Periodic solutions for higher order differential equations with deviating argument

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for higher order differential equations with deviating argument . Some new results on the existence of periodic solutions of the equations are obtained. Meanwhile, an example is given to illustrate our results.     

-loss of derivatives for an evolution type model

This paper is devoted to the study of loss of derivatives for an evolution type model, which allows for non-integer powers of (−Δ). On the one hand, we describe the influence from the orders of principle symbols. On the other hand, we explore the significant impact of ν-related oscillation from the pseudo-differential operators. In the final analysis, to demonstrate the sharpness of our estimates, counter-examples are constructed through the application of Floquet theory and instability arguments.     

Periodic solutions for a kind of Liénard equation with a deviating argument

In this paper, the Liénard equation with a deviating argument

x^″(t)+_f1(t,x(t))x^′(t)+_f2(x(t))(x^′(t))²+g(t,x(t−τ(t)))=p(t)$x^{″}\left(t\right)+f_1(t,x\left(t\right))x^{\prime }\left(t\right)+f_2\left(x\left(t\right)\right){\left(x^{\prime }\left(t\right)\right)}^2+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$

The influence of surfactants on latex flocculation with poly(diallyldimethyl ammonium chloride)

Aqueous latex was flocculated by mixtures of poly(diallyldimethyl ammonium chloride), PDADMAC, and anionic surfactants. Sodium dodecyl sulfate, (SDS), and Aerosol OT influenced flocculation whereas nonionic Tergitol NP-10 did not. The flocculation domains were correlated with properties of the polymer-surfactant complexFlocculation was never observed above the CMC of the corresponding surfactant solution without polymer or latex. At SDS concentrations greater than 10^–3.6 M the flocculation boundary corresponded to the first appearance of insoluble polymer-surfactant complex which was characterized by dynamic light scattering and microelectrophoresis. Under these conditions latex (diameter 570 nm) and dispersed polymer-surfactant complex particles (diameters between 30 and 2 000 nm) displayed simultaneous homo and heteroflocculation. The boundaries of the flocculation domains at low surfactant concentration were determined by the ratio of polymer to latex and by the net electrostatic charge of the soluble polymer-surfactant complex. On the other hand, the mechanisms controlling flocculation boundaries in the dispersed polymer-surfactant domain require further clarification.     

具连续变量非线性差分方程非振动解在脉冲扰动下的保持性

考虑一类具连续变量的非线性时滞差分方程,证明了在一定条件下,其非振动解的存在性在脉冲扰动下仍保持,推广了文献[数学物理学报,2006,26A （4）：595-600]中的相关结论.