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Entire Solutions of Certain Type of Non-Linear Difference Equations

Entire Solutions of Certain Type of Non-Linear Difference Equations In this paper, we study the existence of entire solutions of finite-order of non-linear difference equations of the form $$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$ f n ( z ) + q ( z ) Δ c f ( z ) = p 1 e α 1 z + p 2 e α 2 z , n ≥ 2 and $$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$ f n ( z ) + q ( z ) e Q ( z ) f ( z + c ) = p 1 e λ z + p 2 e - λ z , n ≥ 3 where q, Q are non-zero polynomials, $$c,\lambda ,p_{i},\alpha _{i}(i=1,2)$$ c , λ , p i , α i ( i = 1 , 2 ) are non-zero constants such that $$\alpha _{1}\ne \alpha _{2}$$ α 1 ≠ α 2 and $$\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0$$ Δ c f ( z ) = f ( z + c ) - f ( z ) ≢ 0 . Our results are improvements and complements of Wen et al. (Acta Math Sin 28:1295–1306, 2012), Yang and Laine (Proc Jpn Acad Ser A Math Sci 86:10–14, 2010) and Zinelâabidine (Mediterr J Math 14:1–16, 2017). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Entire Solutions of Certain Type of Non-Linear Difference Equations

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-018-0250-6
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the existence of entire solutions of finite-order of non-linear difference equations of the form $$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$ f n ( z ) + q ( z ) Δ c f ( z ) = p 1 e α 1 z + p 2 e α 2 z , n ≥ 2 and $$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$ f n ( z ) + q ( z ) e Q ( z ) f ( z + c ) = p 1 e λ z + p 2 e - λ z , n ≥ 3 where q, Q are non-zero polynomials, $$c,\lambda ,p_{i},\alpha _{i}(i=1,2)$$ c , λ , p i , α i ( i = 1 , 2 ) are non-zero constants such that $$\alpha _{1}\ne \alpha _{2}$$ α 1 ≠ α 2 and $$\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0$$ Δ c f ( z ) = f ( z + c ) - f ( z ) ≢ 0 . Our results are improvements and complements of Wen et al. (Acta Math Sin 28:1295–1306, 2012), Yang and Laine (Proc Jpn Acad Ser A Math Sci 86:10–14, 2010) and Zinelâabidine (Mediterr J Math 14:1–16, 2017).

Journal

Computational Methods and Function Theory – Springer Journals

Published: Aug 27, 2018

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