On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations

In this paper, sufficient conditions have been obtained under which every solution of

On non-oscillatory solutions of second order differential inequalities

Acta Mathematica Hungarica -     

Out-of-phase solutions to a second order linear differential equation with forced oscillation

A criterion is developed to determine the existence of an out-of-phase solution to a second order linear equation with an oscillatory forcing function. Although the general criterion is difficult to check, if a solution can be exhibited that has zeros at the zeros of the forcing function and the associated homogeneous equation is disconjugate, then the solution must be out-of-phase. In addition, a general class of examples is given which can be used to obtain easily verifiable criteria for specific examples.     

具偏差变元高阶Lienard方程周期解存在性

利用重合度理论,研究了一类具偏差变元高阶L ienard型方程周期解的存在性,获得了该方程至少存在一个周期解的充分条件.     

On the oscillation of solutions of first-order delay differential inequalities and equations

Oscillation criteria generalizing a series of earlier results are established for first-order linear delay differential inequalities and equations.     

On Kneser solutions of higher order nonlinear ordinary differential equations

The equationx ⁽ⁿ⁾(t)=(−1) ⁿ │x(t)│ ^k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t₀)^−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t ₀)^−n/(k−1). It is shown that they do not necessarily coincide withC(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.     

On second order functional differential equations and inequalities with deviating arguments

A necessary and sufficient condition is established in order that (i) the retarded differential equation $$y''(t) = p_0 y(t) + f(y(t - \tau _1 ),...,y(t - \tau _N ))$$ has no bounded nonoscillatory solution and (ii) the advanced differential equation $$y''(t) = p_0 y(t) + f(y(t + \tau _1 ),...,y(t + \tau _N ))$$ has no unbounded nonoscillatory solution, wherep ₀≥0 and τ _j > 0,1 ?i ?N, are constants. Differential inequalities related to (^*) and (^**) are also studied. Finally, an oscillation criterion is given for a class of differential equations containing both retarded and advanced arguments.     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

具无穷时滞高阶中立型微分方程的周期解

利用重合度理论研究了一类具有无穷时滞高阶中立型泛函微分方程周期解的存在性,获得了该方程存在周期解的充分条件,推广和改进了二阶方程的相应结果.     

On zeros of solutions of higher order homogeneous linear differential equations

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if A_k−1 is the dominant coefficient, then every transcendental solution f(z) of equation

f(k)+A_k-1 f(k-1)+?+A₀ f=0$f^{(k)}+A_{k-1}\hspace{0.17em}{f}^{(k-1)}+?+A_0\hspace{0.17em}f=0$

satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z).     

Lyapunov-type integral inequalities for certain higher order differential equations

In this paper, first we will give a short survey of the most basic results on Lyapunov inequality, and next we obtain this-type integral inequalities for certain higher order differential equations. Our results are sharper than some results of Yang (2003) [20].