Nonlinear Analysis 70 (2009) 822–829 www.elsevier.com/locate/na Asymptotic behavior of solutions to an integral equation underlying a second-order differential equation ✩ Crist´ obal Gonz´ alez ∗ , Antonio Jim´ enez-Melado Dept. An´ alisis Matem´ atico, Fac. Ciencias, Univ. M´ alaga, 29071 M´ alaga, Spain Received 7 November 2007; accepted 9 January 2008 Abstract In this paper we propose the study of an integral equation of the type y (t ) = ω(t ) −  ∞ 0 f (t , s, y (s ))ds, t ≥ 0. We investigate which conditions give existence, and which ones uniqueness, of solutions behaving like the function ω(t ) at ∞. In applying our results to second-order nonlinear differential equations, we are able to recover the previous results and some generalizations. c  2008 Elsevier Ltd. All rights reserved. MSC: 45G10; 34A34; 45M05; 47J05 Keywords: Nonlinear integral equation; Asymptotic behavior; Schauder fixed point theorem 1. Introduction In recent papers, much interest has been given to studying conditions that ensure the existence (and uniqueness) of asymptotically constant solutions to the equation: y ′′ (t ) + F (t , y (t )) = 0, t ≥ 0. (1) We have to mention that the pioneering work for this type of research comes from the hands of Atkinson [1], when investigating the existence of non-oscillatory solutions for differential equations of the above type. For just a few references on the subject we indicate the papers [1–10] and the references therein. This paper is motivated by recent works of Dub´ e and Mingarelli [3], Wahl´ en [9], and Ehrnstr¨ om [4,5]. In each of these works, they look for solutions of (1) asymptotically equal to a real number ω (Ehrnstr¨ om also considers ✩ Research partially supported by the Spanish (Grants MTM2007-60854 and MTM2006-26627) and regional Andalusian (Grants FQM210 and P06-FQM01504) Governments. ∗ Corresponding author. E-mail addresses: cmge@uma.es (C. Gonz´ alez), melado@uma.es (A. Jim´ enez-Melado). 0362-546X/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.01.012