Miroslav Bartusek

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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−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)
(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)