Abstract
Singular Kneser solutions of higher-order quasilinear ordinary differential equations
Highlights
For a positive constant α, let D(α) be the first-order differential operator defined by D(α)x =d dt (|x|αsgn x), and for n positive constants α1, α2, . . . , αn let D(αi, αi−1, . . . , α1) be the ith-order iterated differential operator defined byD(αi, αi−1, . . . , α1)x = D(αi)D(αi−1) · · · D(α1)x, i = 0, 1, 2, . . . , n.Here, if i = 0, D(αi, . . . , α1)x is interpreted as x
The main purpose of this paper is to show that Theorem A can be generalized as follows
Letting t → ∞, we find that s(νn+1)σ−μnτ−1 p(s)σds < ∞
Summary
In this paper we give a new sufficient condition in order that all nontrivial Kneser solutions of the quasilinear ordinary differential equation For a positive constant α, let D(α) be the first-order differential operator defined by Naito and Usami ([6, Theorem 4.1]) have proved that, for each A > 0, the equation (1.1) has at least one Kneser solution x(t) on [a, ∞) such that x(a) = A.
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