ESTIMATES FOR THE HYPER-ORDER OF SOLUTIONS OF CERTAIN LINEAR DIFFERENTIAL EQUATIONS

In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Meromorphic Coefficients

In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.

ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

EXISTENCE OF MEROMORPHIC SOLUTIONS OF SOME HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.

INSTABILITY OF SOLUTION FOR THE FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH VARIED COEFFICIENT

In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.

ON THE COMPLEX OSCILLATION OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

In this paper, we investigate the complex oscillation of higher order homogenous and non- homogeneous linear differential equations with meromorphic coefficients of iterated order, and obtain some results which improve and extend those given by Z. X. Chen, L. Kinnunen, etc.

ON THE ITERATED ORDER OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this paper, we investigate complex homogeneous and non-homogeneous higher order linear differential equations with meromorphic coefficients. We obtain several results concerning the iterated order of meromorphic solutions, and the iterated convergence exponent of the zeros of meromorphic solutions.

Oscillation Result for Some Non-homogeneous Linear Differential Equations and Its Application

This paper investigates solutions of some non-homogeneous linear differential equations, which have non-homogeneous term as the small function of solution. Using the similar method, we can generalize the result of G.Gundersen and L.Z.Yang.

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system.At first we point out some questions existing in current study of control theory for the L^(p)linear system with an unbounded control operator and an unbounded observation operator,such as stabilization problem and observer theory that are closely relevant to state feedback operator.After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations.As a principle,we propose a concept of admissible state feedback operator for system(A,B).Finally we give an existence result of admissible state feedback operators,including semigroup generation and the equivalent conditions of admissibility of state feedback operators,for an L^(p)well-posed system.

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differential equation: , where A is a constant matrix, Q(t,?) is quasiperiodic in t, and ? is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to , where R* is exponentially small in ?.

ON THE MEROMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

In this paper, we investigate the growth of meromorphic solutions of higher order linear differential equation f^（k） ＋Ak-1 （z）e^Pk-1^（z） f^（k-1） ＋…＋A1 （z）e^P1（z） f′ ＋Ao（z）e^Po（z） f = 0 （k ≤ 2）, where Pj（z） （j = 0, 1,..., k - 1） are nonconstant polynomials such that deg Pj = n （j = 0, 1,..., k - 1） and Aj（z）（≠ 0） （j = 0, 1,..., k - 1） are meromorphic functions with order p（Aj） 〈 n （j = 0, 1,..., k - 1）.

ZEROS OF SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

We consider the higher order equation f(k) + ak-1,(z)f(k-1) +……+ al(z)f + (Q1(z)ep1(z) + Q2(z)ep2(z) + Q(z))f = 0, where pl(z) = {1zn+''' , p2(z) = '{2zn+''' are non constant polynomials, Q1(z)( 0), Q2(z)( 0), Q(z) and aj(z)(j = 1,''', k - 1) are entire functions and their orders are less than n. In this paper we treat the case when '{1/{1 is not real. It is shown that the exponent of convergence to the zero-sequence of any non-trivial solution of the above equation is infinite.

SOLUTIONS TO LINEAR DIFFERENTIAL EQUATIONS WITH SOME COEFFICIENT BEING LACUNARY SERIES OF [p, q]-ORDER IN THE COMPLEX PLANE

In this paper, we investigate the growth and value distribution of meromorphic solutions to higher order linear differential equations with some dominating coefficient being Lacunary series and the results of this paper improve and extend the previous results of J. Tu, 2013.

DIFFERENTIATOR SERIES SOLUTION OF LINEAR DIFFERENTIAL ORDINARY EQUATION

In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential difference between this method and the others is that by this method special and general solutions can be obtained directly with the operations of the differentor in the differential equation and without the enlightenment of other scientific knowledge.

ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS

This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

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.

LIMITING DIRECTION AND BAKER WANDERING DOMAIN OF ENTIRE SOLUTIONS OF DIFFERENTIAL EQUATIONS

In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

SOME OSCILLATION THEOREMS OF HIGHER ORDER NON－HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH TRANSCENDENTAL MEROMORPIC COEFFICIENTS

In this paper, we investigate the complex oscillation of the higher order differential equation where B0, ...,Bk-1,,F 0 are transcendental meromorpic functions having only finitely many poles. We obtain some precise estimates of the exponent of convergence of the zero sequence of meromorphic solutions for the above equation.