Forced oscillation of second order linear and half-linear difference equations

Oscillation properties of solutions of the forced second order linear difference equation

are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation

1, \end{displaymath}">

are also given.

     

Necessary and sufficient conditions for oscillation of second order linear differential equations

Necessary and sufficient conditions are given for oscillations of second order linear differential equationx″+p(t)x′+q(t)x=0.     

On second order sublinear oscillation

The basic purpose of this paper is to present a new oscillation criterion for second order sublinear ordinary differential equations of the formx(t) +a(t)f[x(t)] = 0,t t ₀>0, wherea is a continuous function on [t ₀, ) without any restriction on its sign andf is a continuous function on the real line, which is continuously differentiable, except possibly at 0, and satisfiesyf(y)>0 andf(y)>0 fory 0, and . The results obtained include the average behavior of the integral of the coefficienta.     

一类二阶非齐次线性微分方程解的复振荡

研究了一类二阶非齐次线性微分方程f″+Ae~(az~n)f′+(B_1e~(bz~n)+B_0e~(dz~n))f=F(z)解的增长性和零点分布,其中F为级小于n的非零整函数,A,B1,B0为非零多项式.在复数a,b,d满足一定条件下,得到该方程的每一个解的超级和二级零点收敛指数的精确估计.     

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.     

An integral criterion for oscillation of linear differential equations of second order

It is proved that if for some n>2 the function x^1–nA_n(x), where A_n(x) is the n-th primitive ofa(x), is not bounded above, then the equation y +a(x)y = 0 oscillates.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 249–251, February, 1978.In conclusion, I thank R. S. Ismagilov for useful discussions about the problem of osillation.     

New oscillation criteria for second order linear matrix differential equations with damping

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system
（P（t）X＇（t））＇ ＋ R（t）X＇（t） ＋ Q（t）X（t） =0, t ≥ to 〉 0
where P（t）, R（t）, Q（t） are n × n real continuous matrix functions, P（t） and R（t） are commutative. Theresults improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.     

Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Summary Reflection principles, analogous to the classicalSchwarz reflection principle for harmonic functions, are obtained for solutions of linear elliptic second order partial differential equations with constant coefficients. The boundary conditions employed are supposed to be satisfied in a limiting sense only, and do not require (a priori) the existence of derivatives on the boundary. To Mauro Picone on his 70^th birth day. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command. The work of this author was sponsored by the Office of Ordnance Research, U.S. Army, under the Contract DA-36-034-ORD-1486.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

Asymptotically unbiased estimation of the second order tail parameter

We introduce a class of asymptotically unbiased estimators for the second order parameter in extreme value statistics. The estimators are constructed by means of an appropriately chosen linear combination of two simple, but biased, kernel estimators for the second order parameter. Asymptotic normality is proven under a third order condition on the tail behavior, some conditions on the kernel functions and for an intermediate number of upper order statistics. A specific member from the proposed class, obtained with power kernel functions, is derived and its finite sample behavior studied in a small simulation experiment.     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua‐type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

二阶中立型差分方程的振动性

研究了一类变系数的二阶中立型时滞差分方程的振动性,得到了该类方程振动及解的一阶差分振动的充分条件。     

On the oscillation of second order quasilinear elliptic equations

By using the Riccati technique, some oscillation criteria for the second order quasilinear elliptic equation $\sum _{i,j=1}^N{D}_i\left[\Phi \left(y\right)A_{ij}\left(x\right){\left|Dy\right|}^{p-2}{D}_{j}y\right]+p\left(x\right)f\left(y\right)=0$ are established. These results contain the known oscillation theorems from the literature as special cases.