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and Applied Analysis 3 If any solution x of 1.1 is either oscillatory, or satisfies the condition 1.7 , or admits the asymptotic representation x i c 1 sin t − α i εi t , i 0, 1, 2, 3 , 1.8 where c / 0 and α are constants, the continuous functions εi i 0, 1, 2, 3 vanish at infinity and ε0 satisfies the inequality cε0 t > 0 for large t, then we say that 1.1… (More)

In the paper a sufficient condition for all solutions of the differential equation with p-Laplacian to be proper. Examples of super-half-linear and sub-half-linear equations (|y|y) +r(t)|y| sgn y = 0, r > 0 are given for which singular solutions exist (for any p > 0, λ > 0, p 6= λ). Consider the differential equation with p-Laplacian ( a(t)|y|y ′ + r(t)f(y)… (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)

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)

- M. Bartušek, M. Cecchi, M. Marini
- 2014

and Applied Analysis 3 Motivated by 14, 15 , here we study the existence of AL-solutions for 1.1 . The approach is completely different from the one used in 15 , in which an iteration process, jointly with a comparison with the linear equation y 4 q t y 2 0, is employed. Our tools are based on a topological method, certain integral inequalities, and some… (More)

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

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.