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Cn - pseudo almost automorphic solutions of class r for neutral partial functional differential equations under the light of measure theory

Napo Miailou, Zabsonre Issa, Bayili Gilbert

Silesian Journal of Pure and Applied Mathematics

2022

Vol. 12, iss. 1

1--32

The aim of this work is to present new approach to study Cn-(µ,v)-pseudo almost automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed. We use the variation of constants formula and the spectral decomposition of the phase space.

Pseudo almost periodic solutions of infinite class under the light of measure theory

Votsia Djokata, Zabsonre Issa

Silesian Journal of Pure and Applied Mathematics

2020

vol. 10, iss. 1

15--43

The aim of this work is to present new approach to study weighted pseudo almost periodic functions with infinite delay using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of (μ, ν)-pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed on ]−∞, 0] using the spectra decomposition of the phase space developed in Adimy and co-authors.

Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

Czernous W.

Opuscula Mathematica

2014

Vol. 34, no. 2

291--310

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain Ω with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of ∂Ω with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on (0,c)×Ω to the initial boundary value problem, for small c. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

Classical solutions of initial problems for quasilinear partial functional differential equations of the first order

Czernous W.

Opuscula Mathematica

2006

Vol. 26, no. 1

13-29

We consider the initial problem for a quasilinear partial functional differential equation of the first order [formula], z(t, x) = varphi(t, x) ((t, x) ∈ [-h0, 0] x Rn) where z(t, x) : [-h0, 0] x [-h, h] → R is a function defined by z(t, x) (τ, ξ) = z(t + τ, + ξ) for (τ, ξ) ∈ [-h0, 0] x [-h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.