Massera Type Theorems for Abstract Functional Differential Equations

Liu, Qing; Minh, Nguyen Van; N’Guérékata, Gaston M.; Yuan, Rong

doi:10.1619/fesi.51.329

Cited by 13 publications

(4 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the spectrum of the matrix G is identical with the set S We refer the reader to [7,8,9,10,12,13,15,16,17,18,19,23] and the references therein for more information on the theory of admissibility of function spaces with respect to several linear equations including ordinary di¤er-ential equations and functional di¤erential equations. In what follows, as an application of the decomposition formula of VCF in the phase space, we will analyze the admissibility with respect to Eq.…”

Section: Theoremmentioning

confidence: 99%

Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

Matsunaga

Murakami

Minh

2012

View full text Add to dashboard Cite

Abstract. The present paper is concerned with linear integral equations of the form xðtÞ ¼ Ð t Ày Kðt À sÞxðsÞds; where KðsÞ is a matrix measurable in s, and satisfies some boundedness condition. We propose a dynamical-systems approach to studying the behavior of the equations by treating them as functional equations with infinite delay. As a result we obtain a decomposition of the phase space corresponding to a set of normal eigenvalues of the generator of solution semigroups associated with the equations. Moreover, we establish a representation formula (which is called ''a variation-of-constants formula'' in the phase space) for solutions of nonhomogeneous integral equations, together with a decomposed formula based on the decomposition of the phase space. Finally, we apply the formula to investigate the admissibility of function spaces with respect to these equations.

show abstract

Section: Theoremmentioning

confidence: 99%

Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

Matsunaga

Murakami

Minh

2012

View full text Add to dashboard Cite

show abstract

“…The existence of periodic solutions to (1.1), (1.2) and their variants has been of great interest for many authors (cf. [1,3,4,5,7,9,8,11,12,13,14,18,19]). To establish the existence of periodic solutions to (1.2), one of the key steps is to consider first the existence of periodic solutions to the linear equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Massera type theorems for abstract non-autonomous evolution equations

Zheng,

Ding

2024

ejde

View full text Add to dashboard Cite

We establish two fixed point theorems for affine maps in Banach spaces, with weaker assumptions than those in the literature. Then we establish some Massera type results for abstract linear evolution equations without assuming the existence of bounded solutions, which is an indispensable condition in the classical Massera theorem and in the earlier literature. As application, we present an existence result on periodic mild solutions to abstract nonautonomous semilinear evolution equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/35/abstr.html

show abstract

“…These milestone results have motivated researchers remarkably, and it is possible to find detailed literature providing Massera-, Bohr-Neugebauer-, and Favard-type theorems for various kind of dynamic equations based on conventional periodicity, almost periodicity, or almost automorphy notions. We refer to [21,[32][33][34][35][36][37][38][39][40] as pioneering studies. However, we must point out that there is a poor research backlog on Massera-or Bohr-Neugebauer-type theorems on the almost automorphic solutions of difference equations unlike the enormous amount of literature on differential equations.…”

Section: Introductionmentioning

confidence: 99%

Almost Automorphic Solutions to Nonlinear Difference Equations

Kostić,

Koyuncuoğlu,

Fedorov

2023

Mathematics

View full text Add to dashboard Cite

In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions.

show abstract

Massera Type Theorems for Abstract Functional Differential Equations

Cited by 13 publications

References 23 publications

Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

Massera type theorems for abstract non-autonomous evolution equations

Almost Automorphic Solutions to Nonlinear Difference Equations

Contact Info

Product

Resources

About