complex differences and difference...

Guest Editors: Zong-Xuan Chen, Kwang Ho Shon, and Zhi-Bo Huang

Complex Differences and Difference Equations

Abstract and Applied Analysis

Abstract and Applied Analysis


Guest Editors: Zong-Xuan Chen, Kwang Ho Shon,and Zhi-Bo Huang

Copyright © 2014 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Ravi P. Agarwal, USABashir Ahmad, Saudi ArabiaM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalM. K. Aouf, EgyptNarcisa C. Apreutesei, RomaniaNatig M. Atakishiyev, MexicoFerhan M. Atici, USAIvan Avramidi, USASoohyun Bae, KoreaChuanzhi Bai, ChinaZhanbing Bai, ChinaDumitru Baleanu, TurkeyJózef Banaś, PolandMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Başar, TurkeyAbdelghani Bellouquid, MoroccoDaniele Bertaccini, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin J. Bohner, USAGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, KoreaFabio M. Camilli, ItalyJinde Cao, ChinaAnna Capietto, ItalyJianqing Chen, ChinaWing-Sum Cheung, Hong KongMichel Chipot, SwitzerlandChangbum Chun, KoreaSoon-Yeong Chung, KoreaJaeyoung Chung, KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, Italy

Juan Carlos Cortés López, SpainGraziano Crasta, ItalyZhihua Cui, ChinaBernard Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranLuis F. Pinheiro de Castro, PortugalToka Diagana, USAJesús I. Dı́az, SpainJosef Dibĺık, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainAlexander Domoshnitsky, IsraelMarco Donatelli, ItalyBoQing Dong, ChinaWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USAAhmed El-Sayed, EgyptLuca Esposito, ItalyKhalil Ezzinbi, MoroccoJulian F. Bonder, ArgentinaDashan Fan, USAAngelo Favini, ItalyMárcia Federson, BrazilStathis Filippas, Equatorial GuineaAlberto Fiorenza, ItalyIlaria Fragala, ItalyXianlong Fu, ChinaMassimo Furi, ItalyJesús G. Falset, SpainGiovanni P. Galdi, USAIsaac Garcia, SpainJ. A. G.a-R.guez, SpainLeszek Gasinski, PolandGyörgy Gát, HungaryVladimir Georgiev, ItalyLorenzo Giacomelli, ItalyJaume Giné, SpainValery Y. Glizer, IsraelJean P. Gossez, BelgiumJose L. Gracia, SpainMaurizio Grasselli, ItalyLuca Guerrini, Italy

Yuxia Guo, ChinaQian Guo, ChinaC. P. Gupta, USAUno Hämarik, EstoniaMaoan Han, ChinaFerenc Hartung, HungaryJiaxin Hu, ChinaZhongyi Huang, ChinaChengming Huang, ChinaGennaro Infante, ItalyIvan Ivanov, BulgariaHossein Jafari, South AfricaJaan Janno, EstoniaAref Jeribi, TunisiaUncig Ji, KoreaZhongxiao Jia, ChinaLucas Jódar, SpainJong S. Jung, Republic of KoreaHenrik Kalisch, NorwayHamid R. Karimi, NorwayChaudry M. Khalique, South AfricaSatyanad Kichenassamy, FranceTero Kilpeläinen, FinlandSung G. Kim, Republic of KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, FinlandVictor Kovtunenko, AustriaRen-Jieh Kuo, TaiwanPavel Kurasov, SwedenMilton C. L. Filho, BrazilMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USAMatti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, ChinaElena Litsyn, IsraelShengqiang Liu, ChinaYansheng Liu, ChinaCarlos Lizama, ChileGuozhen Lu, USAJinhu Lü, China

Grzegorz Lukaszewicz, PolandWanbiao Ma, ChinaNazim I. Mahmudov, TurkeyEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesús Maŕın-Solano, SpainJose M. Martell, SpainM. Mastyło, PolandMing Mei, CanadaTaras Mel’nyk, UkraineAnna Mercaldo, ItalyStanislaw Migorski, PolandMihai Mihǎilescu, RomaniaFeliz Minhós, PortugalDumitru Motreanu, FranceMaria Grazia Naso, ItalyGaston M. N’Guerekata, USAMicah Osilike, NigeriaMitsuharu Ôtani, JapanTurgut Ôziş, TurkeyNikolaos S. Papageorgiou, GreeceSehie Park, KoreaKailash C. Patidar, South AfricaKevin R. Payne, ItalyAdemir F. Pazoto, BrazilShuangjie Peng, ChinaAntonio M. Peralta, SpainSergei V. Pereverzyev, AustriaAllan Peterson, USAAndrew Pickering, SpainCristina Pignotti, ItalySomyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, Mexico

Irena Rachu̇nková, Czech RepublicMaria A. Ragusa, ItalySimeon Reich, IsraelAbdelaziz Rhandi, ItalyHassan Riahi, MalaysiaJuan P. Rincón-Zapatero, SpainLuigi Rodino, ItalyYuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalySatit Saejung, ThailandStefan G. Samko, PortugalMartin Schechter, USAJavier Segura, SpainValery Serov, FinlandNaseer Shahzad, Saudi ArabiaAndrey Shishkov, UkraineStefan Siegmund, GermanyAbdel-Maksoud A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalyHari M. Srivastava, CanadaSvatoslav Staněk, Czech RepublicStevo Stević, SerbiaAntonio Suárez, SpainWenchang Sun, ChinaWenyu Sun, ChinaRobert Szalai, UKSanyi Tang, ChinaChun-Lei Tang, ChinaGabriella Tarantello, ItalyNasser-Eddine Tatar, Saudi ArabiaGerd Teschke, GermanySergey Tikhonov, SpainClaudia Timofte, RomaniaThanh Tran, AustraliaJuan J. Trujillo, SpainGabriel Turinici, FranceMilan Tvrdy, Czech Republic

Mehmet nal, TurkeyCsaba Varga, RomaniaCarlos Vazquez, SpainJesus Vigo-Aguiar, SpainQing-WenWang, ChinaYushun Wang, ChinaShawn X. Wang, CanadaYouyu Wang, ChinaJing P. Wang, UKPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeYonghong Wu, AustraliaZili Wu, ChinaShi-Liang Wu, ChinaShanhe Wu, ChinaTiecheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaGongnan Xie, ChinaFuding Xie, ChinaDaoyi Xu, ChinaZhenya Yan, ChinaXiaodong Yan, USANorio Yoshida, JapanBeong In Yun, KoreaAgacik Zafer, TurkeyJianming Zhan, ChinaWeinian Zhang, ChinaChengjian Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaYong Zhou, ChinaTianshou Zhou, ChinaChun-Gang Zhu, ChinaQiji J. Zhu, USAMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

Complex Differences and Difference Equations, Zong-Xuan Chen, Kwang Ho Shon, and Zhi-Bo HuangVolume 2014, Article ID 124843, 1 page

Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and NonlinearNeumann Boundary Condition, Zhong Bo Fang and Yan ChaiVolume 2014, Article ID 289245, 8 pages

Some Properties on Complex Functional Difference Equations, Zhi-Bo Huang and Ran-Ran ZhangVolume 2014, Article ID 283895, 10 pages

The Regularity of Functions on Dual Split Quaternions in Clifford Analysis,Ji Eun Kim and Kwang Ho ShonVolume 2014, Article ID 369430, 8 pages

Unicity of Meromorphic Functions Sharing Sets withTheir Linear Difference Polynomials,Sheng Li and BaoQin ChenVolume 2014, Article ID 894968, 7 pages

A ComparisonTheorem for Oscillation of the Even-Order Nonlinear Neutral Difference Equation,Quanxin ZhangVolume 2014, Article ID 492492, 5 pages

Difference Equations and Sharing Values Concerning Entire Functions andTheir Difference,Zhiqiang Mao and Huifang LiuVolume 2014, Article ID 584969, 6 pages

Admissible Solutions of the Schwarzian Type Difference Equation, Baoqin Chen and Sheng LiVolume 2014, Article ID 306360, 5 pages

Statistical Inference for Stochastic Differential Equations with Small Noises,Liang Shen and Qingsong XuVolume 2014, Article ID 473681, 6 pages

On the Deficiencies of Some Differential-Difference Polynomials, Xiu-Min Zheng and Hong Yan XuVolume 2014, Article ID 378151, 12 pages

On Growth of Meromorphic Solutions of Complex Functional Difference Equations, Jing Li,Jianjun Zhang, and Liangwen LiaoVolume 2014, Article ID 828746, 6 pages

Unicity of Entire Functions concerning Shifts and Difference Operators, Dan Liu, Degui Yang,and Mingliang FangVolume 2014, Article ID 380910, 5 pages

On Positive Solutions and Mann Iterative Schemes of aThird Order Difference Equation, Zeqing Liu,Heng Wu, Shin Min Kang, and Young Chel KwunVolume 2014, Article ID 470181, 16 pages

Algebroid Solutions of Second Order Complex Differential Equations, Lingyun Gao and Yue WangVolume 2014, Article ID 123049, 4 pages

EditorialComplex Differences and Difference Equations

Zong-Xuan Chen,1 Kwang Ho Shon,2 and Zhi-Bo Huang1

1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Correspondence should be addressed to Zong-Xuan Chen; [email protected]

Received 12 August 2014; Accepted 12 August 2014; Published 22 December 2014

Copyright © 2014 Zong-Xuan Chen et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In more recent years, activity in the area of the complexdifferences and the complex difference equations has fleetlyincreased.

This journal has set up a column of this special issue. Wewere pleased to invite the interested authors to contributetheir original research papers as well as good expositorypapers to this special issue that will make better improvementon the theory of complex differences and difference equa-tions.

In this special issue, many good results are obtained.Difference equations are widely applied to mathematical

physics, economics, and chemistry. In this special issue, Z.-B. Huang and R.-R. Zhang, J. Li et al., and L. Gao and Y.Wang investigate the growth, a Borel exceptional value ofmeromorphic solutions to different types of higher ordernonliear difference equations, respectively; B. Chen and S. Liinvestigate the Schwarzian type difference equation. D. Liuet al., Z. Mao and H. Liu, and S. Li and B. Chen investigateunicity of meromorphic functions concerning different typesof difference operators. Recently, many difference analoguesof the classic Nevanlinna theory are obtained.

In this special issue, X.-M. Zheng and H. Y. Xu obtaina differential difference analogue of Valiron-Mohonko theo-rem. Related topics with complex difference, J. E. Kim andK. H. Shon investigate the regularity of functions on dualsplit quaternions in Clifford analysis and the tensor productrepresentation of polynomials of weak type in a DF-space;Q. Zhang and Z. Liu et al. investigate different types ofreal difference equations, respectively; L. Shen and Q. Xuinvestigate stochastic differential equations.

This special issue stimulates the continuing efforts to thecomplex differences and the complex difference equations.

Zong-XuanChenKwangHo ShonZhi-BoHuang

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 124843, 1 pagehttp://dx.doi.org/10.1155/2014/124843

http://dx.doi.org/10.1155/2014/124843

Research ArticleBlow-Up Analysis for a Quasilinear Parabolic Equation withInner Absorption and Nonlinear Neumann Boundary Condition

Zhong Bo Fang and Yan Chai

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Zhong Bo Fang; [email protected]

Received 24 February 2014; Revised 11 April 2014; Accepted 11 April 2014; Published 30 April 2014

Academic Editor: Zhi-Bo Huang

Copyright © 2014 Z. B. Fang and Y. Chai. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinearNeumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee that 𝑢(𝑥, 𝑡) exists globally orblows up at some finite time 𝑡∗. Moreover, an upper bound for 𝑡∗ is derived. Under somewhat more restrictive conditions, a lowerbound for 𝑡∗ is also obtained.

1. Introduction

We are concerned with the global existence and blow-upphenomenon for a quasilinear parabolic equation with non-linear inner absorption term

𝑢𝑡= [(|∇𝑢|

𝑝

+ 1) 𝑢,𝑖],𝑖

− 𝑓 (𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡∗

) , (1)

subjected to the nonlinear Neumann boundary and initialconditions

(|∇𝑢|𝑝

+ 1)

𝜕𝑢

𝜕]= 𝑔 (𝑢) , (𝑥, 𝑡) ∈ 𝜕Ω × (0, 𝑡

∗

) , (2)

𝑢 (𝑥, 0) = 𝑢0(0) ≥ 0, 𝑥 ∈ Ω, (3)

whereΩ is a bounded star-shaped region of 𝑅𝑁 (𝑁 ≥ 2) withsmooth boundary 𝜕Ω, ] is the unit outward normal vectoron 𝜕Ω, 𝑝 ≥ 0, 𝑡∗ is the blow-up time if blow-up occurs, orelse 𝑡∗ = +∞, the symbol, 𝑖 denotes partial differentiationwith respect to 𝑥

𝑖, 𝑖 = 1, 2, . . . , 𝑁, the repeated index indicates

summation over the index, and ∇ is gradient operator.Many physical phenomena and biological species theo-

ries, such as the concentration of diffusion of some non-Newton fluid through porous medium, the density of somebiological species, and heat conduction phenomena, havebeen formulated as parabolic equation (1) (see [1–3]). Thenonlinear Neumann boundary condition (2) can be physi-cally interpreted as the nonlinear radial law (see [4, 5]).

In the past decades, there have been many works dealingwith existence and nonexistence of global solutions, blow-upof solutions, bounds of blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of solutions to nonlinearparabolic equations; see the books [6–8] and the surveypapers [9–11]. Specially, we would like to know whether thesolution blows up and at which time when blow-up occurs.A variety of methods have been used to study the problemabove (see [12]), and in many cases, these methods, used toshow that solutions blow up, often provide an upper boundfor the blow-up time. However, lower bounds for blow-uptime may be harder to be determined. For the study ofthe initial boundary value problem of a parabolic equationwith homogeneous Dirichlet boundary condition, see [13,14]. Payne et al. [13] considered the following quasilinearparabolic equation:

𝑢𝑡= div (𝜌|∇𝑢|2∇𝑢) + 𝑓 (𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡∗) , (4)

where Ω is a bounded domain in 𝑅3 with smooth boundary𝜕Ω. To get the lower bound for the blow-up time, the authorsassumed that 𝜌 is a positive 𝐶1 function which satisfies

𝜌 (𝑠) + 𝑠𝜌

(𝑠) > 0, 𝑠 > 0. (5)

The lower bound for the blow-up time of solution to (4) withRobin boundary condition was obtained in [15], where 𝜌 is

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 289245, 8 pageshttp://dx.doi.org/10.1155/2014/289245

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2 Abstract and Applied Analysis

also assumed to satisfy the condition (5). However, under thisboundary condition, the best constant of Sobolev inequalityused in [13] is no longer applicable. They imposed suitableconditions on 𝑓 and 𝜌 and determined a lower bound forthe blow-up time if blow-up occurs and determined whenblow-up cannot occur. Marras and Vernier Piro [14] studiedthe nonlinear parabolic problem with time dependent coeffi-cients

𝑘1(𝑡) div (𝑔 (|∇𝑢|2∇𝑢)) + 𝑘

2(𝑡) 𝑓 (𝑢) = 𝑘

3(𝑡) 𝑢𝑡,

(𝑥, 𝑡) ∈ Ω × (0, 𝑡∗

) ,

(6)

where Ω is a bounded domain in 𝑅𝑁 with smooth boundary𝜕Ω. Under some conditions on the data and geometry of thespatial domain, they obtained upper and lower bounds of theblow-up time. Moreover, the sufficient conditions for globalexistence of the solution were derived.

For the study of the initial boundary value problem ofa parabolic equation with Robin boundary condition, werefer to [15–19]. Li et al. [16] investigated the problem of thenonlinear parabolic equation

𝑢𝑡= [(|∇𝑢|

𝑝

+ 1) 𝑢,𝑖],𝑖

+ 𝑓 (𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡∗

) , (7)

where Ω is a bounded domain in 𝑅3 with smooth boundary𝜕Ω.They derived the lower bound for the blow-up timewhenthe blow-up occurs. Clearly, |∇𝑢|𝑝 + 1 does not satisfy thecondition (5). Enache [17] discussed the quasilinear parabolicproblem

𝑢𝑡= (𝑔 (𝑢) 𝑢

,𝑖),𝑖

+ 𝑓 (𝑢) , (8)

where Ω is a bounded domain in 𝑅𝑁 (𝑁 ≥ 2) with smoothboundary 𝜕Ω. By virtue of a first-order differential inequalitytechnique, they showed the sufficient conditions to guaranteethat the solution 𝑢(𝑥, 𝑡) exists globally or blows up. Inaddition, a lower bound for the blow-up time when blow-up occurs was also obtained. Ding [18] studied the nonlinearparabolic problem

(𝑏 (𝑢))𝑡= ∇ ⋅ (𝑔 (𝑢) ∇𝑢) + 𝑓 (𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡

∗

) , (9)

where Ω is a bounded domain in 𝑅3 with smooth boundary𝜕Ω. They derived conditions on the data which guaranteethe blow-up or the global existence of the solution. A lowerbound on blow-up time when blow-up occurs was alsoobtained. For the problem of the nonlinear nonlocal porousmedium equation, we read the paper of Liu [19].

Recently, for the problems with nonlinear Neumannboundary conditions, Payne et al. [20] studied the semilinearheat equation with inner absorption term

𝑢𝑡= Δ𝑢 − 𝑓 (𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡

∗

) . (10)

They established conditions on nonlinearity to guarantee thatthe solution𝑢(𝑥, 𝑡) exists for all time 𝑡 > 0 or blows up at somefinite time 𝑡∗. Moreover, an upper bound for 𝑡∗ was derived.Under somewhat more restrictive conditions, a lower bound

for 𝑡∗ was derived.Thereafter, they considered the quasilinearparabolic equation

𝑢𝑡= ∇ ⋅ (|∇𝑢|

𝑝

∇𝑢) , (𝑥, 𝑡) ∈ Ω × (0, 𝑡∗

) , (11)

and they showed that blow-up occurs at some finite timeunder certain conditions on the nonlinearities and the data;upper and lower bounds for the blow-up time were derivedwhen blow-up occurs; see [21]. Liu et al. The authors [22, 23]studied the reaction diffusion problem with nonlocal sourceand inner absorption terms or with local source and gradientabsorption terms. Very recently, Fang et al. [24] consideredlower bounds estimate for the blow-up time to nonlocalproblemwith homogeneousDirichlet orNeumann boundarycondition.

Motivated by the above work, we intend to study theglobal existence and the blow-up phenomena of problem (1)–(3), and the results of the semilinear equations are extended tothe quasilinear equations. Unfortunately, the techniques usedfor semilinear equation to analysis of blow-up phenomenaare no longer applicable to our problem. As a consequence,by using the suitable techniques of differential inequalities,we establish, respectively, the conditions on the nonlinearities𝑓 and 𝑔 to guarantee that 𝑢(𝑥, 𝑡) exists globally or blows upat some finite time. If blow-up occurs, we derive upper andlower bounds of the blow-up time.

The rest of our paper is organized as follows. In Section 2,we establish conditions on the nonlinearities to guaranteethat 𝑢(𝑥, 𝑡) exists globally. In Section 3, we show the condi-tions on data forcing the solution 𝑢(𝑥, 𝑡) to blow up at somefinite time 𝑡∗ and obtain an upper bound for 𝑡∗. A lowerbound of blow-up time under some assumptions is derivedin Section 4.

2. The Global Existence

In this section,we establish the conditions on the nonlinearity𝑓 and nonlinearity 𝑔 to guarantee that 𝑢(𝑥, 𝑡) exists globally.We state our result as follows.

Theorem 1. Assume that the nonnegative functions 𝑓 and 𝑔satisfy

𝑓 (𝜉) ≥ 𝑘1𝜉𝑞

, 𝜉 ≥ 0,

𝑔 (𝜉) ≤ 𝑘2𝜉𝑠

, 𝜉 ≥ 0,

(12)

where 𝑘1> 0, 𝑘

2≥ 0, 𝑠 > 1, 2𝑠 < 𝑞 + 1, and 𝑠 − 1 < 𝑝 < 𝑞 − 1.

Then the (nonnegative) solution 𝑢(𝑥, 𝑡) of problem (1)-(3) doesnot blow up; that is, 𝑢(𝑥, 𝑡) exists for all time 𝑡 > 0.

Proof. Set

Ψ (𝑡) = ∫

Ω

𝑢2

𝑑𝑥. (13)

Abstract and Applied Analysis 3

Similar to Theorem 2.1 in [20], we get

Ψ

(𝑡) ≤ {2𝛿2

∫

Ω

𝑢2𝑠

𝑑𝑥 − 𝑘1∫

Ω

𝑢𝑞+1

𝑑𝑥}

+ {

2𝑘2𝑁

𝜌0

∫

Ω

𝑢𝑠+1

𝑑𝑥

−2∫

Ω

|∇𝑢|𝑝+2

𝑑𝑥 − 𝑘1∫

Ω

𝑢𝑞+1

𝑑𝑥}

= 𝐼1+ 𝐼2,

(14)

where 𝛿 = 𝑘2(𝑠 + 1)𝑑/2𝜌

0, 𝜌0

= min𝑥∈𝜕Ω

(𝑥 ⋅ ]), 𝑑 =max𝑥∈𝜕Ω

|𝑥|, and

𝐼1≤ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑠+1)/(𝑞+1)

× {𝐴1|Ω|(𝑞−𝑠)/(𝑞+1)

− 𝐴2(∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑞−𝑠)/(𝑞+1)

} ,

(15)

where 𝐴1= 2𝛿2

𝛼𝜀(𝛼−1)/𝛼, 𝐴

2= 𝑘1− 2𝛿2

(1 − 𝛼)𝜀, 𝛼 = (𝑞 + 1 −2𝑠)/(𝑞 − 𝑠) < 1, 𝜀 > 0.

Next, we estimate 𝐼2= (2𝑘2𝑁/𝜌0) ∫Ω

𝑢𝑠+1

𝑑𝑥−2 ∫Ω

|∇𝑢|𝑝+2

𝑑𝑥 − 𝑘1∫Ω

𝑢𝑞+1

𝑑𝑥. Since

∇𝑢(𝑝/2)+1

2

= (

𝑝

2

+ 1)

2

𝑢𝑝

|∇𝑢|2

, (16)

it follows from Hölder inequality that

∫

Ω

∇𝑢(𝑝/2)+1

2

𝑑𝑥 ≤ (

𝑝

2

+ 1)

2

(∫

Ω

|∇𝑢|𝑝+2

𝑑𝑥)

2/(𝑝+2)

× (∫

Ω

𝑢𝑝+2

𝑑𝑥)

𝑝/(𝑝+2)

.

(17)

Furthermore, we have

∫

Ω

𝑢𝑝+2

𝑑𝑥 ≤ [

(𝑝 + 2)2

4𝜆1

]

(𝑝/2)+1

∫

Ω

|∇𝑢|𝑝+2

𝑑𝑥, (18)

which follows from (17) and membrane inequality

𝜆1∫

Ω

𝜔2

𝑑𝑥 ≤ ∫

Ω

|∇𝜔|2

𝑑𝑥, (19)

where 𝜆1is the first eigenvalue in the fixed membrane

problem

Δ𝜔 + 𝜆𝜔 = 0, 𝜔 > 0 in Ω, 𝜔 = 0 on 𝜕Ω. (20)

Combining 𝐼2and (18), we have

𝐼2≤

2𝑘2𝑁

𝜌0

∫

Ω

𝑢𝑠+1

𝑑𝑥 − 2[

4𝜆1

(𝑝 + 2)2]

(𝑝/2)+1

× ∫

Ω

𝑢𝑝+2

𝑑𝑥 − 𝑘1∫

Ω

𝑢𝑞+1

𝑑𝑥.

=

{

{

{

2𝑘2𝑁

𝜌0

∫

Ω

𝑢𝑠+1

𝑑𝑥 − 3[

4𝜆1

(𝑝 + 2)2

]

(𝑝/2)+1

∫

Ω

𝑢𝑝+2

𝑑𝑥

}

}

}

+

{

{

{

[

4𝜆1

(𝑝 + 2)2]

(𝑝/2)+1

∫

Ω

𝑢𝑝+2

𝑑𝑥 − 𝑘1∫

Ω

𝑢𝑞+1

𝑑𝑥

}

}

}

= 𝐼21+ 𝐼22.

(21)

Making use of Hölder inequality, we obtain

∫

Ω

𝑢𝑠+1

𝑑𝑥 ≤ (∫

Ω

𝑢𝑝+2

𝑑𝑥)

(𝑠+1)/(𝑝+2)

|Ω|(𝑝−𝑠+1)/(𝑝+2)

, (22)

Ψ (𝑡) = ∫

Ω

𝑢2

𝑑𝑥 ≤ (∫

Ω

𝑢𝑠+1

𝑑𝑥)

2/(𝑠+1)

|Ω|(𝑠−1)/(𝑠+1)

. (23)

Combining (21), (22) with (23), we get

𝐼21

≤ (∫

Ω

𝑢𝑠+1

𝑑𝑥) {𝐵1− 𝐵2Ψ(𝑝−𝑠+1)/2

} , (24)

with

𝐵1=

2𝑘2𝑁

𝜌0

, 𝐵2= 3[

4𝜆1

(𝑝 + 2)2]

(𝑝/2)+1

|Ω|−(𝑝−𝑠+1)/2

.

(25)

Applying Hölder inequality, we obtain

∫

Ω

𝑢𝑝+2

𝑑𝑥 ≤ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑝+2)/(𝑞+1)

|Ω|(𝑞−𝑝−1)/(𝑞+1)

,

Ψ (𝑡) = ∫

Ω

𝑢2

𝑑𝑥 ≤ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

2/(𝑞+1)

|Ω|(𝑞−1)/(𝑞+1)

.

(26)

It follows from (26) that

𝐼22

≤ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑝+2)/(𝑞+1)

{𝐶1− 𝐶2Ψ(𝑞−𝑝−1)/2

} , (27)

where

𝐶1= [

4𝜆1

(𝑝 + 2)2

]

(𝑝/2)+1

|Ω|(𝑞−𝑝−1)/(𝑞+1)

,

𝐶2= 𝑘1|Ω|(1−𝑞)(𝑞−𝑝−1)/2(𝑞+1)

.

(28)


Combining (14), (15), (21), and (24) with (27), we obtain

Ψ

(𝑡) ≤ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑠+1)/(𝑞+1)

{𝐴1− 𝐴2Ψ(𝑡)(𝑞−𝑠)/2

}

+ (∫

Ω

𝑢𝑠+1

𝑑𝑥) {𝐵1− 𝐵2Ψ(𝑝−𝑠+1)/2

}

+ (∫

Ω

𝑢𝑞+1

𝑑𝑥)

(𝑝+2)/(𝑞+1)

{𝐶1− 𝐶2Ψ(𝑞−𝑝−1)/2

} ,

(29)

with

𝐴1= 𝐴1|Ω|(𝑞−𝑠)/(𝑞+1)

, 𝐴2= 𝐴2|Ω|(1−𝑞)(𝑞−𝑠)/2(𝑞+1)

. (30)

We conclude from (29) that Ψ(𝑡) is decreasing in each timeinterval on which we obtain

Ψ (𝑡) ≥ max{(𝐴1

𝐴2

)

2/(𝑞−𝑠)

, (

𝐵1

𝐵2

)

2/(𝑝−𝑠+1)

, (

𝐶1

𝐶2

)

2/(𝑞−𝑝−1)

} ,

(31)

so that Ψ(𝑡) remains bounded for all time under theconditions in Theorem 1. This completes the proof ofTheorem 1.

3. Blow-Up and Upper Bound of 𝑡∗

In this section, Ω needs not to be star-shaped. We establishthe conditions to assure that the solution of (1)–(3) blowsup at finite time 𝑡∗ and derive an upper bound for 𝑡∗. Moreprecisely we establish the following result.

Theorem 2. Let 𝑢(𝑥, 𝑡) be the classical solution of problem (1)-(3). Assume that the nonnegative and integrable functions 𝑓and 𝑔 satisfy

𝜉𝑓 (𝜉) ≤ 2 (1 + 𝛼) 𝐹 (𝜉) , 𝜉 ≥ 0,

𝜉𝑔 (𝜉) ≥ 2 (1 + 𝛽)𝐺 (𝜉) , 𝜉 ≥ 0,

(32)

with

𝐹 (𝜉) = ∫

𝜉

0

𝑓 (𝜂) 𝑑𝜂, 𝐺 (𝜉) = ∫

𝜉

0

𝑔 (𝜂) 𝑑𝜂, (33)

where 𝛼 ≥ 0,

𝛽 ≥ max (𝑝

2

, 𝛼) . (34)

Moreover assume that Φ(0) ≥ 0 with

Φ (𝑡) = 2∫

𝜕Ω

𝐺 (𝑢) 𝑑𝑆 − ∫

Ω

|∇𝑢|2

(1 +

2

𝑝 + 2

|∇𝑢|𝑝

)𝑑𝑥

− 2∫

Ω

𝐹 (𝑢) 𝑑𝑥.

(35)

Then the solution 𝑢(𝑥, 𝑡) of problem (1)-(3) blows up at somefinite time 𝑡∗ < 𝑇 with

𝑇 =

Ψ (0)

2𝛽 (1 + 𝛽)Φ (0)

, 𝛽 > 0, (36)

where Ψ(𝑡) is defined in (13). If 𝛽 = 0, we have 𝑇 = ∞.

Proof. We compute

Ψ

(𝑡) = 2∫

Ω

𝑢𝑢𝑡𝑑𝑥 = 2∫

Ω

𝑢 [((|∇𝑢|𝑝

+ 1) 𝑢,𝑖),𝑖

− 𝑓 (𝑢)] 𝑑𝑥

= 2∫

𝜕Ω

𝑢 (|∇𝑢|𝑝

+ 1)

𝜕𝑢

𝜕]𝑑𝑆 − 2∫

Ω

(|∇𝑢|𝑝

+ 1) |∇𝑢|2

𝑑𝑥

− 2∫

Ω

𝑢𝑓 (𝑢) 𝑑𝑥

= 2∫

𝜕Ω

𝑢𝑔 (𝑢) 𝑑𝑆 − 2∫

Ω

(|∇𝑢|𝑝

+ 1) |∇𝑢|2

𝑑𝑥

− 2∫

Ω

𝑢𝑓 (𝑢) 𝑑𝑥.

(37)

Making use of the hypotheses stated inTheorem 2, we have

Ψ

(𝑡) ≥ 2 (1 + 𝛽)Φ (𝑡) . (38)

Differentiating (35), we derive

Φ

(𝑡) = 2∫

𝜕Ω

𝑔 (𝑢) 𝑢𝑡𝑑𝑆 − ∫

Ω

(|∇𝑢|𝑝

+ 1) (|∇𝑢|2

)𝑡

𝑑𝑥

− 2∫

Ω

𝑓 (𝑢) 𝑢𝑡𝑑𝑥.

(39)

Integrating the identity ∇ ⋅ (𝑢𝑡(|∇𝑢|𝑝

+1)∇𝑢) = 𝑢𝑡∇ ⋅ ((|∇𝑢|

𝑝

+

1)∇𝑢) + (1/2)(|∇𝑢|𝑝

+ 1)(|∇𝑢|2

)𝑡overΩ, we get

∫

Ω

(|∇𝑢|𝑝

+ 1) (|∇𝑢|2

)𝑡

𝑑𝑥

= 2∫

Ω

∇ ⋅ (𝑢𝑡(|∇𝑢|𝑝

+ 1) ∇𝑢) 𝑑𝑥

− 2∫

Ω

𝑢𝑡∇ ⋅ ((|∇𝑢|

𝑝

+ 1) ∇𝑢) 𝑑𝑥

= 2∫

𝜕Ω

𝑢𝑡(|∇𝑢|𝑝

+ 1) ∇𝑢 ⋅ ]𝑑𝑆

− 2∫

Ω

𝑢𝑡∇ ⋅ ((|∇𝑢|

𝑝

+ 1) ∇𝑢) 𝑑𝑥

= 2∫

𝜕Ω

𝑢𝑡(|∇𝑢|𝑝

+ 1)

𝜕𝑢

𝜕]𝑑𝑆

− 2∫

Ω

𝑢𝑡∇ ⋅ ((|∇𝑢|

𝑝

+ 1) ∇𝑢) 𝑑𝑥.

(40)

Substituting (40) into (39), we have

Φ

(𝑡) = 2∫

Ω

𝑢2

𝑡𝑑𝑥 > 0, (41)

whichwithΦ(0) > 0 implyΦ(𝑡) > 0 for all 𝑡 ∈ (0, 𝑡∗). Makinguse of the Schwarz inequality, we obtain

2 (1 + 𝛽)Ψ

Φ ≤ (Ψ

(𝑡))

2

= 4(∫

Ω

𝑢𝑢𝑡𝑑𝑥)

2

≤ 2Ψ (𝑡)Φ

(𝑡) .

(42)


Multiplying the above inequality by Ψ−2−𝛽, we deduce

(ΦΨ−(1+𝛽)

)

≥ 0. (43)

Arguing as in Theorem 3.1 in [20], we find

𝑡∗

≤ 𝑇 =

1

2𝛽 (1 + 𝛽)

(Ψ (0))−𝛽

=

Ψ (0)

2𝛽 (1 + 𝛽)Φ (0)

(44)

valid for 𝛽 > 0. If 𝛽 = 0, we have

Ψ (𝑡) ≥ Ψ (0) 𝑒2𝑀𝑡 (45)

valid for 𝑡 > 0, implying that 𝑡∗ = ∞. This completes theproof of Theorem 2.

4. Lower Bounds for 𝑡∗

In this section, under the assumption that Ω is a star shapeddomain in 𝑅3, convex in two orthogonal directions, we seek alower bound for the blow-up time 𝑡∗. Now we state the resultas follows.

Theorem 3. Let 𝑢(𝑥, 𝑡) be the nonnegative solution of problem(1)-(3) and 𝑢(𝑥, 𝑡) blows up at 𝑡∗; moreover, the nonnegativefunctions 𝑓 and 𝑔 satisfy

𝑓 (𝜉) ≥ 𝑘1𝜉𝑞

, 𝜉 ≥ 0,

𝑔 (𝜉) ≤ 𝑘2𝜉𝑠

, 𝜉 ≥ 0,

(46)

with 𝑘1> 0, 𝑘

2> 0, 𝑞 > 1, 𝑠 > 1, 𝑞 < 𝑠. Define

𝜑 (𝑡) = ∫

Ω

𝑢𝑛(𝑠−1)

𝑑𝑥, (47)

where 𝑛 is a parameter restricted by the condition

𝑛 > max {4, 2𝑠 − 1

} . (48)

Then 𝜑(𝑡) satisfies inequality

𝜑

(𝑡) ≤ Γ (𝜑) , (49)

for some computable function Γ(𝜑). It follows that 𝑡∗ is boundedfrom below. We have

𝑡∗

≥ ∫

∞

𝜑(0)

𝑑𝜂

Γ (𝜂)

𝑑𝜂. (50)

Proof. Differentiating (47) and making use of the boundarycondition (2) together with the conditions (46), we have

𝜑

(𝑡) = 𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)−1

𝑢𝑡𝑑𝑥

= 𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)−1

× [((|∇𝑢|𝑝

+ 1) 𝑢,𝑖),𝑖

− 𝑓 (𝑢)] 𝑑𝑥

= 𝑛 (𝑠 − 1) ∫

𝜕Ω

𝑢𝑛(𝑠−1)−1

(|∇𝑢|𝑝

+ 1)

𝜕𝑢

𝜕]𝑑𝑆

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1]

× ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1]

× ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|2

𝑑𝑥

− 𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)−1

𝑓 (𝑢) 𝑑𝑥

≤ 𝑘2𝑛 (𝑠 − 1) ∫

𝜕Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑆

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1]

× ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1]

× ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|2

𝑑𝑥

− 𝑘1𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)+𝑞−1

𝑑𝑥.

(51)

Applying inequality (2.7) in [20] to the first term on the righthand side of (51), we have

∫

𝜕Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑆 ≤

3

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥

+

(𝑛 + 1) (𝑠 − 1) 𝑑

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)−1

|∇𝑢| 𝑑𝑥.

(52)


Substituting (52) into (51), we obtain

𝜑

(𝑡) ≤

3𝑘2𝑛 (𝑠 − 1)

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥

+

𝑘2𝑛 (𝑛 + 1) (𝑠 − 1)

2

𝑑

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)−1

|∇𝑢| 𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|2

𝑑𝑥

− 𝑘1𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)+𝑞−1

𝑑𝑥.

(53)

Making use of arithmetic-geometric mean inequality, wederive

∫

Ω

𝑢(𝑛+1)(𝑠−1)−1

|∇𝑢| 𝑑𝑥 ≤

𝜇

2

∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|2

𝑑𝑥

+

1

2𝜇

∫

Ω

𝑢(𝑛+2)(𝑠−1)

𝑑𝑥,

(54)

for all 𝜇 > 0. Choose 𝜇 > 0 such that

𝑘2𝑛 (𝑛 + 1) (𝑠 − 1)

2

𝑑𝜇

2𝜌0

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] = 0. (55)

We rewrite (53) as

𝜑

(𝑡) ≤

3𝑘2𝑛 (𝑠 − 1)

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥

+

𝑘2𝑛 (𝑛 + 1) (𝑠 − 1)

2

𝑑

2𝜇𝜌0

∫

Ω

𝑢(𝑛+2)(𝑠−1)

𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥

− 𝑘1𝑛 (𝑠 − 1) ∫

Ω

𝑢𝑛(𝑠−1)+𝑞−1

𝑑𝑥.

(56)

Using Hölder inequality, we get

∫

Ω

𝑢𝑛(𝑠−1)

𝑑𝑥 ≤ (∫

Ω

𝑢𝑛(𝑠−1)+𝑞−1

𝑑𝑥)

𝑛(𝑠−1)/(𝑛(𝑠−1)+𝑞−1)

× |Ω|(𝑞−1)/(𝑛(𝑠−1)+𝑞−1)

.

(57)

Combining (56) with (57), we obtain

𝜑

(𝑡) ≤

3𝑘2𝑛 (𝑠 − 1)

𝜌0

∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥

+

𝑘2𝑛 (𝑛 + 1) (𝑠 − 1)

2

𝑑

2𝜇𝜌0

∫

Ω

𝑢(𝑛+2)(𝑠−1)

𝑑𝑥

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1]

× ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥

− 𝑘1𝑛 (𝑠 − 1) |Ω|

(1−𝑞)/𝑛(𝑠−1)

𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

=

3𝑘2𝑛 (𝑠 − 1)

𝜌0

𝐽1(𝑡)

+

𝑘2𝑛 (𝑛 + 1) (𝑠 − 1)

2

𝑑

2𝜇𝜌0

𝐽2(𝑡)

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] 𝜔 (𝑡)

− 𝑘1𝑛 (𝑠 − 1) |Ω|

(1−𝑞)/𝑛(𝑠−1)

𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

,

(58)

where

𝐽1(𝑡) = ∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥,

𝐽2(𝑡) = ∫

Ω

𝑢(𝑛+2)(𝑠−1)

𝑑𝑥,

𝜔 (𝑡) = ∫

Ω

𝑢𝑛(𝑠−1)−2

|∇𝑢|𝑝+2

𝑑𝑥.

(59)

Using Sobolev type inequality (A.5) derived by Payne et al.[21], we obtain

𝐽1(𝑡) = ∫

Ω

𝑢(𝑛+1)(𝑠−1)

𝑑𝑥

≤ {

3

𝜌0

∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)

𝑑𝑥 +

(𝑛 + 1) (𝑠 − 1)

3

×(1 +

𝑑

𝜌0

)∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)−1

|∇𝑢| 𝑑𝑥}

3/2

.

(60)

We now make use of Hölder inequality to bound the secondintegral on the right hand side of (60) as follows:

∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)−1

|∇𝑢| 𝑑𝑥

≤ (∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)(1−𝛿

1)

𝑑𝑥)

(𝑝+1)/(𝑝+2)

𝜔1/(𝑝+2)

,

(61)

with

𝛿1=

(𝑛 − 2) (𝑠 − 1) + 3𝑝

2 (𝑛 + 1) (𝑠 − 1) (𝑝 + 1)

. (62)


Wenote that 𝛿1< 1 for 𝑛 > (3𝑝−2(𝑠−1)(𝑝+2))/(𝑠−1)(2𝑝+1),

an inequality satisfied in view of (48). Using again Hölder’sinequality, we obtain

∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)(1−𝛿

1)

𝑑𝑥

≤ 𝜑2(𝑛+1)(1−𝛿

1)/3𝑛

|Ω|1−(2(𝑛+1)(1−𝛿

1)/3𝑛)

,

∫

Ω

𝑢(2/3)(𝑛+1)(𝑠−1)

𝑑𝑥 ≤ 𝜑2(𝑛+1)/3𝑛

|Ω|1−(2(𝑛+1)/3𝑛)

,

(63)

where |Ω| = ∫Ω

𝑑𝑥 is the volume of Ω. Substituting (61) and(63) in (60), we obtain the following inequality:

𝐽1(𝑡) ≤ {𝑐

1𝜑2(𝑛+1)/3𝑛

+ 𝑐2𝜑(2(𝑛+1)(1−𝛿

1)/3𝑛)((𝑝+1)/(𝑝+2))

× 𝜔1/(𝑝+2)

}

3/2

≤ 𝑐1𝜑(𝑛+1)/𝑛

+ 𝑐2𝜑((𝑛+1)(1−𝛿

1)/𝑛)((𝑝+1)/(𝑝+2))

𝜔3/2(𝑝+2)

,

(64)

where 𝑐1, 𝑐2are computable positive constants. Note that the

last inequality in (64) follows from Hölder inequality underthe particular form (𝑎 + 𝑏)3/2 ≤ √2(𝑎3/2 + 𝑏3/2). Similarly, wecan bound 𝐽

2and get

𝐽2(𝑡) ≤ 𝑐

3𝜑(𝑛+2)/𝑛

+ 𝑐4𝜑((𝑛+2)(1−𝛿

2)/𝑛)((𝑝+1)/(𝑝+2))

𝜔3/2(𝑝+2)

,

(65)

where 𝑐3, 𝑐4are computable positive constants,

𝛿2=

(𝑛 − 4) (𝑠 − 1) + 3𝑝

2 (𝑛 + 1) (𝑠 − 1) (𝑝 + 1)

. (66)

Wenote that 𝛿2< 1 for 𝑛 > (3𝑝−4(𝑠−1)(𝑝+2))/(𝑠−1)(2𝑝+1),

an inequality satisfied in view of (48). Inserting (64) and (65)in (58), we arrive at

𝜑

(𝑡) ≤̃𝑑1𝜑(𝑛+1)/𝑛

+̃𝑑2𝜑((𝑛+1)(1−𝛿

1)/𝑛)𝜆

𝜔3/2(𝑝+2)

+ 𝑑3𝜑(𝑛+2)/𝑛

+̃𝑑4𝜑((𝑛+2)(1−𝛿

2)/𝑛)𝜆

𝜔3/2(𝑝+2)

− 𝑛 (𝑠 − 1) [𝑛 (𝑠 − 1) − 1] 𝜔 (𝑡)

− 𝑘1𝑛 (𝑠 − 1) |Ω|

(1−𝑞)/𝑛(𝑠−1)

𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

,

(67)

where 𝜆 = (𝑝 + 1)/(𝑝 + 2), 𝑑3and ̃𝑑

𝑗(𝑗 = 1, 2, 4) are

computable positive constants. Next, we want to eliminatethe quantity 𝜔(𝑡) in inequality (67). By using the followinginequality:

𝜑𝛼

𝜔𝛽

= (𝛾𝜔)𝛽

{

𝜑𝛼/(1−𝛽)

𝛾𝛽/(1−𝛽)

}

1−𝛽

≤ 𝛾𝛽𝜔 + (1 − 𝛽) 𝛾𝛽/(𝛽−1)

+ (1 − 𝛽) 𝛾𝛽/(𝛽−1)

𝜑𝛼/(1−𝛽)

,

(68)

valid for 0 < 𝛽 < 1, where 𝛾 is an arbitrary positive constant,then we have

̃𝑑2𝜑((𝑛+1)(1−𝛿

1)/𝑛)𝜆

𝜔3/2(𝑝+2)

≤ 𝛾1𝜔 (𝑡) + 𝑑

2𝜑(2(𝑛+1)(1−𝛿

1)(𝑝+2)/𝑛(2𝑝+1))𝜆

,

̃𝑑4𝜑((𝑛+2)(1−𝛿

2)/𝑛)𝜆

𝜔3/2(𝑝+2)

≤ 𝛾2𝜔 (𝑡) + 𝑑

4𝜑(2(𝑛+2)(1−𝛿

2)(𝑝+2)/𝑛(2𝑝+1))𝜆

,

(69)

with arbitrary positive constants 𝛾1, 𝛾2and computable

positive constants 𝑑2, 𝑑4. Substitute (69) in (67) and choose

the arbitrary (positive) constants 𝛾1, 𝛾2such that 𝛾

1+𝛾2−𝑛(𝑠−

1)[𝑛(𝑠 − 1) − 1] = 0. We obtain

𝜑

(𝑡) ≤̃𝑑1𝜑(𝑛+1)/𝑛

+ 𝑑2𝜑(2(𝑛+1)(1−𝛿

1)(𝑝+2)/𝑛(2𝑝+1))𝜆

+ 𝑑3𝜑(𝑛+2)/𝑛

+ 𝑑4𝜑(2(𝑛+2)(1−𝛿

2)(𝑝+2)/𝑛(2𝑝+1))𝜆

− 𝑘1𝑛 (𝑠 − 1) |Ω|

(1−𝑞)/𝑛(𝑠−1)

𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

.

(70)

We eliminate the last term in (70), by using the followinginequality:

𝜑(𝑛+1)/𝑛

= {𝑚𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

}

(2𝑛−1)(𝑠−1)/((2𝑛−1)(𝑠−1)+𝑠−𝑞)

× {𝑚(2𝑛−1)(1−𝑠)/(𝑠−𝑞)

𝜑3

}

(𝑠−𝑞)/((2𝑛−1)(𝑠−1)+𝑠−𝑞)

≤

(2𝑛 − 1) (𝑠 − 1)

(2𝑛 − 1) (𝑠 − 1) + 𝑠 − 𝑞

𝑚𝜑(𝑛(𝑠−1)+𝑞−1)/𝑛(𝑠−1)

+

𝑠 − 𝑞

(2𝑛 − 1) (𝑠 − 1) + 𝑠 − 𝑞

𝑚(2𝑛−1)(1−𝑠)/(𝑠−𝑞)

𝜑3

,

(71)

valid for 𝑞 < 𝑠 and arbitrary𝑚 > 0, and choose𝑚 such that

(2𝑛 − 1) (𝑠 − 1)

(2𝑛 − 1) (𝑠 − 1) + 𝑠 − 𝑞

̃𝑑1𝑚 − 𝑘

1𝑛 (𝑠 − 1) |Ω|

(1−𝑞)/𝑛(𝑠−1)

= 0.

(72)

Then (70) can be rewritten as

𝜑

(𝑡) ≤ 𝑑1𝜑3

+ 𝑑2𝜑(2(𝑛+1)(1−𝛿

1)(𝑝+2)/𝑛(2𝑝+1))𝜆

+ 𝑑3𝜑(𝑛+2)/𝑛

+ 𝑑4𝜑(2(𝑛+2)(1−𝛿

2)(𝑝+2)/𝑛(2𝑝+1))𝜆

.

(73)

Integrating (73) over [0, 𝑡], we conclude

𝑡∗

≥ ∫

∞

𝜑(0)

𝑑𝜂

× (𝑑1𝜂3

+ 𝑑2𝜂(2(𝑛+1)(1−𝛿

1)(𝑝+2)/𝑛(2𝑝+1))𝜆

+ 𝑑3𝜂(𝑛+2)/𝑛

+ 𝑑4𝜂(2(𝑛+2)(1−𝛿

2)(𝑝+2)/𝑛(2𝑝+1))𝜆

)

−1

.

(74)

This completes the proof of Theorem 3.


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Authors’ Contribution

All authors contributed equally to the paper and read andapproved the final paper.

Acknowledgments

This work is supported by the Natural Science Foundationof Shandong Province of China (ZR2012AM018) and theFundamental Research Funds for the Central Universities(no. 201362032). The authors would like to deeply thank allthe reviewers for their insightful and constructive comments.

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Research ArticleSome Properties on Complex Functional Difference Equations

Zhi-Bo Huang1,2 and Ran-Ran Zhang3

1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China2Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland3Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China

Correspondence should be addressed to Zhi-Bo Huang; [email protected]

Received 15 January 2014; Accepted 13 March 2014; Published 24 April 2014

Academic Editor: Zong-Xuan Chen

Copyright © 2014 Z.-B. Huang and R.-R. Zhang.This is an open access article distributed under theCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form∑𝜆∈𝐼𝛼𝜆(𝑧)(∏

𝑛

𝑗=0𝑓(𝑧 + 𝑐

𝑗)𝜆𝑗) = 𝑅(𝑧, 𝑓 ∘𝑝) = ((𝑎

0(𝑧) +𝑎

1(𝑧)(𝑓 ∘𝑝)+ ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘𝑝)

𝑠

)/(𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘𝑝)+ ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘𝑝)

𝑡

)),where 𝐼 is a finite set of multi-indexes 𝜆 = (𝜆

0, 𝜆1, . . . , 𝜆

𝑛), 𝑐0= 0, 𝑐

𝑗∈ C \ {0} (𝑗 = 1, 2, . . . , 𝑛) are distinct complex constants, 𝑝(𝑧) is

a polynomial, and 𝛼𝜆(𝑧) (𝜆 ∈ 𝐼), 𝑎

𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠), and 𝑏

𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small meromorphic functions relative to 𝑓(𝑧).

We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero andpole.

1. Introduction and Main Results

In this paper, a meromorphic function means meromorphicin the whole complex plane C. For a meromorphic function𝑦(𝑧), let 𝜎(𝑦) be the order of growth and 𝜇(𝑦) the lowerorder of𝑦(𝑧). Further, let 𝜆(𝑦) (resp., 𝜆(1/𝑦)) be the exponentof convergence of the zeros (resp., poles) of 𝑦(𝑧). We alsoassume that the reader is familiar with the fundamentalresults and the standard notations of Nevanlinna theory ofmeromorphic functions (see, e.g., [1]). Given a meromorphicfunction 𝑦(𝑧), we call a meromorphic function 𝑎(𝑧) a smallfunction relative to 𝑦(𝑧) if 𝑇(𝑟, 𝑎(𝑧)) = 𝑆(𝑟, 𝑦) = 𝑜(𝑇(𝑟, 𝑦))as 𝑟 → ∞, possibly outside of an exceptional set of finitelogarithmic measure. Moreover, if 𝑅(𝑧, 𝑦) is rational in 𝑦(𝑧)with small functions relative to 𝑦(𝑧) as its coefficients, we usethe notation 𝑑 = deg

𝑦𝑅(𝑧, 𝑦) for the degree of 𝑅(𝑧, 𝑦) with

respect to𝑦(𝑧). Inwhat follows, we always assume that𝑅(𝑧, 𝑦)is irreducible in 𝑦(𝑧).

Meromorphic solutions of complex difference equationshave recently gained increasing interest, due to the problemof integrability of difference equations. This is related to theactivity concerning Painlevé differential equations and theirdiscrete counterparts in the last decades. Ablowitz et al. [2]considered discrete equations to be delay equations in thecomplex plane. This allowed them to analyze these equations

with the methods from complex analysis. In regard to relatedpapers concerning a more general class of complex differenceequations, we may refer to [3–5]. These papers mainly dealtwith equations of the form

∑

{𝐽}

𝛼𝐽(𝑧)(∏

𝑗∈𝐽

𝑓 (𝑧 + 𝑐𝑗)) = 𝑅 (𝑧, 𝑓) , (1)

where {𝐽} is a collection of all nonempty subsets of{1, 2, . . . , 𝑛}, 𝑐

𝑗(𝑗 ∈ 𝐽) are distinct complex constants, 𝑓(𝑧) is

a transcendental meromorphic function, 𝛼𝐽(𝑧) (𝐽 ∈ {𝐽}) are

small functions relative to𝑓(𝑧), and𝑅(𝑧, 𝑓) is a rational func-tion in 𝑓(𝑧) with small meromorphic coefficients. Moreover,if the right-hand side of (1) is essentially like the compositefunction 𝑒 ∘𝑓 of 𝑓(𝑧) and a rational function 𝑒(𝑧), Laine et al.reversed the order of composition; that is, they considered thecomposite function 𝑓 ∘ 𝑒 of 𝑓(𝑧) and a rational function 𝑒(𝑧),which resulted in a complex functional difference equation.The following theorem [5, Theorem 2.8] gives an example.

Theorem A (see [5, Theorem 2.8]). Suppose that 𝑓(𝑧) is atranscendental meromorphic solution of equation

∑

{𝐽}

𝛼𝐽(𝑧)(∏

𝑗∈𝐽

𝑓 (𝑧 + 𝑐𝑗)) = 𝑓 (𝑝 (𝑧)) , (2)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 283895, 10 pageshttp://dx.doi.org/10.1155/2014/283895

http://dx.doi.org/10.1155/2014/283895


where 𝑝(𝑧) is a polynomial of degree 𝑘 ≥ 2. Moreover, oneassumes that the coefficients 𝛼

𝐽(𝑧) are small functions relative

to 𝑓(𝑧) and that 𝑛 ≥ 𝑘. Then

𝑇 (𝑟, 𝑓) = 𝑂 ((log 𝑟)𝛼+𝜀) , (3)

where 𝛼 = (log 𝑛)/(log 𝑘).At this point, we briefly introduce some notations used in

this paper. A difference monomial of a meromorphic function𝑓(𝑧) is defined as

𝑓(𝑧)𝜆0𝑓(𝑧 + 𝑐

1)𝜆1

⋅ ⋅ ⋅ 𝑓(𝑧 + 𝑐𝑛)𝜆𝑠

:=

𝑠

∏

𝑗=0

𝑓(𝑧 + 𝑐𝑗)

𝜆𝑗

, (4)

where 𝑐0= 0, 𝑐

𝑗∈ C \ {0} (𝑗 = 1, 2, . . . , 𝑠) are distinct

constants, and 𝜆𝑗(𝑗 = 0, 1, . . . , 𝑠) are natural numbers. A

difference polynomial 𝐻(𝑧, 𝑓(𝑧)) of a meromorphic function𝑓(𝑧), a finite sum of difference monomials, is defined as

𝐻(𝑧, 𝑓 (𝑧)) = ∑

𝜆∈𝐼

𝛼𝜆(𝑧)(

𝑛

∏

𝑗=0


𝜆𝑗

) , (5)

where 𝐼 is a finite set of multi-indexes 𝜆 = (𝜆0, 𝜆1, . . . , 𝜆

𝑛),

𝛼𝜆(𝑧) (𝜆 ∈ 𝐼) are small functions relative to 𝑓(𝑧). The degree

and the weight of the difference polynomial (5), respectively, aredefined as

deg𝑓(𝐻) = max

𝜆∈𝐼

{

{

{

𝑛

∑

𝑗=0

𝜆𝑗

}

}

}

, 𝜅𝑓(𝐻) = max

𝜆∈𝐼

{

{

{

𝑛

∑

𝑗=1

𝜆𝑗

}

}

}

.

(6)

Consequently, 𝜅𝑓(𝐻) ≤ deg

𝑓(𝐻). For instance, the degree and

the weight of the difference polynomial 𝑓2(𝑧)𝑓(𝑧−1)𝑓(𝑧+1)+𝑓(𝑧)𝑓(𝑧 + 1)𝑓(𝑧 + 2) + 𝑓

2

(𝑧 − 1)𝑓(𝑧 + 2), respectively, arefour and three. Moreover, a difference polynomial (5) is said tobe homogeneous with respect to 𝑓(𝑧) if the degree ∑𝑛

𝑗=0𝜆𝑗of

each monomial in the sum of (5) is nonzero and the same forall 𝜆 ∈ 𝐼.

In the following, we proceed to prove generalizations ofTheorem A and investigate some new results for the first time.We permit more general expressions on both sides of (1).

Theorem 1. Let 𝑓(𝑧) be a transcendental meromorphic solu-tion of equation

𝐻(𝑧, 𝑓 (𝑧)) = 𝑅 (𝑧, 𝑓 ∘ 𝑝)

=

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡,

(7)

where𝐻(𝑧, 𝑓(𝑧)) is defined as (5), 𝑝(𝑧) = 𝑑𝑘𝑧𝑘

+ ⋅ ⋅ ⋅+𝑑1𝑧+𝑑

0

is a polynomial with constant coefficients 𝑑𝑘( ̸= 0), . . . , 𝑑

1, 𝑑0

and of the degree 𝑘 ≥ 2, and 𝑎𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠) and

𝑏𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small meromorphic functions relative

to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡ 0. Set 𝑑 = max{𝑠, 𝑡}. If 𝑘𝑑 ≤

(𝑛 + 1)deg𝑓(𝐻), then

𝑇 (𝑟, 𝑓) = 𝑂 ((log 𝑟)𝛼+𝜀) , (8)

where 𝛼 = (log(𝑛 + 1) + log deg𝑓(𝐻) − log𝑑)/(log 𝑘).

Similar to the proof of Theorem 1, we easily obtain thefollowing result, which is a generation of Theorem A.

Theorem 2. Let 𝑐𝑖∈ C (𝑖 = 1, 2, . . . , 𝑛) be distinct

constants and 𝑓(𝑧) be a transcendental meromorphic solutionof equation

∑

{𝐽}

𝛼𝐽(𝑧)(∏

𝑗∈𝐽

𝑓 (𝑧 + 𝑐𝑗))

= 𝑅 (𝑧, 𝑓 ∘ 𝑝)

=

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡,

(9)

where 𝑝(𝑧) = 𝑑𝑘𝑧𝑘

+ ⋅ ⋅ ⋅ + 𝑑1𝑧 + 𝑑

0is a polynomial with

constant coefficients 𝑑𝑘( ̸= 0), . . . , 𝑑

1, 𝑑0and of the degree 𝑘 ≥ 2

and 𝑎𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠) and 𝑏

𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small

functions relative to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡ 0. If 𝑘𝑑 =

𝑘max{𝑠, 𝑡} ≤ 𝑛, then

𝑇 (𝑟, 𝑓) = 𝑂 ((log 𝑟)𝛼+𝜀) , (10)

where 𝛼 = (log 𝑛 − log 𝑑)/(log 𝑘).

We then proceed to consider the distribution of zeros andpoles of meromorphic solutions of (7). The following resultindicates that solutions having Borel exceptional zeros andpoles appear only in special situations.

Theorem 3. Let 𝑐0= 0, let 𝑐

𝑖∈ C \ {0} (𝑖 = 1, 2, . . . , 𝑛) be

distinct constants, and let 𝑓(𝑧) be a finite order transcendentalmeromorphic solution of equation

𝑛

∏

𝑗=0


𝜆𝑗

= 𝑅 (𝑧, 𝑓 ∘ 𝑝)

=

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡,

(11)


+ ⋅ ⋅ ⋅ + 𝑑1𝑧 + 𝑑




and 𝑎𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠) and 𝑏

𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small

meromorphic functions relative to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡

0. If

max{𝜆 (𝑓) , 𝜆 ( 1𝑓

)} < 𝜎 (𝑓) , (12)

then (11) is either of the form𝑛

∏

𝑗=0


𝜆𝑗

= 𝛼

𝑎𝑠(𝑧)

𝑏0(𝑧)

(𝑓 ∘ 𝑝)𝑠

(13)

or𝑛

∏

𝑗=0


𝜆𝑗

= 𝛼

𝑎0(𝑧)

𝑏𝑡(𝑧)

1

(𝑓 ∘ 𝑝)𝑡, (14)

where 𝛼 ∈ C \ {0} is some constant.


Example 4. 𝑓(𝑧) = cos 𝑧 solves difference equation

4𝑓(𝑧)2

𝑓(𝑧 + 𝜋)2

= 𝑓(2𝑧)2

+ 2𝑓 (2𝑧) + 1. (15)

Here 𝑝(𝑧) = 2𝑧. Clearly, 𝜆(1/𝑓) = 0 < 1 = 𝜆(𝑓) = 𝜎(𝑓). Thisexample shows that condition (12) is necessary and cannot bereplaced by

min{𝜆 (𝑓) , 𝜆 ( 1𝑓

)} < 𝜎 (𝑓) . (16)

Moreover, we obtain a result parallel toTheorem 5.4 in [6]for the difference case.

Theorem 5. Suppose that the equation𝑛

∏

𝑗=0


𝜆𝑗

=

𝑐 (𝑧)

𝑓(𝑧)𝑚, 𝑚 ∈ N, (17)

has a meromorphic solution of finite order, where 𝑐0= 0, 𝑐

𝑗∈

C \ {0} (𝑗 = 1, 2, . . . , 𝑛) are distinct constants, and 𝑐(𝑧) is anontrivialmeromorphic function. If𝑓(𝑧)has only finitelymanypoles, then𝑓(𝑧) = 𝐷(𝑧)𝑒𝐸(𝑧), where𝐷(𝑧) is a rational functionand𝐸(𝑧) is a polynomial, if and only if 𝑐(𝑧) = 𝐺(𝑧)𝑒𝑀(𝑧), where𝐺(𝑧) is a rational function and𝑀(𝑧) is a polynomial.

Example 6. Difference equation

𝑓(𝑧)2

𝑓 (𝑧 + 1) 𝑓 (𝑧 − 1) = (

1

𝑧4(𝑧2− 1)

𝑒6𝑧

) ⋅

1

𝑓(𝑧)2

(18)

of the type (17) is solved by 𝑓(𝑧) = 𝑒𝑧/𝑧. Here, 𝑓(𝑧) = 𝑒𝑧/𝑧and 𝑐(𝑧) = 𝑒6𝑧/𝑧4(𝑧2 − 1) satisfy Theorem 5.

As an application of Theorem 3, we obtain the following.

Theorem 7. Let 𝑐 ∈ C \ {0} and let 𝑓(𝑧) be a finite ordertranscendental meromorphic solution of equation

𝑓 (𝑧 + 𝑐) = 𝑅 (𝑧, 𝑓 ∘ 𝑝)

=

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡,

(19)


+ ⋅ ⋅ ⋅ + 𝑑1𝑧 + 𝑑




and 𝑎𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠) and 𝑏

𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small

meromorphic functions relative to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡

0. Then 𝑓(𝑧) has at most one Borel exceptional value.

If the degree 𝑘 of polynomial 𝑝(𝑧) is 1 in Theorem 7, theresult does not hold. For example, we have the following.

Example 8. 𝑓(𝑧) = tan 𝑧 solves difference equation

𝑓 (𝑧 + 1) =

𝑓 (𝑧) + tan 11 − (tan 1) 𝑓 (𝑧)

(20)

of the type (19). Obviously, 𝑓(𝑧) has two Borel exceptionalvalues ±𝑖.

If we remove the assumption max{𝜆(𝑓), 𝜆(1/𝑓)} < 𝜎(𝑓)used in Theorem 3, we obtain a result similar to Theorem 12in [4].

Theorem 9. Let 𝑓(𝑧) be a transcendental meromorphic solu-tion of equation

𝐻(𝑧, 𝑓 (𝑧)) = 𝑅 (𝑧, 𝑓)

=

𝑎0(𝑧) + 𝑎

1(𝑧) 𝑓 (𝑧) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) 𝑓(𝑧)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) 𝑓 (𝑧) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) 𝑓(𝑧)

𝑡,

(21)

where 𝐻(𝑧, 𝑓(𝑧)) is defined as (5) and 𝑎𝑖(𝑧) (𝑖 = 0, 1, . . . , 𝑠)

and 𝑏𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small meromorphic functions

relative to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡ 0. If 𝑑 := max{𝑠, 𝑡} >

(𝑛 + 1) deg𝑓(𝐻), then 𝜎(𝑓) = ∞.

In fact, the following examples show that the assertion ofTheorem 9 does not remain valid identically if 𝑑 ≤ (𝑛 +1)deg

𝑓(𝐻).

Example 10. 𝑓(𝑧) = exp{𝑒𝑧}/𝑧 solves the difference equation

(𝑧 − 𝜋𝑖) (𝑧 + log 2 − 𝜋𝑖) 𝑓 (𝑧) 𝑓 (𝑧 − 𝜋𝑖) 𝑓 (𝑧 + log 2 − 𝜋𝑖)

+ (𝑧 + log 8) 𝑓 (𝑧 + log 8) =1 + 𝑧

11

𝑓(𝑧)10

𝑧3𝑓(𝑧)

2.

(22)

Clearly, 𝑑 = 10 < (3+1) ⋅ 3 = (𝑛+1) deg𝑓(𝐻) and 𝜎(𝑓) = ∞.

Example 11. 𝑓(𝑧) = tan 𝑧 satisfies the difference equation

𝑓(𝑧 +

𝜋

4

) + 𝑓(𝑧 −

𝜋

4

) =

4𝑓 (𝑧)

1 − 𝑓(𝑧)2

. (23)

Obviously, 𝑑 = 2 < (2+1)×1 = (𝑛+1) deg𝑓(𝐻) and𝜎(𝑓) = 1.

Example 12 (see [7, pages 103–106] and [8, page 8]). Thefollowing difference equation,

𝑓 (𝑧 + 1) = 𝛼𝑓 (𝑧) (1 − 𝑓 (𝑧)) , 𝛼 ̸= 0, (24)

derives from a well-known discrete logistic model in biology.It has been proved that all other meromorphic solutions areof infinite order, apart from the constant solutions 𝑓(𝑧) ≡ 0and 𝑓(𝑧) = (𝛼 − 1)/𝛼. For instance, (24) has one-parameterfamilies of entire solutions of infinite order:

𝑓 (𝑧) =

1

2

(1 − exp (𝐴𝑒𝑧 log 2)) , 𝐴 ∈ C \ {0} , 𝛼 = 2,

𝑓 (𝑧) = sin2 (𝐵𝑒𝑧 log 2) , 𝐵 ∈ C \ {0} , 𝛼 = 4.(25)

Here, 𝑑 = 2 = (1 + 1) × 1 = (𝑛 + 1)deg𝑓(𝐻).

Example 13. 𝑓(𝑧) = 𝑧 solves the difference equation

𝑓 (𝑧 + 1) =

1 − 𝑓(𝑧)2

−𝑧2− 𝑧 + 1 + 𝑓(𝑧)

2. (26)

We get 𝑑 = 2 = (1 + 1) × 1 = (𝑛 + 1)deg𝑓(𝐻) and 𝜎(𝑓) = 0.

If the difference polynomial in the left-hand side of (21)is homogeneous, we further obtain the following theorem.


Theorem 14. Let 𝑓(𝑧) be a transcendental meromorphic solu-tion of (21), where 𝐻(z, 𝑓(𝑧)) is defined as (5) and 𝑎

𝑖(𝑧) (𝑖 =

0, 1, . . . , 𝑠) and 𝑏𝑗(𝑧) (𝑗 = 0, 1, . . . , 𝑡) are small meromorphic

functions relative to 𝑓(𝑧) such that 𝑎𝑠(𝑧)𝑏

𝑡(𝑧) ̸≡ 0. Suppose

that 𝐻(𝑧, 𝑓) is homogeneous and has at least one differencemonomial of type

𝑛

∏

𝑗=0


𝜆𝑗

, (𝜆𝑗∈ N

+, 𝑗 = 0, 1, . . . , 𝑛) . (27)

If 𝑑 := max{𝑠, 𝑡} > 3deg𝑓(𝐻), then 𝜎(𝑓) = ∞.

2. Proof of Theorem 1

We need some preliminaries to proveTheorem 1.

Lemma 15 (see [9, Lemma 4]). Let 𝑓(𝑧) be a transcendentalmeromorphic function and let 𝑝(𝑧) = 𝑑

𝑘𝑧𝑘

+ ⋅ ⋅ ⋅ + 𝑑1𝑧 +

𝑑0(𝑑𝑘̸= 0)be a polynomial of degree 𝑘. Given 0 < 𝛿 < |𝑑

𝑘|,

denote ] := |𝑑𝑘| + 𝛿 and 𝜇 := |𝑑

𝑘| − 𝛿. Then, given 𝜀 > 0 and

𝑎 ∈ C ∪ {∞}, one has, for all 𝑟 ≥ 𝑟0> 0,

𝑘𝑛 (𝜇𝑟𝑘

, 𝑎, 𝑓) ≤ 𝑛 (𝑟, 𝑎, 𝑓 ∘ 𝑝) ≤ 𝑘𝑛 (]𝑟𝑘, 𝑎, 𝑓) ,

𝑁 (𝜇𝑟𝑘

, 𝑎, 𝑓) + 𝑂 (log 𝑟) ≤ 𝑁 (𝑟, 𝑎, 𝑓 ∘ 𝑝) ≤ 𝑁(]𝑟𝑘, 𝑎, 𝑓)

+ 𝑂 (log 𝑟) ,

(1 − 𝜀) 𝑇 (𝜇𝑟𝑘

, 𝑓) ≤ 𝑇 (𝑟, 𝑓 ∘ 𝑝) ≤ (1 + 𝜀) 𝑇 (]𝑟𝑘, 𝑓) .(28)

Lemma16 (see [10,Theorem B.16]). Given distinctmeromor-phic functions 𝑓

1, . . . , 𝑓

𝑛, let {𝐽} denote the collection of all

nonempty subsets of {1, 2, . . . , 𝑛}, and suppose that 𝛼𝐽∈ C for

each 𝐽 ∈ {𝐽}. Then

𝑇(𝑟,∑

{𝐽}

𝛼𝐽(∏

𝑗∈𝐽

𝑓𝑗)) ≤

𝑛

∑

𝑘=1

𝑇 (𝑟, 𝑓𝑘) + 𝑂 (1) . (29)

By denoting 𝑓𝑖+1= 𝑓(𝑧 + 𝑐

𝑖)𝜆𝑖(𝑖 = 0, 1, . . . , 𝑛) below, it is

an easy exercise to prove the following result from Lemma 16.

Lemma 17. Let 𝑓(𝑧) be a meromorphic function, let 𝐼 be afinite set of multi-indexes 𝜆 = (𝜆

0, 𝜆1, . . . , 𝜆

𝑛), and let 𝛼

𝜆(𝑧)

be small functions relative to 𝑓(𝑧) for all 𝜆 ∈ 𝐼. Then thecharacteristic function of the difference polynomial (5) satisfies

𝑇(𝑟, ∑

𝜆∈𝐼

𝛼𝜆(𝑧)(

𝑛

∏

𝑗=0


𝜆𝑗

))

≤ (𝑛 + 1) deg𝑓(𝐻) 𝑇 (𝑟 + 𝐶, 𝑓) + 𝑆 (𝑟, 𝑓) ,

(30)

where 𝐶 = max{|𝑐1|, |𝑐

2|, . . . , |𝑐

𝑛|}.

Lemma18 (see [11, Lemma5]). Let𝑔(𝑟) andℎ(𝑟) bemonotonenondecreasing functions on [0,∞) such that 𝑔(𝑟) ≤ ℎ(𝑟) for all𝑟 ∉ 𝐸 ∪ [0, 1], where 𝐸 ⊂ (1,∞) is a set of finite logarithmicmeasure. Let 𝛼 > 1 be a given constant. Then there exists an𝑟0= 𝑟

0(𝛼) > 0 such that 𝑔(𝑟) ≤ ℎ(𝛼𝑟) for all 𝑟 ≥ 𝑟

0.

Lemma 19 (see [12, Lemma 3]). Let 𝜓(𝑟) be a function of𝑟 (𝑟 ≥ 𝑟

0), positive and bounded in every finite interval.

(i) Suppose that 𝜓(𝜇𝑟𝑚) ≤ 𝐴𝜓(𝑟) + 𝐵 (𝑟 ≥ 𝑟0), where

𝜇 (𝜇 > 0),𝑚 (𝑚 > 1),𝐴 (𝐴 ≥ 1), and 𝐵 are constants.Then 𝜓(𝑟) = 𝑂((log 𝑟)𝛼) with 𝛼 = (log𝐴)/(log𝑚),unless 𝐴 = 1 and 𝐵 > 0; and if 𝐴 = 1 and 𝐵 > 0, thenfor any 𝜀 > 0, 𝜓(𝑟) = 𝑂((log 𝑟)𝜀).

(ii) Suppose that (with the notation of (i)) 𝜓(𝜇𝑟𝑚) ≥𝐴𝜓(𝑟) (𝑟 ≥ 𝑟

0). Then for all sufficiently large values of

𝑟, 𝜓(𝑟) ≥ 𝐾(log 𝑟)𝛼 with 𝛼 = (log𝐴)/(log𝑚) for somepositive constant 𝐾.

Proof of Theorem 1. For any 𝜀 (0 < 𝜀 < 1), we may applyValiron-Mohon’ko lemma, Lemmas 15 and 17, and (5) and (7)to conclude that𝑑 (1 − 𝜀) 𝑇 (𝜇𝑟

𝑘

, 𝑓)

≤ 𝑑𝑇 (𝑟, 𝑓 ∘ 𝑝) + 𝑆 (𝑟, 𝑓)

= 𝑇(𝑟,

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡)

+ 𝑆 (𝑟, 𝑓)

= 𝑇(𝑟, ∑

𝜆∈𝐼

𝛼𝜆(𝑧)(

𝑛

∏

𝑗=0


𝜆𝑗

)) + 𝑆 (𝑟, 𝑓)

≤ (𝑛 + 1) deg𝑓(𝐻) 𝑇 (𝑟 + 𝐶, 𝑓) + 𝑆 (𝑟, 𝑓)

≤ (𝑛 + 1) deg𝑓(𝐻) (1 + 𝜀) 𝑇 (𝑟 + 𝐶, 𝑓)

(31)holds for all sufficiently large 𝑟, possibly outside of anexceptional set of finite logarithmic measure, where 𝐶 =max{|𝑐

1|, |𝑐

2|, . . . , |𝑐

𝑛|} and 𝜇 is defined as Lemma 15. Now, we

may apply Lemma 18 to deal with the exceptional set andconclude that, for every 𝜂 > 1, there exists an 𝑟

0> 0 such

that𝑑 (1 − 𝜀) 𝑇 (𝜇𝑟

𝑘

, 𝑓) ≤ (𝑛 + 1) deg𝑓(𝐻) (1 + 𝜀) 𝑇 (𝜂𝑟, 𝑓)

(32)holds for all 𝑟 ≥ 𝑟

0. Denote 𝜔 = 𝜂𝑟. Then (32) can be written

in the form

𝑇(

𝜇

𝜂𝑘

𝜔𝑘

, 𝑓) ≤

(𝑛 + 1) deg𝑓(𝐻) (1 + 𝜀)

𝑑 (1 − 𝜀)

𝑇 (𝜔, 𝑓) . (33)

Since 𝑑𝑘 ≤ (𝑛 + 1)deg𝑓(𝐻), we get ((𝑛 + 1)deg

𝑓(𝐻)(1 +

𝜀))/(𝑑(1 − 𝜀)) > 1 for all 0 < 𝜀 < 1. Thus, we now applyLemma 19(i) to conclude that

𝑇 (𝑟, 𝑓) = 𝑂 ((log 𝑟)𝛼+𝜀) ,

𝛼 =

log ((𝑛 + 1) deg𝑓(𝐻) (1 + 𝜀) /𝑑 (1 − 𝜀))

log 𝑘

=

log (𝑛 + 1) + log deg𝑓(𝐻) − log𝑑

log 𝑘+ 𝑜 (1) .

(34)

The proof of Theorem 1 is completed.


3. Proof of Theorems 3 and 5

We again need some preliminaries.

Lemma 20 (see [13, Theorem 1.5]). Suppose that 𝑓𝑗(𝑧) (𝑗 =

1, 2, . . . , 𝑛) (𝑛 ≥ 2) are meromorphic functions and 𝑔𝑗(𝑧) (𝑗 =

1, 2, . . . , 𝑛) are entire functions satisfying the following condi-tions.

(1) ∑𝑛𝑗=1𝑓𝑗(𝑧)𝑒

𝑔𝑗(𝑧)

= 0.

(2) 𝑔𝑗(𝑧) − 𝑔

𝑘(𝑧) are not constants for 1 ≤ 𝑗 < 𝑘 ≤ 𝑛.

(3) For 1 ≤ 𝑗 ≤ 𝑛, 1 ≤ ℎ < 𝑘 ≤ 𝑛,

𝑇 (𝑟, 𝑓𝑗) = 𝑜 {𝑇 (𝑟, 𝑒

𝑔ℎ−𝑔𝑘)} (𝑟 → +∞, 𝑟 ∉ 𝐸) , (35)

where 𝐸 ⊂ (1, +∞) is of finite linear measure or finitelogarithmic measure.

Then 𝑓𝑗(𝑧) ≡ 0 (𝑗 = 1, 2, . . . , 𝑛).

Lemma21 (see [14,Theorem4]). Let𝐹(𝑧),𝑃𝑛(𝑧), . . . , 𝑃

0(𝑧) be

polynomials such that 𝐹𝑃𝑛𝑃0̸≡ 0 and then every finite order

transcendental meromorphic solution 𝑓(𝑧) of equation

𝑃𝑛(𝑧) 𝑓 (𝑧 + 𝑛) + ⋅ ⋅ ⋅ + 𝑃

1(𝑧) 𝑓 (𝑧 + 1) + 𝑃

0(𝑧) 𝑓 (𝑧) = 𝐹 (𝑧)

(36)

satisfies 𝜆(𝑓) = 𝜎(𝑓) ≥ 1.

Remark 22. Replacing 𝑗 by 𝑐𝑗(𝑗 = 1, 2, . . . , 𝑛), where 𝑐

𝑗(𝑗 =

1, 2, . . . , 𝑛) are distinct nonzero complex constants, Lemma 21remains valid.

Proof of Theorem 3. Let 𝜏 be the multiplicity of pole of 𝑓(𝑧)at the origin, and let 𝑞(𝑧) be a canonical product formed withnonzero poles of 𝑓(𝑧). Since max{𝜆(𝑓), 𝜆(1/𝑓)} < 𝜎(𝑓), thenℎ(𝑧) = 𝑧

𝜏

𝑞(𝑧) is an entire function such that

𝜎 (ℎ) = 𝜆(

1

𝑓

) < 𝜎 (𝑓) < +∞, (37)

and 𝑔(𝑧) = ℎ(𝑧)𝑓(𝑧) is a transcendental entire function with

𝑇 (𝑟, 𝑔) = 𝑇 (𝑟, 𝑓) + 𝑆 (𝑟, 𝑓) , 𝜎 (𝑔) = 𝜎 (𝑓) ,

𝜆 (𝑔) = 𝜆 (𝑓) .

(38)

If 𝑞(𝑧) is a polynomial, we obtain quickly that 𝜎(ℎ ∘ 𝑝) =0 < 𝜎(𝑔 ∘ 𝑝). Otherwise, we conclude from the last assertionof Lemma 15, (37), and (38) that

𝜎 (ℎ ∘ 𝑝) = 𝑘𝜎 (ℎ) = 𝑘𝜆(

1

𝑓

) < 𝑘𝜎 (𝑔) = 𝜎 (𝑔 ∘ 𝑝) . (39)

Therefore,

𝑇 (𝑟, ℎ ∘ 𝑝) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) . (40)

Now, substituting 𝑓(𝑧) = 𝑔(𝑧)/ℎ(𝑧) into (11), we concludethat

(ℎ ∘ 𝑝)𝑠−𝑡

∏𝑛

𝑗=0ℎ(𝑧 + 𝑐

𝑗)

𝜆𝑗

(

𝑛

∏

𝑗=0

𝑔(𝑧 + 𝑐𝑗)

𝜆𝑗

)

= (𝑎0(𝑧) (ℎ ∘ 𝑝)

𝑠

+ 𝑎1(𝑧) (ℎ ∘ 𝑝)

𝑠−1

(𝑔 ∘ 𝑝)

+ ⋅ ⋅ ⋅ + 𝑎𝑠(𝑧) (𝑔 ∘ 𝑝)

𝑠

)

× (𝑏0(𝑧) (ℎ ∘ 𝑝)

𝑡

+ 𝑏1(𝑧) (ℎ ∘ 𝑝)

𝑡−1

(𝑔 ∘ 𝑝)

+ ⋅ ⋅ ⋅ + 𝑏𝑡(𝑧) (𝑔 ∘ 𝑝)

𝑡

)

−1

.

(41)

Obviously, it follows from (37)–(40) and Lemma 15 that

𝑇(𝑟,

1

∏𝑛

𝑗=0ℎ(𝑧 + 𝑐

𝑗)

𝜆𝑗

) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) ,

𝑇 (𝑟, (ℎ ∘ 𝑝)𝑠−𝑡

) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) ,

𝑇 (𝑟, 𝑎𝑢(𝑧) (ℎ ∘ 𝑝)

𝑠−𝑢

) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) , 𝑢 = 0, 1, . . . , 𝑠,

𝑇 (𝑟, 𝑏V (𝑧) (ℎ ∘ 𝑝)𝑡−V) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) , V = 0, 1, . . . , 𝑡.

(42)

Denoting𝐴(𝑧) = (ℎ ∘ 𝑝)𝑠−𝑡/∏𝑛𝑗=0ℎ(𝑧 + 𝑐

𝑗)𝜆𝑗 , we get from (42)

that𝑇 (𝑟, 𝐴) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) . (43)

Since zeros and poles are Borel exceptional values of 𝑓(𝑧) by(12), we may apply a result due to Whittaker; see [15, Satz13.4], to deduce that 𝑓(𝑧) is of regular growth. Thus, we useLemma 15 and (12) again to get

𝑇(𝑟,

𝑓

𝑓

) = 𝑁(𝑟, 𝑓) + 𝑁(𝑟,

1

𝑓

) + 𝑆 (𝑟, 𝑓) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) .

(44)

Similarly, if we set 𝐵(𝑧) = 𝐴(𝑧)(∏𝑛𝑗=0𝑔(𝑧 + 𝑐

𝑗)𝜆𝑗), we

also deduce from the lemma of the logarithmic derivative,Lemma 15, (12), (38), and (43) that

𝑇(𝑟,

𝐵

𝐵

) = 𝑇(𝑟,

𝐴

𝐴

+

𝑛

∑

𝑗=0

𝜆𝑗

𝑔

(𝑧 + 𝑐𝑗)

𝑔 (𝑧 + 𝑐𝑗)

) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) .

(45)Denoting 𝐹(𝑧) = 𝑔 ∘ 𝑝,

𝑃 (𝑧, 𝐹) =

𝑎0(𝑧)

𝑎𝑠(𝑧)

(ℎ ∘ 𝑝)𝑠

+

𝑎1(𝑧)

𝑎𝑠(𝑧)

(ℎ ∘ 𝑝)𝑠−1

𝐹 (𝑧) + ⋅ ⋅ ⋅ + 𝐹(𝑧)𝑠

,

𝑄 (𝑧, 𝐹) =

𝑏0(𝑧)

𝑏𝑡(𝑧)

(ℎ ∘ 𝑝)𝑡

+

𝑏1(𝑧)

𝑏𝑡(𝑧)

(ℎ ∘ 𝑝)𝑡−1

𝐹 (𝑧) + ⋅ ⋅ ⋅ + 𝐹(𝑧)𝑡

.

(46)


Therefore, we deduce from Lemma 15 and (42) that thecoefficients of 𝑃(𝑧, 𝐹) and𝑄(𝑧, 𝐹) are small functions relativeto 𝐹(𝑧). Thus, (41) can be written in the form

𝑏𝑡(𝑧)

𝑎𝑠(𝑧)

𝐵 (𝑧) =

𝑃 (𝑧, 𝐹)

𝑄 (𝑧, 𝐹)

:= 𝑢 (𝑧, 𝐹) . (47)

Denoting

𝜓 (𝑧) =

𝐹

(𝑧)

𝐹 (𝑧)

, 𝑈 (𝑧) =

𝑢

(𝑧, 𝐹)

𝑢 (𝑧, 𝐹)

, (48)

we get 𝑇(𝑟, 𝑈) = 𝑆(𝑟, 𝑔 ∘ 𝑝) from (45) and (47). Wealso conclude from the lemma of logarithmic derivative,Lemma 15, and (12) that

𝑇 (𝑟, 𝜓) = 𝑇(𝑟,

𝐹

𝐹

) = 𝑚(𝑟,

𝐹

𝐹

) + 𝑁(𝑟,

𝐹

𝐹

)

≤ 𝑁 (𝑟, 𝐹) + 𝑁(𝑟,

1

𝐹

) + 𝑆 (𝑟, 𝐹)

= 𝑁 (𝑟, 𝑔 ∘ 𝑝) + 𝑁(𝑟,

1

𝑔 ∘ 𝑝

) + 𝑆 (𝑟, 𝑔 ∘ 𝑝)

≤ 𝑁(𝑟,

1

𝑔 ∘ 𝑝

) + 𝑆 (𝑟, 𝑔 ∘ 𝑝)

≤ 𝑁(]𝑟𝑘,1

𝑔

) + 𝑆 (𝑟, 𝑔 ∘ 𝑝) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) ,

(49)

where ] is defined as Lemma 15.Since

𝑃

𝑄 − 𝑃𝑄

𝑄2

= 𝑢

= 𝑈𝑢 =

𝑈𝑃

𝑄

, (50)

we conclude that

𝑃

𝑄 − 𝑃𝑄

= 𝑈𝑃𝑄. (51)

Now, writing 𝐹 = 𝜓𝐹 in (51), regarding then (51) as analgebraic equation in𝐹with coefficients of growth 𝑆(𝑟, 𝐹), andcomparing the leading coefficients, we deduce that

(𝑠 − 𝑡) 𝜓 = 𝑈. (52)

By integrating both sides of the last equality above, weconclude that

𝑢 (𝑧, 𝐹) = 𝛼𝐹(𝑧)𝑠−𝑡

, (53)

for some 𝛼 ∈ C \ {0}. Therefore, by combining therepresentations of 𝐹, 𝐵, 𝐴, 𝑔 with (53), we conclude that

𝑛

∏

𝑗=0


𝜆𝑗

= 𝛼

𝑎𝑠(𝑧)

𝑏𝑡(𝑧)

(𝑓 ∘ 𝑝)𝑠−𝑡

. (54)

If 𝑠𝑡 ̸= 0, we deduce from (11) and (54) that

𝛼

𝑎𝑠(𝑧)

𝑏𝑡(𝑧)

(𝑓 ∘ 𝑝)𝑠−𝑡

= 𝑅 (𝑧, 𝑓 ∘ 𝑝)

=

𝑎0(𝑧) + 𝑎

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑎

𝑠(𝑧) (𝑓 ∘ 𝑝)

𝑠

𝑏0(𝑧) + 𝑏

1(𝑧) (𝑓 ∘ 𝑝) + ⋅ ⋅ ⋅ + 𝑏

𝑡(𝑧) (𝑓 ∘ 𝑝)

𝑡.

(55)

From this, we get that 𝑅(𝑧, 𝑓 ∘ 𝑝) is not irreducible in 𝑓 ∘ 𝑝,a contradiction. Thus, 𝑡 = 0 or 𝑠 = 0. Therefore, we deducefrom (54) that

𝑛

∏

𝑗=0


𝜆𝑗

= 𝛼

𝑎𝑠(𝑧)

𝑏0(𝑧)

(𝑓 ∘ 𝑝)𝑠

(56)

or𝑛

∏

𝑗=0


𝜆𝑗

= 𝛼

𝑎0(𝑧)

𝑏𝑡(𝑧)

1

(𝑓 ∘ 𝑝)𝑡. (57)

The proof of Theorem 3 is completed.

Proof of Theorem 5. Assume first that 𝑓(𝑧) = 𝐷(𝑧)𝑒𝐸(𝑧),where 𝐷(𝑧) is a rational function and 𝐸(𝑧) is a polynomial.One can see from (17) that

𝑐 (𝑧) = 𝑓(𝑧)𝑚

𝑛

∏

𝑖=0

𝑓(𝑧 + 𝑐𝑖)𝜆𝑖

= [𝐷(𝑧)𝑚

𝑛

∏

𝑖=0

𝐷(𝑧 + 𝑐𝑖)𝜆𝑖

] 𝑒𝑚𝐸(𝑧)+∑

𝑛

𝑖=0𝜆𝑖𝐸(𝑧+𝑐

𝑖)

:= 𝐺 (𝑧) 𝑒𝑀(𝑧)

,

(58)

where 𝐺(𝑧) = 𝐷(𝑧)𝑚∏𝑛𝑖=0𝐷(𝑧 + 𝑐

𝑖)𝜆𝑖 is rational and𝑀(𝑧) =

𝑚𝐸(𝑧) + ∑𝑛

𝑖=0𝜆𝑖𝐸(𝑧 + 𝑐

𝑖) is a polynomial.

Suppose next that 𝑐(𝑧) = 𝐺(𝑧)𝑒𝑀(𝑧), where 𝐺(𝑧) is arational function and 𝑀(𝑧) is a polynomial. Since 𝑓(𝑧) hasonly finitely many poles, we conclude from (17) that

𝑁(𝑟,

1

𝑓

) ≤ 𝑁(𝑟,

1

𝑓𝑚

) = 𝑁(𝑟,

∏𝑛

𝑖=0𝑓(𝑧 + 𝑐

𝑖)𝜆𝑖

𝑐 (𝑧)

)

= 𝑂 (log 𝑟) .

(59)

Thus, 𝑓(𝑧) has only finitely many zeros and poles, and𝑓(𝑧) = 𝐷(𝑧)𝑒

𝐸(𝑧), where𝐷(𝑧) is rational and 𝐸(𝑧) is an entirefunction. In the following,we only prove𝐸(𝑧) is a polynomial.Now, substituting𝑓(𝑧) = 𝐷(𝑧)𝑒𝐸(𝑧) and 𝑐(𝑧) = 𝐺(𝑧)𝑒𝑀(𝑧) into(17), we get

𝑛

∏

𝑖=0

{𝐷(𝑧 + 𝑐𝑖)𝜆𝑖 exp (𝜆

𝑖𝐸 (𝑧 + 𝑐

𝑖))}

=

𝐺 (𝑧)

𝐷(𝑧)𝑚exp (𝑀 (𝑧) − 𝑚𝐸 (𝑧)) ,

(60)

(

𝑛

∏

𝑖=0

𝐷(𝑧 + 𝑐𝑖)𝜆𝑖

) exp(𝑛

∑

𝑖=0

𝜆𝑖𝐸 (𝑧 + 𝑐

𝑖))

=

𝐺 (𝑧)

𝐷(𝑧)𝑚exp (𝑀 (𝑧) − 𝑚𝐸 (𝑧)) .

(61)

Thus, we deduce from Lemma 20 that two exponents in (61)cancel each other to a constant 𝜏 ∈ C such that

𝑛

∑

𝑖=0


𝑖) = 𝑀 (𝑧) − 𝑚𝐸 (𝑧) + 𝜏; (62)


that is,𝑛

∑

𝑖=1


𝑖) + (𝜆

0+ 𝑚)𝐸 (𝑧) = 𝑀 (𝑧) + 𝜏. (63)

Suppose that 𝐸(𝑧) is not a polynomial. If 𝐸(𝑧) is a transcen-dental entire function of finite order, we get from Lemma 21,Remark 22, and (63) that 𝜎(𝐸) ≥ 1. Otherwise, 𝐸(𝑧) is atranscendental entire function of infinite order. These bothshow that 𝜎(𝑓) = ∞, contradicting the assumption that𝑓(𝑧) is finite order. Thus, 𝐸(𝑧) is a polynomial. The proof ofTheorem 5 is completed.

4. Proof of Theorem 7

Lemmas 23 and 25 reveal some properties of the maximalmodule of the polynomial in composite function 𝑓 ∘ 𝑝 with ameromorphic function 𝑓(𝑧) and a polynomial 𝑝(𝑧), whichare useful for proving the existence of Borel exceptionalvalue of finite order meromorphic solutions of functionaldifference equation of type (19).

Lemma 23. Let 𝑔(𝑧) be a nonconstant entire function of order𝜎(𝑔) = 𝜎 < ∞. Suppose that 𝛼

𝑗(𝑧) (𝑗 = 1, 2, . . . , 𝑚) are small

meromorphic functions relative to 𝑔(𝑧). Then there exists a set𝐸 ⊂ (1,∞) of lower logarithmic density 1 such that

𝑀(𝑟, 𝛼𝑗)

𝑀 (𝑟, 𝑔)

→ 0, 𝑗 = 1, 2, . . . , 𝑚, (64)

hold simultaneously for all 𝑟 ∈ 𝐸 as 𝑟 → ∞, where the lowerlogarithmic density of set 𝐸 is defined by

logdens (𝐸) = lim inf𝑟→∞

∫[1,𝑟]∩𝐸

(𝑑𝑡/𝑡)

log 𝑟. (65)

Remark 24. The proof of Lemma 23 is similar to the proof ofLemma 2.4 and Remark 2.5 in [16]. Here, we omit it.

Lemma 25. Let 𝑓(𝑧) be a finite order transcendental mero-morphic function satisfying (12), and 𝑝(𝑧) = 𝑑

𝑘𝑧𝑘

+ ⋅ ⋅ ⋅+𝑑1𝑧+

𝑑0is a polynomial with constant coefficients 𝑑

𝑘( ̸= 0), . . . , 𝑑

1, 𝑑0

and of the degree 𝑘 ≥ 1. Suppose that

𝐻(𝑧) = 𝑎𝑛(𝑧) (𝑓 ∘ 𝑝)

𝑛

+ 𝑎𝑛−1(𝑧) (𝑓 ∘ 𝑝)

𝑛−1

+ ⋅ ⋅ ⋅ + 𝑎1(𝑧) (𝑓 ∘ 𝑝) + 𝑎

0(𝑧)

(66)

is a polynomial in 𝑓 ∘𝑝, where 𝑛 (≥ 1) is a positive integer and𝑎𝑛(𝑧) ( ̸≡ 0), 𝑎

𝑛−1(𝑧), . . . , 𝑎

1(𝑧), 𝑎

0(𝑧) are small meromorphic

functions relative to 𝑓(𝑧). Then there exists a set 𝐸1of lower

logarithmic density 1 such that

log+𝑀(𝑟,𝐻) ≥ (𝑛 − 2𝜀) 𝑇 (𝜇𝑟𝑘, 𝑓) (67)

for all 𝑟 ∈ 𝐸1as 𝑟 → ∞, where 0 < 𝜇 < |𝑑

𝑘|. Hence,𝐻(𝑧) ̸≡

0.

Proof of Lemma 25. Let 𝜏 be the multiplicity of pole of 𝑓(𝑧)at the origin, and let 𝑞(𝑧) be a canonical product formed

with the nonzero poles of 𝑓(𝑧). Since 𝑓(𝑧) satisfies (12), thenℎ(𝑧) = 𝑧

𝜏

𝑞(𝑧) is an entire function. Thus, 𝑔(𝑧) = ℎ(𝑧)𝑓(𝑧) isentire, and (37), (38), and (40) also hold.

Now, substituting 𝑓(𝑧) = 𝑔(𝑧)/ℎ(𝑧) into (66), we con-clude that

𝐻(𝑧) = 𝑎𝑛(𝑧) ⋅

(𝑔 ∘ 𝑝)𝑛

(ℎ ∘ 𝑝)𝑛+ 𝑎

𝑛−1(𝑧) ⋅

(𝑔 ∘ 𝑝)𝑛−1

(ℎ ∘ 𝑝)𝑛−1

+ ⋅ ⋅ ⋅

+ 𝑎1(𝑧) ⋅

(𝑔 ∘ 𝑝)

(ℎ ∘ 𝑝)

+ 𝑎0(𝑧)

=

𝑎𝑛(𝑧)

(ℎ ∘ 𝑝)𝑛(𝑔 ∘ 𝑝)

𝑛

× [1 +

𝑎𝑛−1(𝑧) (ℎ ∘ 𝑝)

𝑎𝑛(𝑧)

(𝑔 ∘ 𝑝)−1

+ ⋅ ⋅ ⋅

+

𝑎1(𝑧) (ℎ ∘ 𝑝)

𝑛−1

𝑎𝑛(𝑧)

(𝑔 ∘ 𝑝)1−𝑛

+

𝑎0(𝑧) (ℎ ∘ 𝑝)

𝑛

𝑎𝑛(𝑧)

(𝑔 ∘ 𝑝)−𝑛

] .

(68)

We note from Lemma 15 and (40) that

𝑇(𝑟,

𝑎𝑛(𝑧)

(ℎ ∘ 𝑝)𝑛) = 𝑆 (𝑟, 𝑔 ∘ 𝑝) ,

𝑇(𝑟,

𝑎𝑗(𝑧) (ℎ ∘ 𝑝)

𝑛−𝑗

𝑎𝑛(𝑧)

) = 𝑆 (𝑟, 𝑔 ∘ 𝑝)

for 𝑗 = 0, 1, . . . , 𝑛 − 1.

(69)

Therefore, we deduce from Lemma 23 that there exists a set𝐸 ⊂ (1,∞) of lower logarithmic density 1 such that

𝑀(𝑟, 𝑎𝑛(𝑧) /(ℎ ∘ 𝑝)

𝑛

)

𝑀 (𝑟, 𝑔 ∘ 𝑝)

→ 0,

𝑀(𝑟, 𝑎𝑗(𝑧) (ℎ ∘ 𝑝)

𝑛−𝑗

/𝑎𝑛(𝑧))

𝑀 (𝑟, 𝑔 ∘ 𝑝)

→ 0,

(𝑗 = 0, 1, . . . , 𝑛 − 1) .

(70)

Moreover, according to the choosing of 𝐸 in the proofof Lemma 23, we know that 𝑎

𝑗(𝑧)(ℎ ∘ 𝑝)

𝑛−𝑗

/𝑎𝑛(𝑧) for


𝑗 = 0, 1, . . . , 𝑛 − 1 have no zeros and poles for all |𝑧| = 𝑟 ∈ 𝐸.Thus, we conclude from (68) and (70) that, for any 𝜀 > 0,

𝑀(𝑟,𝐻) ≥ 𝑀(𝑟, 𝑔 ∘ 𝑝)𝑛−𝜀

× [1 −

𝑎𝑛−1(𝑧) (ℎ ∘ 𝑝)

𝑎𝑛(𝑧)

𝑀(𝑟, 𝑔 ∘ 𝑝)−1

− ⋅ ⋅ ⋅

−

𝑎1(𝑧) (ℎ ∘ 𝑝)

𝑛−1

𝑎𝑛(𝑧)

𝑀(𝑟, 𝑔 ∘ 𝑝)1−𝑛

−

𝑎0(𝑧) (ℎ ∘ 𝑝)

𝑛

𝑎𝑛(𝑧)

𝑀(𝑟, 𝑔 ∘ 𝑝)−𝑛

]

≥ (1 − 𝜀)𝑀(𝑟, 𝑔 ∘ 𝑝)𝑛−𝜀

,

(71)

and so

log+𝑀(𝑟,𝐻) ≥ (𝑛 − 32

𝜀) log+𝑀(𝑟, 𝑔 ∘ 𝑝) (72)

for all |𝑧| = 𝑟 ∈ 𝐸 and |𝑔 ∘ 𝑝(𝑧)| = 𝑀(𝑟, 𝑔 ∘ 𝑝).Therefore, we deduce from Lemma 15 and (38) that

log+𝑀(𝑟,𝐻) ≥ (𝑛 − 2𝜀) 𝑇 (𝜇𝑟𝑘, 𝑓) (73)

for all |𝑧| = 𝑟 ∈ 𝐸1= 𝐸 ∩ (𝑟

0, +∞), where 𝑟

0> 0. It is

obvious that 𝐸1has lower logarithmic density 1. The proof of

Lemma 25 is completed.

Proof of Theorem 7. Suppose that 𝑓(𝑧) has two finite Borelexceptional values 𝑎 and 𝑏 ( ̸= 0, 𝑎). For the case where oneof 𝑎 and 𝑏 is infinite, we can use a similar method to prove it.Set

𝑔 (𝑧) =

𝑓 (𝑧) − 𝑎

𝑓 (𝑧) − 𝑏

. (74)

Then 𝜎(𝑔) = 𝜎(𝑓) and

𝜆 (𝑔) = 𝜆 (𝑓 − 𝑎) < 𝜎 (𝑔) , 𝜆 (

1

𝑔

) = 𝜆 (𝑓 −

complex differences and difference...

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