Boundedness of semilinear Duffing equations at resonance in a critical situation

Boundedness in Asymmetric Oscillations at Resonance in a Critical Situation

“…The proof can be obtained by direct calculations similar to that of Lemma 3.1 in [14]. Thus we omit it here.…”

“…Recently, Xing, Wang and Wang [14] succeeded in answer the question. They obtained a certain sufficient and necessary condition for the boundedness for (1.6) in the critical situation, that is,…”

“…In this article, we are going to study the analogical problem of [14] for the asymmetric equation (1.4). The corresponding critical situation should be (1.9)…”

The existence of unbounded solutions of asymmetric oscillations in the degenerate resonant case

“…In fact, according to the result of Alonso and Ortega in [1], here we only need to consider the situation that for all . The main idea for proving theorem 1.1 is similar to [15] and as follows. By means of a series of transformations, the original system is transformed into a normal form, for which the twist condition is violated.…”