Monotone solutions of some differential equations

On Kneser-type solutions of sublinear ordinary differential equations

“…12) (-l)-/(r,x.,...,x,)^ff(0 holds on D n (a, r, r 0 ) where g satisfies(3.3), only the existence of a KS with the property (3.4) is guaranteed. This result generalizes Theorem 5 of the paper[5] of M. Svec. The example of the equation u" = -t~3u' all the KS-s of which are constant shows that in general one cannot claim more, and the necessary conditions for the existence of a NKS of (0.1) given in Section 4 imply that (3.3) cannot be omitted, either.Accordingto Theorem 2.3 and its corollary, Theorem 3.1 implies Corollary.…”

“…This definition is motivated by the fact that the problem of finding a solution of (0.1) satisfying (0.2) together with the additional condition u(a) = u 0 > 0 was for the first time considered by A. Kneser [1] in the case n = 2. Later this problem was studied in [2] - [4] for the case n = 2 and in [5] - [8] for the general case.…”

Global Non-monotonicity of Solutions to Nonlinear Second-Order Differential Equations

“…As demonstrated by Liberto Jannelli [19], extensions of monotonicity theorems from linear to nonlinear differential equations are essentially nontrivial. Several important results on monotone solutions to different classes of nonlinear differential equations and systems of differential equations have been obtained by Elias and Kreith [8], Kreith [14], Marini [20],Švec [35] and, more recently, by Cecchi et al [3], Evtukhov and Klopot [9], Li and Fan [18], Tanigawa [36], Wang [40]; see also the bibliography in the cited papers.…”

An equivalence theorem on properties A, B for third order differential equations

“…Then (1 + ) is oscillatory and by [23] it has a Kneser solution tending to a nonzero constant, thus (1 +) does not have property A, see also [3]. EXAMPLE 2.…”