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We study vanishing at infinity solutions of a fourth-order nonlinear differential equation. We state sufficient and/or necessary conditions for the existence of the positive solution on the half-line [0, ∞) which is vanishing at infinity and sufficient conditions ensuring that all eventually positive solutions are vanishing at infinity. We also discuss an… (More)

We describe the nonlinear limit-point/limit-circle problem for the n-th order differential equation y (n) + r(t)f (y, y ′ ,. .. , y (n−1)) = 0. The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated. 1 Background In 1910, H. Weyl… (More)

- Miroslav Bartušek, Eva Pekárková
- 2007

In the paper, the nonlinear differential equation (a(t)|y | p−1 y) + b(t)g(y) + r(t)f (y) = e(t) is studied. Sufficient conditions for the nonexistence of singular solutions of the first and second kind are given. Hence, sufficient conditions for all nontrivial solutions to be proper are derived. Sufficient conditions for the nonexistence of weakly… (More)

- Miroslav Bartusek, Zuzana Doslá
- Appl. Math. Lett.
- 2014

The author considers the n-th order nonlinear differential equation with delays. He presents sufficient conditions for this equation to have (not to have) noncontinuable solutions. The Cauchy problem and a boundary value problem are investigated.

In the paper, sufficient conditions are given under which all nontrivial solutions of (g(a(t)y)) +r(t)f (y) = 0 are proper where a > 0, r > 0, f (x)x > 0, g(x)x > 0 for x = 0 and g is increasing on R. A sufficient condition for the existence of a singular solution of the second kind is given.

The aim of the paper is to study a global structure of solutions of four differential inequalities α i y i (t)y i+1 ≥ 0, y i+1 (t) = 0 ⇒ y i (t) = 0, i = 1, 2, 3, 4 , α i ∈ {−1, 1}, α 1 α 2 α 3 α 4 = −1 with respect to their zeros. The structure of an oscillatory solution is described, and the number of points with trivial Cauchy conditions is investigated.

Sufficient conditions are given under which the nonlinear n-th order differential equation with quasiderivatives has oscillatory solutions.

In this paper, the authors consider both the forced and unforced versions of a second order nonlinear differential equation with a p-Laplacian. They give sufficient conditions as well as necessary and sufficient conditions for the unforced equation to be of the strong nonlinear limit-circle type, and they give sufficient conditions for the forced equation… (More)

- Miroslav Bartušek, Zuzana DošláB, John R. Graef
- 2014

We study the oscillation of a fourth order nonlinear differential equation with a middle term. Using a certain energy function, we describe the properties of oscillatory solutions. The paper extends oscillation criteria stated for equations with the operator x (4) + x and completes the results stated for super-linear and sub-linear case. Oscillation results… (More)