PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 — 27, 2002, Wilmington, NC, USA pp. 647—655 EXISTENCE OF SQUARE INTEGRABLE SOLUTIONS OF PERTURBED NONLINEAR DIFFERENTIAL EQUATIONS Octavian G. Mustafa Centre for Nonlinear Analysis University of Craiova, Romania Yuri V. Rogovchenko Department of Mathematics Eastern Mediterranean University Famagusta, TRNC, Mersin 10, Turkey Abstract. We give constructive proof of the existence of square integrable solutions for a class of n-th order nonlinear differential equations and discuss properties of square integrable solutions, obtaining as by-products an efficient estimate for the rate of decay of the L 2 norm of the solution and the nonexistence result due to Grammatikopoulos and Kulenovic [On the nonexistence of L 2 -solutions of n-th order differential equations, Proc. Edinburgh Math. Soc. (2) 24 (1981), 131—136]. 1. Introduction. The problem of nonexistence of nontrivial L 2 -solutions to var- ious classes of n-th order linear and nonlinear ordinary differential equations has been discussed in a series of papers by Burlak [1], Grammatikopoulos and Kulen- ovic [2], Hallam [3], and Wong [10, 11] which followed the pioneering contribution of Wintner [9]. In the interesting paper by Grammatikopoulos and Kulenovic [2], sharp conditions for the nonexistence of nontrivial L 2 -solutions to n-th order linear and nonlinear differential equations have been established. Further results in this direction were obtained recently by Mustafa and Rogovchenko [7]. The purpose of this paper is to study existence of square integrable solutions to a large class of perturbed n-th order differential equations under the hypotheses imposed by Grammatikopoulos and Kulenovic [2]. The main contribution is to show that even very small perturbations of the equation which does not possess nontrivial L 2 -solutions generate such solutions. Let n ≥ 2 be an integer. Consider the nonlinear differential equation u (n) + f (t, u)= b(t), t ≥ t 0 ≥ 1, (1.1) where the nonlinearity f (t, u) is a real-valued function satisfying |f (t, u)| ≤ h(t) |u| , t ≥ t 0 , u ∈ R. (1.2) In what follows, we assume that h(t) is continuous, nonnegative and Z ∞ t 2n−1 h 2 (t)dt < +∞. (1.3) 1991 Mathematics Subject Classification. 34A12, 34C11, 34D10. Key words and phrases. Nonlinear differential equations, square integrable solutions, existence, perturbation. 647