Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 103, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EQUI-ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS AND APPLICATIONS TO FUNCTIONAL INTEGRAL EQUATIONS HUI-SHENG DING, QING-LONG LIU, GASTON M. N’GU ´ ER ´ EKATA Abstract. In this article, we introduce the notion of equi-asymptotically al- most periodicity, and investigate some properties of equi-asymptotically almost periodic functions. In addition, by applying the results on equi-asymptotically almost periodic functions, we obtain an existence theorem for asymptotically almost periodic solutions to a class of functional integral equation. 1. Introduction and preliminaries In 1940s, M. Fr´ echet introduced the notion of asymptotically almost periodicity, which turns out to be one of the most interesting and important generalizations of almost periodicity. In fact, asymptotically almost periodic functions now have been of great interest for many mathematicians and have been applied to various branches of pure and applied mathematics, especially to differential equations and dynamical systems. For example, we refer the reader to [9, 10, 11, 2] and references therein for some recent progress on asymptotically almost periodic functions and their applications to differential equations. In a recent work, when studying the existence of asymptotically almost periodic solutions to a class of Volterra-type difference equations, Long et al [12] introduced the notion of equi-asymptotically almost periodic sequences. To the best of our knowledge, there are only a few publications about equi-asymptotically almost pe- riodic functions. This motivates the publication of this work. For the rest of this paper, if there is no special statement, we denote by R the set of real numbers, by X a Banach space, and by C(R,X) the set of all continuous functions from R to X. In addition, we denote C 0 (R,X)= {f ∈ C(R,X): lim |t|→∞ f (t)=0}. Next, let us recall some notation and basic results of almost periodic functions and asymptotically almost periodic functions (for more details, see [4, 8, 15, 14]). 2000 Mathematics Subject Classification. 34K14, 45G10. Key words and phrases. Asymptotically almost periodic; equi-asymptotically almost periodic; functional integral equation. c 2013 Texas State University - San Marcos. Submitted February 6, 2013. Published April 24, 2013. 1