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On Meromorphic Solutions of the Systems of Linear Differential Equations with Meromorphic Coefficients
Abstract
For systems of linear differential equations whose dimension can be lowered, we estimate the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of the coefficients of the system.
... Recently, the problems of meromorphic solutions of linear differential equations with meromorphic coefficients have been considered in [16] and some significative results have been proposed based on Nevanlinna value distribution theory. Especially, in [5] the problems of growth of meromorphic solutions have been discussed on the differential equations in the form of f (n) + a n−1 f (n−1) + · · · + a 1 f + a 0 f = 0 and the authors have proposed some important results which stated that if there existed meromorphic function a j such that the relations max j=1,··· ,n−1 ...
... Till now, Nevanlinna value distribution theory has still been a main and useful tool for dealing with the problems of the growth of infinite order solutions. Motivated by the existing results [5] on the growth of solutions of linear differential equations in the references [5,16,18] and the references cited therein, the problems on the complex oscillation of solutions with meromorphic coefficient a j which has a structure of A j e Q(Z) will be considered by Nevanlinna value distribution theory. Some sufficient conditions for estimating the growth of all meromorphic solutions with infinite order have been proposed. ...
The authors address the complex oscillation problems of all solutions of homogenous linear differential equations with meromorphic coefficients. Sufficient conditions for estimating the growth of meromorphic solution with infinite order have been proposed based on Nevanlinna value distribution theory. Compared with the existing results, the proposed hyper-order of all meromorphic solutions with infinite order can be estimated in terms of a bounded interval which includes information of order of growth of meromorphic functions and meromorphic polynomial coefficients.
... One can also investigate the question if the number a = 0 is an exceptional value of the function σ(z) in the Borel's sense and the question on the Julia's rays for the functions ϑ j+1 (z)(j = 0, 3) similarly to how it was done in [10] for the function σ(z). The obtained asymptotic formulas can be applied for an investigation of properties for the solutions of differential equations and their systems, in which the functions ϑ j+1 (z), ϑ j+1 (z)/ϑ j+1 (z)(j = 0, 3) play a role, similar to the main facts of the Nevanlinna theory used in the papers [13][14][15][16]. ...
A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.
UDC 517.925.7 For a system of linear differential equations of the mth order with coefficients holomorphic in a neighborhood of we obtain an a priori estimate of growth for the n-valued () analytic vector solutions of the system via the estimate for the order of growth of its coefficients.
This book offers a modern introduction to Nevanlinna theory and its intricate relation to the theory of normal families, algebraic functions, asymptotic series, and algebraic differential equations.
Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value distribution of differential polynomials and illustrates how value- and pair-sharing problems are linked to algebraic curves and Briot–Bouquet differential equations. In addition to discussing classical applications of Nevanlinna theory, the book outlines state-of-the-art research, such as the effect of the Yosida and Zalcman–Pang method of re-scaling to algebraic differential equations, and presents the Painlevé–Yosida theorem, which relates Painlevé transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes of meromorphic functions.
Aimed at graduate students interested
in recent developments in the field and researchers working on related problems,Nevanlinna Theory, Normal Families, and Algebraic Differential Equations will also be of interest to complex analysts looking for an introduction to various topics in the subject area. With examples, exercises and proofs seamlessly intertwined with the body of the text, this book is particularly suitable for the more advanced reader.
The Malmquist theorem (1913) on the growth of meromorphic solutions of the differential equation f = P(z,f) / Q(z,f), where P(z,f) and Q(z,f) are polynomials in all variables, is proved for the case of meromorphic solutions with logarithmic singularity at infinity.
Estimation of the Nevanlinna characteristics for some classes of meromorphic functions and their applications to differential equations
- A Z Mokhonko
- V D Mokhonko
Distribution of Values of Meromorphic Functions
- I V Ostrovskii