The boundedness of all solutions is shown for Duffing-type equations $\frac{{d^2 x}}{{dt^2 }} + x^{2n + 1} + \sum\limits_{j = 0}^{2n} {x^j p_j (t) = 0, n \geqslant 1,} $ wherep 1,p 2,...,p 2n are of period 1 and of Li...The boundedness of all solutions is shown for Duffing-type equations $\frac{{d^2 x}}{{dt^2 }} + x^{2n + 1} + \sum\limits_{j = 0}^{2n} {x^j p_j (t) = 0, n \geqslant 1,} $ wherep 1,p 2,...,p 2n are of period 1 and of Lipschitzian continuity andp n+1,...,p 2n are of Zygmundian continuity. This conclusion implies that the boundedness phenomenon for the Duffing-type equations does not require the smoothness in the time-variable, thus answering the question posed by Dieckerhoff and Zehnder.展开更多
MOTIVATED by various significant applications to non-Newtonian fluid theory, diffusion offlows in porous media, nonlinear elasticity, and theory of capillary surfaces, several authors(see refs.[1,2] and references cit...MOTIVATED by various significant applications to non-Newtonian fluid theory, diffusion offlows in porous media, nonlinear elasticity, and theory of capillary surfaces, several authors(see refs.[1,2] and references cited therein ) have recently studied the existence of periodicsolutions and other properties for the following differential equation:展开更多
We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA0. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we g...We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA0. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than ErdSs- Moser theorem under RCA0.展开更多
We prove the boundedness of all solutions for the equation x＂ ＋ V＇（x） = DxG（x,t）, where V（x） is of singular potential, i.e., limx→-1 Y（x） = ∞, and G（x, t） is bounded and periodic in t. We give sufficien...We prove the boundedness of all solutions for the equation x＂ ＋ V＇（x） = DxG（x,t）, where V（x） is of singular potential, i.e., limx→-1 Y（x） = ∞, and G（x, t） is bounded and periodic in t. We give sufficient conditions on V（x） and G（x, t） to ensure that all solutions are bounded.展开更多
文摘The boundedness of all solutions is shown for Duffing-type equations $\frac{{d^2 x}}{{dt^2 }} + x^{2n + 1} + \sum\limits_{j = 0}^{2n} {x^j p_j (t) = 0, n \geqslant 1,} $ wherep 1,p 2,...,p 2n are of period 1 and of Lipschitzian continuity andp n+1,...,p 2n are of Zygmundian continuity. This conclusion implies that the boundedness phenomenon for the Duffing-type equations does not require the smoothness in the time-variable, thus answering the question posed by Dieckerhoff and Zehnder.
文摘MOTIVATED by various significant applications to non-Newtonian fluid theory, diffusion offlows in porous media, nonlinear elasticity, and theory of capillary surfaces, several authors(see refs.[1,2] and references cited therein ) have recently studied the existence of periodicsolutions and other properties for the following differential equation:
基金Acknowledgements The author thanks Prof. Wei Wang for his valuable insights and helpful comments. He also thanks Profs. Chitat Chong, Qi Feng, and Yue Yang for providing chances to participate in a series of logic programs held by MCM of CAS and IMS of NUS. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11001281） and the Basic Research Foundation of Jilin University, China （No. 450060502080）.
文摘We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA0. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than ErdSs- Moser theorem under RCA0.
文摘We prove the boundedness of all solutions for the equation x＂ ＋ V＇（x） = DxG（x,t）, where V（x） is of singular potential, i.e., limx→-1 Y（x） = ∞, and G（x, t） is bounded and periodic in t. We give sufficient conditions on V（x） and G（x, t） to ensure that all solutions are bounded.
基金supported by the Project of Science and Technology of the Educational Department of Shandong Province(J07WH01)Binzhou University (BZCYL200416)(BZXYQMG200622)(BZXYNLG200618)
