Almost Periodic Solutions of Differential Equations in Banach Spaces

Hino, Yoshiyuki; Naito, Toshiki; Minh, Nguyễn Văn; Shin, Jong Son

doi:10.1201/b16833

Cited by 82 publications

(86 citation statements)

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“…and asymptotically almost periodic functions we refer to [21]. Furthermore, the reader can consult [6,11,12] for recent developments about the existence of almost periodic and asymptotically almost periodic solutions, and the extension to the admissibility of functions spaces, of the abstract Cauchy problem. In what follows, we denote by C b ðX Þ the space consisting of bounded continuous functions from ½0; yÞ into X endowed with the norm of uniform convergence, and the notation APðX Þ (resp.…”

Section: Existence Of Asymptotically Almost Periodic Solutionsmentioning

confidence: 99%

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

Hernández

Henrı́quez²

2009

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Section: Existence Of Asymptotically Almost Periodic Solutionsmentioning

confidence: 99%

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

Hernández

Henrı́quez²

2009

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“…We note that (see e.g. [2,12,20]) if A generates a semigroup ðTðtÞÞ tb0 , then it is a mild solution in the sense that it satisfies the following integral equation…”

Section: Resultsmentioning

confidence: 99%

Massera Type Theorems for Abstract Functional Differential Equations

Liu

Minh

N’Guérékata

et al. 2008

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Abstract. The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional di¤erential equation _ u uðtÞ ¼ AuðtÞ þ ½BuðtÞ þ f ðtÞ; where A is a closed operator in a Banach space X, 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 autonomous condition. We develop a general procedure to carry out the decomposition that does not need the wellposedness of the equations. The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator A þ B and the spectrum of f . Moreover, we give conditions for the equation not to have quasi-periodic solutions with di¤erent structures of spectrum. The obtained results extend previous ones.

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“…If A is the generator of a C 0 -semigroup, by [20,Lemma 2.11] this condition is equivalent to the condition that, for every s,…”

Section: Commutingmentioning

confidence: 99%

“…On the other hand, it arises naturally from recent studies on the existence of (almost) periodic solutions of evolution equations (see e.g. [6,7,10,17,20,22,24,25,26,27,31,32,33,38,39,40]). By the superposition principle, it is closely related to the conditions for the inhomogeneous equations to have at least one periodic solution, and for all solutions of the corresponding homogeneous equations to be (asymptotic) periodic solutions.…”

Section: Introductionmentioning

confidence: 99%