On Some Classes of Continuable Solutions of a Nonlinear Differential Equation

On nonoscillatory solutions tending to zero of third-order nonlinear differential equations

“…Namely, we give the necessary and sufficient conditions for the existence of nonoscillatory solutions in the subclasses N Obtained results complement those in [17] where the existence of bounded nonoscillatory solutions of (N) in the classes N 1 and N 2 has been investigated. Moreover, our results complement and extend some other ones that have been stated in [7] and [12], respectively.…”

Global attraction to the origin in a parametrically driven nonlinear oscillator

“…Moreover the search for periodic solutions in time-dependent dynamical systems has proven very fruitful by using a number of techniques such as the calculus of variations [5] and the Poincaré-Birkhoff theorem [6,7,8]. Some work has also been done on the classification of the asymptotic behaviour of solutions of nonlinear differential equations; see [9] and references therein. Another natural question one may ask is: do there exist time-dependent potentials such that all the solutions of the corresponding differential equations are bounded in phase space?…”

Global monotonicity and oscillation for second order differential equation

“…assumptions (11) and (15) for equation (8) are equivalent to the assumption (6) for equation (1). Applying Lemmas 1 and 4 to (8) and going back to (1) we obtain that every solution x of (1) with a zero is oscillatory and any nonoscillatory solution x of (1) satisfies lim t→∞ x(t) = 0.…”

“…In this paper we are concerned with the oscillatory behavior of solutions for the second order nonlinear differential equation (1) (r(t)x ) + q(t)f (x) = 0, We will restrict our attention to those solutions x of (1) that exist on some interval [t 0 , b x ), b x ∞, and not continuable for t b x , whereby b x may depend on the particular solution involved. A solution x is called oscillatory if there exists {t n }, t 0 t n < b x , such that lim n→∞ t n = b x and x(t n ) = 0, otherwise it is called nonoscillatory.…”

“…b x = ∞, as will be pointed out in the next section. Two problems have been widely studied in literature: the existence of nonoscillatory solutions and the oscillation of all solutions, see, e.g., [1], [2], [6], [7], [14] and [5], [15], [17], [18], respectively, and references therein. More difficult problems, especially those motivated by some physical applications, deal with: (i) the existence of at least one oscillatory solution; (ii) the global monotonicity of nonoscillatory solutions x, i.e.…”