scholarly journals Complex Oscillation of Higher-Order Linear Differential Equations with Coefficients Being Lacunary Series of Finite Iterated Order

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jin Tu ◽  
Hong-Yan Xu ◽  
Hua-ming Liu ◽  
Yong Liu
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Lacunary Series ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Iterated Order ◽  
Complex Oscillation ◽  
Linear Differential ◽  
Growth Of Solutions

The authors introduce the lacunary series of finite iterated order and use them to investigate the growth of solutions of higher-order linear differential equations with entire coefficients of finite iterated order and obtain some results which improve and extend some previous results of Belaidi, 2006, Cao and Yi, 2007, Kinnunen, 1998, Laine and Wu, 2000, Tu and Chen, 2009, Tu and Deng, 2008, Tu and Deng, 2010, Tu and Liu, 2009, and Tu and Long, 2009.

scholarly journals Fast growth and fixed points of solutions of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 6 (10) ◽  
pp. 10833-10845
Author(s):  
Yu Chen ◽  
◽  
Guantie Deng ◽  
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Exponent Of Convergence ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential

<abstract><p>In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.</p></abstract>

scholarly journals Some Oscillation Results of Higher-Order Linear Differential Equations with Meromorphic Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.

scholarly journals Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu Chen ◽  
Guan-Tie Deng ◽  
Zhan-Mei Chen ◽  
Wei-Wei Wang
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential ◽  
Derivatives Of

AbstractIn this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

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