J. Nonl. Evol. Equ. Appl. 2021 (4), pp. 63-94, published on April 30, 2021:

Stepanov-like pseudo almost automorphic solutions of infinite class under the light of measure theory

D. Votsia, I. Zabsonre

UFR Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP 7021 Ouagadougou 03, BURKINA FASO

Received on June 28, 2018, revised version on October 19, 2020
Accepted on October 30, 2020

Communicated by Gisèle M. Mophou

Abstract.  The aim of this work is to study weighted Stepanov-like pseudo almost automorphicfunctions 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 also study the existence and uniqueness of (μ, ν)-Weighted Stepanov-like pseudo almost automorphic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition, the delayed part are assumed to be pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.
Keywords:measure theory; ergodicity; Weighted Stepanov-like pseudo almost automorphic function; evolution equations; partial functional differential equations.
2010 Mathematics Subject Classification:   34K30; 35B15; 35K57; 44A35; 42A85; 42A75.

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