The Asymptotic Behavior of a Class of Nonlinear Differential Equations of Second Order

Abstract

Abstract.Let u" + fit, u) = 0 be a nonlinear differential equation. If there are two nonnegative continuous functions ü(r), <p(r) for t > 0, and a continuous function g(u) for u > 0, such that (i) / J° v(t)<p{t) dt < oo; (ii) for u > 0, g(u) is positive and nondecreasing; (iii) \fit, u)\ < t>(i)<p(0?(l"l/0 f°r ' > '. -oo < u < oo, then the equation has solutions asymptotic to a + bt, where a, b are constants and b =£ 0. Our result generalizes a theorem of D. S. Cohen [3].Consider the nonlinear differential equation (1) u"+f(t,u) = 0. (U-3)\f(t,u)\<fu(t,0)\u\onD.In addition, suppose that (2) f" tfu(t, 0) dt < oo.Then equation (I) has solutions which are asymptotic to a + bt as t -» oo, where a, b are constants and b ¥= 0.In the proof of Theorem A, Cohen used R. Bellman's method [1, pp. 114-115] and Gronwall's inequality. In this paper we use the same method and Bihari's inequality [2] to generalize Theorem A.Theorem B. Let fit, u) be continuous on D: t > 0, -oo < u < oo. If there are two nonnegative continuous functions v(t), rjp(f) for t > 0, and a continuous function g (u) for u > 0, such that (i) ff v(t)<p(t) dt < oo, (ii) for u > 0, g(u) is positive and nondecreasing, (iii) ¡fit, u)\ < v(t)<p(t)g(\u\/t)for t > I, -oo < u < oo, then the equation (1) has solutions which are asymptotic to a + bt, where a, b are constants and b 9* 0.

On the Asymptotic Behaviour of the Solutions to a Class of Second Order Nonlinear Differential Equations

“…where a ∈ ‫.ޒ‬ Theorem A generalizes results obtained in [1], [2], [5], [6], [13], [14], and, though it does not ensure that all solutions are asymptotic to lines (see Example 3), it represents a starting point for our approach. = ∞.…”

“…where f ∈ C([1, ∞) × ‫,ޒ‬ ‫)ޒ‬ have been widely investigated by Cohen [1], Constantin [2], Dannan [5], Fan Wei Meng [6], S. Rogovchenko and Y. Rogovchenko [12], Tong [13] and Trench [14]. Using integral inequalities, these papers were concerned with sufficient conditions which ensure that all solutions of (1) will approach those of u = 0.…”

On the Existence of Solutions With Prescribed Asymptotic Behaviour for Perturbed Nonlinear Differential Equations of Second Order

“…Recently, Lipovan [12] demonstrated the existence of a global solution u(t) of Equation (1) that is asymptotic to a given straight line L(t) = at + b, where a, b ∈ ‫,ޒ‬ i.e. Similar and related results have been obtained in [20], [14], [2], [23], [3], [19], [13], [15], [21], [22]. We mention also the pioneering contribution [1].…”

Asymptotic behavior of a class of nonlinear differential equations of 𝑛th order

“…Here we quote the Theorem B in [1] as a proposition, according to the author, which includes the Theorem in [2] as its special case.…”