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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)

This paper presents necessary and sufficient conditions for an n-th order differential equation to have a non-continuable solution with finite limits of its derivatives up to the orders n − 2 at the right-hand end point of the definition interval.

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

The aim of the paper is to study a global structure of solutions of four differential inequalities αiy i(t)yi+1 ≥ 0, yi+1(t) = 0 ⇒ y ′ i(t) = 0, i = 1, 2, 3, 4 , αi ∈ {−1, 1}, α1α2α3α4 = −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.

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)

- 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)

- Miroslav Bartušek, Eva Pekárková
- 2007

In the paper, the nonlinear differential equation (a(t)|y′|p−1y′)′ + 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)

By a solution of () we mean a function x ∈ C[Tx,∞), Tx ≥ , which satisfies () on [Tx,∞). A solution is said to be nonoscillatory if x(t) = for large t; otherwise, it is said to be oscillatory. Observe that if λ≥ , according to [, Theorem .], all nontrivial solutions of () satisfy sup{|x(t)| : t ≥ T} > for T ≥ Tx, on the contrary to the case λ… (More)

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 6= 0 and g is increasing on R. A sufficient condition for the existence of a singular solution of the second kind is given.

(1) where n ≥ 2, f is a continuous function defined on R+ × R , R+ = [0,∞), R = (−∞,∞), τi ∈ C (R+) and τi(t) ≤ t for t ∈ R+ and i = 0, 1, . . . , n− 1 . We will suppose for the simplicity that inf t∈R+ τi(t) > −∞ for i = 0, 1, . . . , n − 1. Note, that C(I), s ∈ {0, 1, . . .}, I ⊂ R+ is the set of continuous functions on I that have continuous derivatives… (More)