奇异点附近的复微分方程解的增长性与系数下级的关系（英文）

We investigate the growth of solutions of the following complex linear differential equation f'+ A(z)f'+ B(z)f = 0,where A(z) and B(z) are analytic functions in C-{z_0}, z_0 ∈ C. Some estimations of lower bounded of growth of solutions of the differential equation are obtained by using the concept of lower order.     

二阶周期线性微分方程解的几个性质

In this paper,the zeros of solutions of periodic second order linear differential equation y ＋ Ay = 0,where A（z） = B（e z ）,B（ζ） = g（ζ） ＋ p j=1 b ？j ζ ？j ,g（ζ） is a transcendental entire function of lower order no more than 1/2,and p is an odd positive integer,are studied.It is shown that every non-trivial solution of above equation satisfies the exponent of convergence of zeros equals to infinity.     

LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

We study the local analytic solutions f of the functional equation f(ψ(zf(z)))=(f(z)) for z in some neighborhood of the origin.Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation.In this paper,we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠0.When assuming f(0) =0,for technical reasons,we just get the result for f’(0)≠0.Then when assuming f(0)=ω0≠0,ψ’(0)=s≠0,ψ(z) is analytic at z=0 and(z)is analytic at z=ω0,we give the existence of local analytic solutions f in the case of 0<|sω0|<1 and the case of |sω0|=1 with the Brjuno condition.     

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations f~n+ P_d(z, f) = p_1 e~(α1 z)+ p_2 e~(α2 z) are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p_1, p_2 are small meromorphic functions of ez, and α_1, α_2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.     

Some Sharpening and Generalizations of a Result of T. J. Rivlin

Let p(z)=a_0+a_1z+a_2z~2+a_3z~3+···+a_nz~n be a polynomial of degree n.Rivlin[12]proved that if p(z)≠0 in the unit disk,then for 0r≤1,max|z|=r|p(z)|≥((r+1)/2)~nmax|p(z)||z|=1.In this paper,we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin’s Theorem.     

Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.     

Some Integral Mean Estimates for Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ k. Fork = 1,it is known that for each r 0 and |α|≥ 1,n(|α|- 1) {∫2π0|P(eiθ)|rdθ}1/r 0r≤ {∫2π0|1+ eiθ|rdθ}1/rmax|z|=|Dα P(z)|.In this paper, we shall first consider the case when k ≥ 1 and present certain generalizations of this inequality. Also for k ≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.     

SINGULAR PERTURBATIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

In this paper we obtain some results about the convergence of aolutions of the boundary value problems of the third order nonlinear ordinary differential equation with a small parameter ε>0: (i=0, 1, 2) to a solution of their reduced problem as ε→0, hero z=ψ(t, x, y) is a root of the equation f(t, x, y, z, 0)=0, and about the existence of solutions of the reduced problem. In addition, under certain conditions we prove the existence of solutions of the boundary value problems (1), (2_i) (i=1, 2), and give their asymptotic estimations.     

Integral Operators Between Fock Spaces∗

In this paper, the authors study the integral operator Sφf(z) = Z C φ(z, w)f(w)dλα(w) induced by a kernel function φ(z, ·) ∈ F ∞α between Fock spaces. For 1 ≤ p ≤ ∞, they prove that Sφ : F 1 α → F p α is bounded if and only if sup a∈C kSφkakp,α < ∞, (?) where ka is the normalized reproducing kernel of F 2 α; and, Sφ : F 1 α → F p α is compact if and only if lim |a|→∞ kSφkakp,α = 0. When 1 < q ≤ ∞, it is also proved that the condition (?) is not sufficient for boundedness of Sφ : F q α → F p α . In the particular case φ(z, w) = eαzw?(z ? w) with ? ∈ F 2 α, for 1 ≤ q < p < ∞, they show that Sφ : F p α → F q α is bounded if and only if ? = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator Sφ : F p α → F q α.     

ON ALMOST AUTOMORPHIC SOLUTIONS OF THIRD-ORDER NEUTRAL DELAY-DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

We present some conditions for the existence and uniqueness of almost automorphic solutions of third order neutral delay-differential equations with piecewise constant of the form(x(t) + px(t-1))′′′= a0x([t]) + a_1x([t-1]) + f(t),where [·] is the greatest integer function,p,a0 and a1are nonzero constants,and f(t) is almost automorphic.     

On the growth of solutions of second order linear differential equations with extremal coefficients

In this paper, we consider the differential equation f + A(z)f + B(z)f = 0, where A and B ≡ 0 are entire functions. Assume that A is extremal for Yang's inequality, then we will give some conditions on B which can guarantee that every non-trivial solution f of the equation is of infinite order.     

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

ON THE NEVANLINNA DIRECTION OF ALGEBROID FUNCTIONS

In this paper, we prove that for an algebroid function w(z), the singular direction arg z =φ0 , satisfying that for arbitrary ε(0 < ε < π 2 ) and any given a ∈ C, lim r → + ∞ n(r,φ0-ε,φ0 +ε,w=a)/ log r = +∞ holds with at most 2v possible exceptional values of a, is the Nevanlinna direction of w(z).     

A Nontrivial Homotopy Element of Order p~2 Detected by the Classical Adams Spectral Sequence

Let p be an odd prime.The authors detect a nontrivial element p of order p~2 in the stable homotopy groups of spheres by the classical Adams spectral sequence.It is represented by a_0~(p-2)h_1 ∈ Ext_A~(p-1,pq+p-2)(Z/p,Z/p) in the E_2-term of the ASS and meanwhile p · p is the first periodic element αp.     

M_φ-type Submodules over the Bidisk

Let H~2(D~2) be the Hardy space over the bidisk D~2,and let M_φ=[(z-φ(w))~2] be the submodule generated by(z-φ(w))~2,where φ(w) is a function in H∞(w).The related quotient module is denoted by N_φ=H~2(D~2)ΘM_φ.In the present paper,we study the Fredholmness of compression operators S_z,S_w on N_φ.When φ(w) is a nonconstant inner function,we prove that the Beurling type theorem holds for the fringe operator F_w on [(z-w)~2]Θ z[(z-w)~2] and the Beurling type theorem holds for the fringe operator Fz on M_φΘwM_φ if φ(0)=0.Lastly,we study some properties of F_w on[(z-w~2)~2]Θz[(z-w~2)~2].     

ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

Asymptotic behaviour of solutions is studied for some second order equations including the model case x(t) + γ x˙(t) + ■Φ(x(t)) = h(t) with γ > 0 and h ∈ L 1(0,+∞;H),Φ being continuouly differentiable with locally Lipschitz continuous gradient and bounded from below.In particular when Φ is convex,all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.     

SUPERCONVERGENCE OF RITZ-VOLTERRA PROJECTION ON FINITE ELEMENT SPACES

The purpose of this paper is to study the superconvergence properties of Ritz-Volterra projection.Through construction a new type of Green function and making use of its properties and the principle of duality,the paper proves that the Ritz-Volterra projection defined on r-1 order finite element spaces of Lagrange type in one and two space variable cases possesses O(h2r~2)order and O(h4+1|Inh|)order nodal superconvergence,respectively,and the same type of superconver-gence results are demonstrated for the semidiscrete finite dement approximate solutions of Soboleve-quations.     

On Representing the General Solution with the Special Solutions for the Differential Equation y''''=sum from i=0 to n(a_i(x)y~i)

It is well known that the n+1 coefficients of the equation y′=a_n(x)y~n+a_(n-1)(x)y~(n-1)+…+a_1(x)y+a_o(X) (1) can be completely determined by any n+1 different special solutions of the same equation. Hence, any solution of the same equation can be completely determined by its initial value and the n+1 special solutions. In addition, when 0≤n≤2, the general solution of Eq. (1) can be represented with n+1 different special solutions and an integral constant, and the representation is independent of the concrete forms of the coefficients a_i(x). Therefore, we give     

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS

In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ■= JHz_{t, z}, where the Hamiltonian function H possesses the form H(t, z) =1/2L(t)z·z + G(t, z), and G(t, z) is only locally defined near the origin with respect to z. Under some mild conditions on L and G, we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadratici...