.Vo,%ne:y .4nn~y:rr. 7’kheory. .Merko& & Applicnnonr. Vol. 3. So. 3. pp. ?61-271. 1981. n36?-26x% 53 !xl - 00 Printed m Great Bntam. c 19E4 Pergmon Press Lrd lMONOTONE SOLUTIONS OF A CLASS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS MAURO MARINI Istituto di Matematica Applicata .‘G. Sansone”, via S. Xlarta, 3. 50139 Firenze, Italy zyxwvutsrqponmlkjihgf (Received 29 April 1983. Received for publicafion 20 June 1983) Key words andphrases: Ordinary differential equations, asymptotic behaviour, boundedness of solutions. asymptotic comparison. SECT1051 MANY physical systems are modelled by second order nonlinear differential equations of the type lX~)f(~(~))x’(01’ = &)&+(t)). (I) For example both the Thomas-Fermi and the Schr6dinger-Persico equations, which occur in the study of atomic fields, are of the type (1) [7]. M oreover equation (1) arises both in the study of chemically reacting systems [l] and in celestial mechanics where it represents the law of angular momentum conservation when the field strength is time dependent [2]. The behaviour of (1) has been investigated by several authors. In particular it is worth mentioning [2] for continuability problems, [3]-[5] for the oscillatory character and the asymptotic behaviour. For other references the reader is referred to the above quoted papers. Our object is to study the asymptotic behaviour of the solutions of (1). We give some necessary and sufficient conditions for boundedness of these solutions which extend a known result to the linear case [4]. Furthermore, we obtain some asymptotic relationships between the solutions of (1) and the second order linear differential equation b(4u’(Ol’= d440. (2) Conditions are found that lead to an equivalence between some of the solutions of (1) and some of the solutions of (2). This last result extends an asymptotic comparison theorem of [6] to the nonlinear case. Equation (1) can be reduced by a transformation of variables (see example, [3]) to z”(t) + h(r)G(z(t)) = 0. In analyzing oscillatory problems this approach is useful, however it is not suitable considering the asymptotic behaviour. Indeed, under our hypotheses, equation (1) has no oscillatory solutions. Moreover, we consider here equation (1) in its self-adjoint form. In fact it is more adequate than the normal form, because it simplifies the proofs and because our hypotheses are expressed in a more elegant form. 261