scholarly journals On entire solutions of a certain type of nonlinear differential equation

2001 ◽  
Vol 64 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Chung-Chun Yang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Finite Order ◽  
Nonlinear Differential Equation ◽  
Entire Solution ◽  
Differential Polynomial ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

Entire solutions of certain type of non-linear differential equations

2020 ◽  
Vol 70 (1) ◽  
pp. 87-94
Author(s):  
Bo Xue
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

AN ENTIRE FUNCTION SHARING TWO VALUES WITH ITS LINEAR DIFFERENTIAL POLYNOMIAL

2017 ◽  
Vol 97 (2) ◽  
pp. 265-273
Author(s):  
INDRAJIT LAHIRI
Keyword(s):  
Entire Function ◽  
Differential Polynomial ◽  
Value Sharing ◽  
Linear Differential Polynomial ◽  
Linear Differential ◽  
Function Sharing

We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].

Growth and oscillation theory of non-homogeneous linear differential equations

2000 ◽  
Vol 43 (2) ◽  
pp. 343-359 ◽  
Author(s):  
Gary G. Gundersen ◽  
Enid M. Steinbart ◽  
Shupei Wang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Differential Equations ◽  
Finite Order ◽  
Oscillation Theory ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Homogeneous Linear

AbstractWe investigate the growth and the frequency of zeros of the solutions of the differential equation f(n) + Pn–1 (z) f(n–1) + … + P0 (z) f = H (z), where P0 (z), P1(z), …, Pn–1(z) are polynomials with P0 (z) ≢ 0, and H (z) ≢ 0 is an entire function of finite order.

scholarly journals Radial distributions of Julia sets of difference operators of entire solutions of complex differential equations

2022 ◽  
Vol 7 (4) ◽  
pp. 5133-5145
Author(s):  
Jingjing Li ◽  
◽  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Lower Bound ◽  
Julia Sets ◽  
Differential Polynomial ◽  
Difference Operators ◽  
Transcendental Entire Function ◽  
Entire Solutions ◽  
Complex Differential Equation ◽  
Complex Differential Equations

<abstract><p>In this paper, we mainly investigate the radial distribution of Julia sets of difference operators of entire solutions of complex differential equation $ F(z)f^{n}(z)+P(z, f) = 0 $, where $ F(z) $ is a transcendental entire function and $ P(z, f) $ is a differential polynomial in $ f $ and its derivatives. We obtain that the set of common limiting directions of Julia sets of non-trivial entire solutions, their shifts have a definite range of measure. Moreover, an estimate of lower bound of measure of the set of limiting directions of Jackson difference operators of non-trivial entire solutions is given.</p></abstract>

scholarly journals An entire function sharing fixed points with its linear differential polynomial

2018 ◽  
Vol 22 (1) ◽  
pp. 125-136
Author(s):  
Imrul Kaish ◽  
Indrajit Lahiri
Keyword(s):  
Entire Function ◽  
Fixed Points ◽  
Entire Functions ◽  
Differential Polynomial ◽  
Differential Polynomials ◽  
Linear Polynomial ◽  
Linear Differential Polynomial ◽  
Linear Differential Polynomials ◽  
Linear Differential ◽  
Function Sharing

We study the uniqueness of entire functions, when they share a linear polynomial, in particular, fixed points, with their linear differential polynomials.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

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