scholarly journals Oscillation of Solutions of LDE’s in Domains Conformally Equivalent to Unit Disc

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
I. Chyzhykov ◽  
J. Gröhn ◽  
J. Heittokangas ◽  
J. Rättyä
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Conformal Transformation ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Free Solution ◽  
Order Linear ◽  
Linear Differential ◽  
Oscillation Of Solutions

AbstractOscillation of solutions of $$f^{(k)} + a_{k-2} f^{(k-2)} + \cdots + a_1 f' +a_0 f = 0$$ f ( k ) + a k - 2 f ( k - 2 ) + ⋯ + a 1 f ′ + a 0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.

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 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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