Some results on q-difference equations

Zhang Junchao, Wang Gang, Chen Junjie, Zhao, Zhao Rongxiang

Description

In this paper, we consider the q -difference analogue of the Clunie theorem. We obtain there is no zero-order entire solution of f n ( z ) + ( ∇ q f ( z ) ) n = 1 when n ≥ 2 ; there is no zero-order transcendental entire solution of f n ( z ) + P ( z ) ( ∇ q f ( z ) ) m = Q ( z ) when n > m ≥ 0 ; and the equation f n + P ( z ) ∇ q f ( z ) = h ( z ) has at most one zero-order transcendental entire solution f if f is not the solution of ∇ q f ( z ) = 0 , when n ≥ 4 . MSC: 30D35, 30D30, 39A13, 39B12.

Sujets

• Uniqueness quantification
• List of zero terms
• Nevanlinna theory

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 16
• Langue: Français

Extrait

Zhang et al.Advances in Difference Equations2012,2012:191 http://www.advancesindifferenceequations.com/content/2012/1/191

R E S E A R C H

Some results onq-difference equations 1 2 1*3 Junchao Zhang, Gang Wang, Junjie Chenand Rongxiang Zhao

* Correspondence: [e-mail] 1 College of Computer Science and Technology, Taiyuan University of Technology, Taiyuan, 030024, China Full list of author information is available at the end of the article

Open Access

Abstract In this paper, we consider theq-difference analogue of the Clunie theorem. We n n obtain there is no zero-order entire solution off(z) + (∇ q f(z1 when)) =n≥2; there n m is no zero-order transcendental entire solution off(z) +P(z)(∇ q f(z)) =Q(z) when n n>m≥0; and the equationf+P(z)∇ q f(z) =h(z) has at most one zero-order transcendental entire solutionfiffis not the solution of∇ q f(z) = 0, whenn≥4. MSC:Primary 30D35; secondary 30D30; 39A13; 39B12 Keywords:uniqueness;q-shift;q-difference equations; entire functions; zero order; Nevanlinna theory

1 Introductionand main results It is well known that Clunie’s theorem (see [], Lemma ; also see [], p., Lemma ..) is a useful tool in studying complex differential equations. It states thatQ n (f) is a polyno- mial of total degreenat most in the meromorphic functionfand its derivatives having meromorphic functions as coefficients. IfT(r) is the maximum of the characteristics of the coefficients, then     + –n   logf Q n (f)dϕ=Ologr+logT(r,f) +T(ras) n.e.r→ ∞. π |f|>

Later, Clunie’s theorem has been improved into many forms (see [], pp.-) which are valuable tools for studying meromorphic solutions of Painlevé and other non-linear differential equations; see,e.g., []. In , Laine and Yang [] obtained a discrete version of Clunie’s theorem.

Theorem ALet fbe a transcendental meromorphic solution of finite-orderρof a differ- ence equation of the form

U(z,f)P(z,f) =Q(z,f),

where U(z,f),P(z,f),and Q(z,f)are difference polynomials such that the total degree degU(z,f) =n in f(z)and its shifts,anddegQ(z,f)≤n.Moreover,we assume that U(z,f) contains just one term of maximal total degree in f(z)and its shifts.Then for eachε> ,

   ρ–+ε m r,P(z,f) =O r+S(r,f)

possibly outside of an exceptional set of finite logarithmic measure.

©2012 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.