On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

On the zeros of solutions of certain linear differential equations

Suppose that the linear differential equation $$w^{(k)}(z)+{\mathop \sum^{k-2}\limits_{j=0}}A_{j}(z)w^{(j)}(z)=0$$ is such that the A_j are entire of finite order, and that A₀ is the dominant coefficient in terms of growth. The existence of a fundamental set of solutions each having few zeros is shown to imply that the order of A₀ is a positive integer.     

On the complex zeros of solutions of linear differential equations

Summary A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w + P(z) w + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=_j, for j=1,..., m, such that for any solution w=f(z) 0 and any > 0, all but finitely many zeros off lie in the union of the sectors ¦ arg z - _j¦ < for j=1,..., m. In this paper, we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients. We prove that for any such equation, one of the following two properties must hold: (a) for any ray, arg z=, and any > 0, there is a solution f 0 of the equation having infinitely many zeros in the sector ¦arg z - ¦ <, or (b) there exist finitely many rays, arg z=_j, for j= 1,..., m, such that for any >0, all but finitely many zeros of any solution f 0 must lie in the union of the sectors ¦ arg z - _j¦ < for j=1, ..., m. In addition, our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation, and in the case when (b) holds, our method will produce the rays, arg z=_j. We emphasize that our result applies to all equations having polynomial coefficients, without exception. In addition, we mention that if the coefficients are only assumed to be rational functions, our results will still give precise information on the possible location of the bulk of the zeros of any solution.This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).     

On stationary zeros of solutions of linear elliptic equations

For a nontrivial solution of a linear homogeneous elliptic equation, we study the dimension of the set of zeros whose multiplicity is not less than the order of the equation. In the case of a linear homogeneous differential operator P = P(D) with constant coefficients and three variables, we show that if, for a solution of the equation Pu = 0, a point x ⁰ is a zero of multiplicity not less than the order of the equation, then the intersection of a sufficiently small neighborhood of the point x ⁰ with the set of all other zeros of this kind is a finite set of segments with common endpoint x ⁰.     

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).     

On the sequences of zeros of holomorphic solutions of linear second-order differential equations

We find necessary and sufficient conditions under which a finite or infinite sequence of complex numbers is the sequence of zeros of a holomorphic solution of the linear differential equation f″ + a ₀ f = 0 with a meromorphic coefficient a ₀ that has second-order poles. In addition, we present a criterion for all solutions of second-order linear equations to be meromorphic.     

On the distribution of zeros of solutions of first order delay differential equations

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation

x^′(t)+p(t)x(t−τ)=0     

On determining the location of complex zeros of solutions of certain linear differential equations

Summary We investigate the location of zeros of solutions for a class of second-order linear differential equations. This class had previously been investigated to determine the frequency of zeros of solutions AMS(MOS): 34A20.This research was supported in part by the National Science Foundation (DMS 84-20561).     

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:

u⁽⁴⁾(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,

where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.     

广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter

In this paper the following half-linear ordinary differential equation is considered:

where 0$"> is a constant, 0$"> is a parameter, and is a continuous function on , 0$">, and 0$"> for . The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions of (H such that

It is shown that, if and if (H is strongly nonoscillatory, then there exists a sequence such that ,   as ; and with has exactly zeros in the interval and ; and with has exactly zeros in and . For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.

     

Asymptotic behaviour and zeros of solutions of linear differential equations

Summary. Certain classes of linear differential equations are investigated for which the distribution of zeros of their solutions determines their asymptotic behaviour. These results generalize those already obtained for the second order linear differential equations to equations of arbitrary order.     

On the baire classification of Sergeev frequencies of zeros and roots of solutions of linear differential equations

We show that the upper and lower characteristic frequencies of zeros and the upper frequency of roots of a solution of a linear differential equation treated as functions on the direct product of the space of equations with the compact-open topology by the space of initial vectors of solutions belong to the third Baire class and that the lower characteristic frequency of roots belongs to the second Baire class. As a corollary, we show that the ranges of the considered frequencies on the solutions of a given equation are Suslin (analytic) sets. In addition, we prove the Lebesgue measurability and the Baire property of the extreme characteristic frequencies of zeros and roots of an equation treated as functions of a real parameter on which the coefficients of the equation depend continuously.     

一阶时滞微分方程解的零点分布

Abstract. The paper gives two estimates of the distance between adjacent zeros of solutions