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Massera Type Theorem for Abstract Functional Differential Equations

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Qing Liu, Nguyen Van Minh, G. Nguerekata, Rong Yuan

Functional Analysis · Submitted 7 Dec 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${\cal A}+{\cal B}$ and the spectrum of $f$.

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arXiv IDmath/0612197 (version 2)
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CitationG. Nguerekata, Nguyen Van Minh, Qing Liu, Rong Yuan. "Massera Type Theorem for Abstract Functional Differential Equations." arXiv:math/0612197 (2006). https://pith.science/paper/ZRMW5GZK
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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.76 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $ \dot u(t) = Au(t) + [{\cal B}u](t) +f(t), $ where $A$ is a closed operator in a Banach space $\X$, $\cal B$ is a general bounded linear operator in the function space of all $\X$-valued bounded and uniformly continuous functions that satisfies a so-called {\it autonomous} condition. We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${\cal A}+{\cal B}$ and the spectrum of $f$. Moreover, we give conditions for the equation not to have quasi-periodic solutions with different structures of spectrum. The obtained results extend previous ones.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $ \dot u(t) = Au(t) + [{\cal B}u](t) +f(t), $ where $A$ is a closed operator in a Banach space $\X$, $\cal B$ is a general bounded linear operator in the function space of all $\X$-valued bounded and uniformly continuous functions that satisfies a so-called {\it autonomous} condition. background 0.89 0.23 0.05 asserted 0.95
s1 We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. method 0.89 0.94 0.08 asserted 1.00
s2 The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${\cal A}+{\cal B}$ and the spectrum of $f$. result 0.88 0.90 0.63 asserted 1.00
s3 Moreover, we give conditions for the equation not to have quasi-periodic solutions with different structures of spectrum. result 0.85 0.94 0.11 asserted 1.00
s4 The obtained results extend previous ones. interpretation 0.65 0.68 0.10 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $ \dot u(t) = Au(t) + [{\cal B}u](t) +f(t), $ where $A$ is a closed operator in a Banach space $\X$, $\cal B$ is a general bounded linear operator in the function space of all $\X$-valued bounded and uniformly continuous functions that satisfies a so-called {\it autonomous} condition. We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${\cal A}+{\cal B}$ and the spectrum of $f$. Moreover, we give conditions for the equation not to have quasi-periodic solutions with different structures of spectrum. The obtained results extend previous ones.

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