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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, 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)
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.