a graphical method (so-called the method of D-partitions) by combining with the delay sequence obtained by conditions for purely imaginary roots. By (i), we obtain another proof of Hayes' and Sakata's results, which reveals the nature of imaginary $w$ case in Eq.~$(*)$. By (ii), we propose a method combining the analytic one and geometric one. This combination is important because it will be helpful in studying characteristic equations having higher-dimensional parameters.

math.DS↗

arXiv2020

Asymptotic compactness in topological spaces

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we prove that the asymptotic compactness and the property that the limit set is a nonempty compact set to which the net converges from above are equivalent in uniformizable spaces. We also study the sequential version of the notion of asymptotic compactness by introducing the notion of sequentiality of directed sets.

math.GN↗

arXiv2019

$C^1$-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in $L^p$

The objective of this paper is to clarify the relationship between the $C^1$-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of $C^1$-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the $C^1$-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in $L^p$ plays an important role.

math.CA↗

arXiv2019

Some discontinuous functional differential equation and its connection to smoothness of composition operators in $L^p$

The objective of this paper is to deepen the understanding of the connection between the continuous and smooth dependence of solutions on initial conditions and the regularity of the history functionals for retarded functional differential equations. We consider some differential equation with a single constant delay with the history space of $L^p$-type and obtain the above dependence result by assuming the growth rate of the nonlinearity and its derivative. The corresponding history functional is discontinuous, and it becomes clear that there are the continuity and the smoothness of the composition operators (also called the superposition operators or the Nemytskij operators) between $L^p$-spaces behind the dependence results.

math.CA↗

arXiv2018

Theory of well-posedness for delay differential equations via prolongations and $C^1$-prolongations: its application to state-dependent delay

In this paper, we establish a theory of well-posedness for delay differential equations (DDEs) via notions of \textit{prolongations} and \textit{$C^1$-prolongations}, which are continuous and continuously differentiable extensions of histories to the right, respectively. In this sense, this paper serves as a continuation and an extension of the previous paper by this author (\cite{Nishiguchi 2017}). The results in \cite{Nishiguchi 2017} are applicable to various DDEs, however, the results in \cite{Nishiguchi 2017} cannot be applied to general class of state-dependent DDEs, and its extendability is missing. We find this missing link by introducing notions of ($C^1$-) prolongabilities, regulation of topology by ($C^1$-) prolongations, and Lipschitz conditions about ($C^1$-) prolongations, etc. One of the main result claims that the continuity of the semiflow with a parameter generated by the trivial DDEs $\dot{x} = v$ plays an important role for the well-posedness. The results are applied to general class of state-dependent DDEs.

math.CA↗