Growth of solutions of second order linear differential equations

This paper is devoted to studying the growth of solutions of equations of type f^″+h(z)eazf^′+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.     

On solutions of linear differential equations with entire coefficients

The first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions. Subsequent results involve the separation of the zeros of a Bank-Laine function.     

On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

In this paper, the authors investigate the growth of solutions of a class of higher order linear differential equations

f(k)+Ak−1f(k−1)+?+A₀f=0     

Zeros of solutions of certain second order linear differential equation

In this paper, we investigate the exponent of convergence of the zero-sequence of solutions of the differential equation

(1.3)     

Factorization of entire solutions of some algebraic differential equations

We investigate the factorization of entire solutions of the following algebraic differential equations:

bn(z)finjn(f^′)+bn−1(z)fin−1jn−1(f^′)+?+b₀(z)fi₀j₀(f^′)=b(z),     

Unicity of entire functions and their linear differential polynomials

We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.     

Universal perturbations of linear differential equations

Let be a fundamental solution of with and bounded on . We prove that there exist arbitrary small matrix functions with limit as such that has solutions with dense in .

     

On the growth of entire solutions of algebraic differential equations

This is the opening memorial plenary lecture on memorial meeting in honor of Professor Shlomo Strelitz on Conference of Differential Equations and Complex Analysis, University of Haifa, Israel, December 2000. Shlomo Strelitz: 7.1.1923–27.9.1999, teached at Vilnius University 1946–1973, professor 1967–1973.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 57–63, January–March, 2005.     

The growth of meromorphic solutions for a class of higher-order linear differential equations

In this article, the growth of meromorphic solutions for a class of higher-order linear differential equation with meromorphic coefficients is investigated by applying the method of value distribution, and some estimates for the hyper-order of their solutions are obtained.     

Dichotomous solutions of linear impulsive differential equations

The notion of L_p(h,k) solutions of linear impulsive differential equations in Banach spaces is introduced. Sufficient conditions for existence of such solutions are derived. Possible applications to linear control systems with impulses are considered. An illustrative example is given.     

Limiting directions of julia sets of entire solutions to complex differential equations

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.     

On multiply-connected Fatou components in iteration of meromorphic functions

Let be a transcendental meromorphic function with at most finitely many poles. We mainly investigated the existence of the Baker wandering domains of f(z) and proved, among others, that if f(z) has a Baker wandering domain U, then for all sufficiently large n, fn(U) contains a round annulus whose module tends to infinity as n→∞ and so for some 0<d<1,

     

代数微分方程的代数解

Using value distribution theory and techniques,the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that the results are sharp.     

A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials

We have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations. Specifically, we deal with entire solutions of the equations that are represented in convergent series of Bessel polynomials, and determine orders and types of the solutions, in terms of their Taylor coefficients, by establishing an analogue of Lindelöf-Pringsheim theorem as well as Wiman-Valiron type theory for such functions. Finally, by using value distribution theory of holomorphic functions, we are able to exhibit some uniqueness theorems of the entire (or meromorphic) solutions.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Dual exponential polynomials and linear differential equations

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

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.     

A method for constructing solutions to linear systems of partial differential equations

We obtain some conditions of solvability in Sobolev spaces for the systems of linear partial differential equations and deduce the corresponding formulas for solutions to these systems. The solutions are given as the sum of the series whose terms are the iterations of some pseudodifferential operators constructed explicitly.