IJPAM: Volume 79, No. 2 (2012)

AN EXTENSION OF BESICOVITCH'S THEOREM TO
THE CLASS OF WEAKLY ALMOST PERIODIC AND
PSEUDO ALMOST PERIODIC FUNCTIONS

Farouk Chérif$^1$, Zouhir ben Nahia$^2$
$^1$ISSAT Sousse and
Laboratory of Math Physics; Specials Functions and Applications
LR11ES35, Ecole Supérieure des Sciences et de Technologie
4002, Sousse, TUNISIA
$^2$Ecole Supérieure des Sciences et de Technologie
Hammam-Sousse
4002, Sousse, TUNISIA


Abstract. In this paper we are concerned with the Eberlein's weakly almost periodic functions. First, we shall use The Ryll-Nardzewski fixed point theorem to prove that $%
\mathcal{M}\left\{ f(t)\right\}$, the generalized mean in time, of a weakly almost periodic function $f$ is the unique common fixed point of the operators of translation. Next, we give a generalization of a Besicovitch's theorem for the class of weakly almost periodic functions and the class of pseudo almost periodic functions. Next, we apply our result for the study of some abstract differential equation.

Received: October 18, 2011

AMS Subject Classification: 42A75, 43A60, 42A16, 47H10

Key Words and Phrases: pseudo almost periodic functions, weakly almost periodic functions, fourier-Bohr series, fixed point theorems

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 79
Issue: 2