A Stepanov version for Favard theory

Tarallo, Massimo

doi:10.1007/s00013-007-2435-5

Cited by 17 publications

(7 citation statements)

References 8 publications

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“…If the condition of Favard's theorem does not hold, such problem has been considered by many authors [24,38,[44][45][46]56], and some fascinating counter-examples have been constructed to explain that sometimes almost periodic solutions do not exist, see Zhikov and Levitan [56], Johnson [24], Ortega and Tarallo [38], and Sell [44], which show the optimality of Favard's separation condition. Ortega and Tarallo [38] unified the two situations in [24,56].…”

Section: Theorem 13 (Seementioning

confidence: 97%

On Favard's theorems

Yuan

2010

Journal of Differential Equations

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Section: Theorem 13 (Seementioning

confidence: 97%

On Favard's theorems

Yuan

2010

Journal of Differential Equations

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“…The function F is not almost periodic, because it is not bounded. However, F 2 C 1 .R, R/ \ SAP 1 .R, R/ [21].…”

Section: Applicationmentioning

confidence: 99%

“…The function

F : double-struckR \to double-struckR

is given by

F (t) = \sum_{n \geq 0} F_{n} (t),

where F n are defined for every integer n ≥1 by

F_{n} (t) = \sum_{k \in P_{n}} H (n^{2} (t - k)),

with

P_{n} = 3^{n} ((), 2 double-struckZ + 1) = ({}, 3^{n} (2 k + 1), k \in double-struckZ)

and

H \in C_{0}^{\infty} ((), double-struckR, double-struckR)

, with support in

((), - \frac{1}{2}, \frac{1}{2})

such that

H \geq 0, H (0) = 1 and \int_{\frac{- 1}{2}}^{\frac{1}{2}} H (s) d s = 1 .

The function F is not almost periodic, because it is not bounded. However,

F \in C^{\infty} ((), double-struckR, double-struckR) \cap S A P^{1} ((), double-struckR, double-struckR)

.…”

Section: Applicationmentioning

confidence: 99%

See 1 more Smart Citation

Behavior of bounded solutions for some almost periodic neutral partial functional differential equations

Dads

Es-sebbar

Ezzinbi

et al. 2016

Math Methods in App Sciences

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We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on double-struckR are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Stepanov almost automorphic sequence is studied by Abbas et al [3]. A very good paper on Stepanov version of Favard theory is discussed by Tarallo [38]. The concept of Stepanov weighted pseudo almost automorphic functions are introduced by Zhang et al [44].…”

Section: Introductionmentioning

confidence: 98%

Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

2015

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In this article, we study the concept of Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. We establish the results with Lipschitz condition and without Lipschitz condition on the forcing term. An interesting example is presented to illustrate the main findings. The results proven are new and complement the existing ones.Keywords. Fractional order abstract integro-differential equations; Stepanov-like weighted pseudo almost automorphy.

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A Stepanov version for Favard theory

Abstract: Some recent papers suggest that the classical Favard theory may be improved, by using the weak version of almost periodicity due to Stepanov: this note is to say that the improvement is just apparent.

Cited by 17 publications

References 8 publications

On Favard's theorems

On Favard's theorems

Behavior of bounded solutions for some almost periodic neutral partial functional differential equations

Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

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