On Favard's theorems

In this work, we use an approach due to Favard (Acta Math 51:31-81, 1928) to study the existence of weakly almost periodic and almost automorphic solutions for some evolution equation whose linear part generates a C 0 -group satisfying the Favard condition in uniformly convex Banach spaces. When this C 0 -group is bounded, which is a condition stronger than Favard's condition, we prove the equivalence between almost automorphy and weak almost automorphy of solutions.

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“…x t + t n → y (t) , (32) in the weak sense as n → ∞. For each ϕ ∈ X * with |ϕ| ≤ 1, we have that ϕ, x t + t n ≤ |ϕ| x t + t n ≤ |x| ∞ .…”

Section: Theorem 28 Assume That X Is Uniformly Convex the C 0 -Groupmentioning

confidence: 98%

“…Recently, Almost periodic and almost automorphic solutions for Eq. (2) via Favard's approach has been largely investigated in the literature [9,10,20,26,29,32]. More results about almost periodic differential equations in finite dimensional spaces can be found in [17].…”

Section: Introductionmentioning

confidence: 99%

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

Es-sebbar

Ezzinbi

2016

Semigroup Forum

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“…There exists an intensive theoretical research in DEPCAG equations (see, for instance, the monographies [1,11,34]), which has been accompanied with applications in engineering, life sciences and numerical analysis of ODE-DDE systems [3,9,13,14,21,26,29,32,33,37]. 1.1.…”

Section: Introductionmentioning

confidence: 99%

A Grobman–Hartman Theorem for Differential Equations with Piecewise Constant Arguments of Mixed Type

Pinto

Robledo

2018

Z. Anal. Anwend.

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We obtain sufficient conditions ensuring the existence of a uniformly continuous and Hölder continuous homeomorphism between the solutions of a linear system of differential equations with piecewise constant argument of generalized type and the solutions of the quasilinear corresponding system. We use a definition (recently introduced by M. Akhmet) of exponential dichotomy for those systems combined with technical assumptions on the nonlinear part. Our result generalizes a previous work of G. Papaschinopoulos.

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“…For some applications of this equation we refer the interested reader to [1,[8][9][10]. Several important properties of the analytic solution of EPCA as well as numerical methods have been studied in [11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Analytical Approach to Differential Equations with Piecewise Continuous Arguments via Modified Piecewise Variational Iteration Method

Wang¹

2014

JAMP

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In the present article, we apply the modified piecewise variational iteration method to obtain the approximate analytical solutions of the differential equations with piecewise continuous arguments. This technique provides a sequence of functions which converges to the exact solution of the problem. Moreover, this method reduces the volume of calculations because it does not need discretization of the variables, linearization or small perturbations. The results seem to show that the method is very reliable and convenient for solving such equations.

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On Favard's theorems

Cited by 20 publications

References 52 publications

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

A Grobman–Hartman Theorem for Differential Equations with Piecewise Constant Arguments of Mixed Type

Analytical Approach to Differential Equations with Piecewise Continuous Arguments via Modified Piecewise Variational Iteration Method

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