An oscillation criterion for forced half-linear differential equations with mixed nonlinearities

In this note, a new oscillation criterion for forced half-linear second order differential equations with mixed nonlinearities is obtained by using a generalized Riccati transformation. The result of this note generalizes and improves some previous results in the literature.     

Comparison of nonlinearities in oscillation theory of half-linear differential equations

Several comparison theorems with respect to powers in nonlinearities for half-linear differential equations are presented. The Riccati transformation and the reciprocity principle are utilized. Some examples and an integral extension of the classical comparison result are presented as well.     

半线性Emden-Fowler微分方程的振动性

主要研究了一类半线性Emden-Fowler微分方程的振动性.利用广义Riccati变换和积分平均技巧建立新的振动准则,推广和改进了一些文献中的结果.此外,给出每个定理所相对应的例子,用来说明其相对于已有文献中的定理具有一定的优越性.     

Perturbations of the half-linear Euler-Weber type differential equation

We investigate oscillatory properties of the half-linear second order differential equation

     

A remark on local-global principles for conjugacy classes

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL(n), SL(n), U(n), SU(n), and for all semisimple conjugacy classes in any inner form of Sp(n), over fieldsk with vcd(k)≤1. Over number fields such a principle is known to hold for any inner form of GL(n) and U(n), and for the split forms of Sp(n), O(n), as well as for SL(p) but not for SL(n),n non-prime. The principle holds for all conjugacy classes in any inner form of GL(n), but not even for semisimple conjugacy classes in Sp(n), over fieldsk with vcd(k)≤2. The principle for conjugacy classes is related to that for centralizers.     

Ordinary differential equations in the oscillation theory of partial half-linear differential equation

In the paper we study the damped half-linear partial differential equation

     

关于一类偶数阶半线性微分方程解的振动性的注记

通过运用分析方法,对微分不等式进行估计,得到了一类偶数阶半线性微分方程解的振动性的若干判别准则.所得结果减弱了已有文献的条件,推广和改进了已有文献的结果.     

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities a(t) x ′ (t) γ′ + p 0 (t) x γ (g 0 (t)) + n i =1 p i (t) | x (g i (t)) | α i sgn x (g i (t)) = e(t), where γ is a quotient of odd positive integers, α i > 0, i = 1, 2, ··· , n, a, e, and p i ∈ C ([0, ∞ ) , R), a (t) > 0, gi : R → R are positive continuous functions on R with lim t →∞ g i (t) = ∞ , i = 0, 1, ··· , n. Our results generalize and improve the results in a recent article by Sun and Wong [29].     

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

where τ0, p(t)C¹[0,∞), q(t),q_i(t),e(t)C[0,∞), p(t)>0, . Without assuming that q(t), q_i(t) and e(t) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.     

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117-121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.     

Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities

We present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities:

     

Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities

In this paper, by arithmetic-geometric mean inequality and Riccati transformation, interval oscillation criteria are established for second-order forced impulsive differential equation with mixed nonlinearities of the form

     

A remark on the porous medium equation with nonlinear source

In the present paper, we obtain a new a priori estimate of the solution of the initial-boundary value problem for the porous medium equation with nonlinear source and formulate the conditions guaranteeing the global solvability of this problem.     

Triviality of totally conservative solutions of a stochastic partial differential equation with power-law nonlinearities

We obtain conditions under which a totally conservative solution of the Cauchy problem for a stochastic partial differential equation of parabolic type with nonlinearities of power-law type can only be the identically zero solution.