复合函数-微分方程亚纯解的性质

Abstract

Abstract:

The aim of this paper is twofold. Firstly, we consider the existence of solutions to a type of complex functional-differential equations

(w')ⁿw⁽ⁿ⁾=awⁿ⁺¹(g)+bw+d

in complex variables. We obtain g is linear when w is a transcendental meromorphic function and a≠0, b, d are constants. In addition, due to the different properties between equations and system of equations, it is meaningful to research systems of equations, this paper is also concerned with a type of system of functional equations, properties of meromorphic solutions are obtained under some proper conditions. Examples are constructed to show that our results are accurate.

Key words: Meromorphic solutions, Complex functional differential equations, Growth order, Nevanlinna theory

CLC Number:

O174

Manli Liu,Lingyun Gao. Behaviour of Meromorphic Solutions of Complex Functional-Differential Equations[J].Acta mathematica scientia,Series A, 2020, 40(5): 1121-1131.

Trendmd

References 26

1	Bellman R , Cooke K L . Differential-Difference Equations. New York:Academic Press, 1963
2	Brunt B van , Marshall J C , Wake G C . Holomorphic solutions to pantograph type equations with neutral fixed points. J Math Anal Appl, 2004, 295, 557- 569 doi: 10.1016/j.jmaa.2004.03.019
3	Brunt B van , Wake G C , Kim H K . On a singular Sturm-Liouville problem involving an advanced functional differential equation. European J Appl Math, 2001, 12, 625- 644 doi: 10.1017/S0956792501004624
4	Carr J , Dyson J . The matrix functional equation $ y'(x) = Ay(\lambda x)+By(x) $. Proc Roy Soc Edinburgh Sect A, 1975, 75, 5- 22
5	Chen Z X . Complex Differences and Difference Equations. Beijing:Science Press, 2013
6	Chen Z X , Shon K H . On zeros and fixed points of differences of meromorphic functions. J Math Anal Appl, 2008, 344, 373- 383 doi: 10.1016/j.jmaa.2008.02.048
7	Chen Z X . Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl, 2011, 373, 235- 241 doi: 10.1016/j.jmaa.2010.06.049
8	Fox L , Mayers D F , Ockendon J R , Tayler A B . On a functional differential equation. J Inst Math Appl, 1971, 8, 271- 307 doi: 10.1093/imamat/8.3.271
9	Fredrickson P O. Dirichlet Series Solution for Certain Functional Differential Equations//Urabe M. Japan-United States Seminar on Ordinary Differential and Functional Equations. Berlin: Springer, 1971: 249-254
10	Gao L Y . On meromorphic solutions of a type system of composite functional equations. Acta Mathematica Scientia, 2012, 32B, 800- 806
11	Gao L Y , Liu M L . Riccati-type result for meromorphic solutions of systems of composite functional equations. Acta Mathematica Scientia, 2017, 37B, 1685- 1694
12	Gross F , Yang C . On meromorphic solution of a certain class of functional-differential equations. Annales Polonici Mathematici, 1973, 27, 305- 311 doi: 10.4064/ap-27-3-305-311
13	Hall A J , Wake G C . A functional differential equation arising in the modelling of cell-growth. J Austral Math Soc Ser B, 1989, 30, 424- 435 doi: 10.1017/S0334270000006366
14	Iserles A . On the generalized pantograph functional-differential equation. European J Appl Math, 1993, 4, 1- 38 doi: 10.1017/S0956792500000966
15	Iserles A , Liu Y . On neutral functional-differential equations with proportional delays. J Math Anal Appl, 1997, 207, 73- 95 doi: 10.1006/jmaa.1997.5262
16	Kato T , McLeod J B . The functional differential equation $ y'(x) = ay(\lambda x)+by(x) $. Bull Amer Math Soc, 1971, 77, 891- 937 doi: 10.1090/S0002-9904-1971-12805-7
17	Laine I . Nevanlinna Theory and Complex Differential Equations. Berlin:Walter de Gruyter, 1993
18	Li B Q , Saleeby E . On solutions of functional-differential equations $ f'(x) = a(x)f(g(x))+b(x)f(x)+c(x) $ in the large. Israel J of Math, 2007, 162, 335- 348 doi: 10.1007/s11856-007-0101-z
19	Liu M L , Gao L Y . Transcendental solutions of systems of complex differential-difference equations (in Chinese). Sci Sin Math, 2019, 49 (11): 1633- 1654
20	Mohon'ko V D . Meromorphic solutions of linear defferential equation with meromorphic coefficients. Differentsial'nye Uravneniya, 1973, 9, 1534- 1536
21	Ockendon J R , Tayler A B . The dynamics of a current collection system for an electric locomotive. Proc Roy Soc London Ser A, 1971, 322, 447- 468
22	Silvennoinen H . Meromorphic solutions of some composite functional equations. Ann Acad Sci Fenn Mathematica Dissertations, 2003, 133, 1- 39
23	Valiron G . Sur la dérivée des fonctions algébroides. Bull Soc Math France, 1931, 59, 17- 39
24	Wake G C , Cooper S , Kim H K , Brunt B van . Functional differential equations for cell-growth models with dispersion. Comm Appl Anal, 2000, 4, 561- 573
25	Xu H Y , Liu S Y , Li Q P . The existence and growth of solutions for several systems of complex nonlinear difference equations. Mediterr J Math, 2019, 16, 1- 30 doi: 10.1007/s00009-018-1275-9
26	Yi H X , Yang C C . Theory of the Uniqueness of Meromorphic Functions (in Chinese). Beijing:Science Press, 1995

[1]	LV Feng, ZHOU Feng. A New Study on Normal Families Concerning Continuous Functions [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1348-1352.
[2]	QI Xiao-Guang, YANG Lian-Zhong. Value Sharing for a Meromorphic Function with Its q-Shift Difference [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1372-1380.
[3]	QI Xiao-Guang, LIU Yong, YANG Lian-Zhong. Nevanlinna Theory for the f(qz+c) and Its Applications [J]. Acta mathematica scientia,Series A, 2013, 33(5): 819-828.
[4]	QI Jian-Ming, LI Ye-Zhou, YUAN Wen-Jun. Further Results of Gol'dberg's Theorem Concerning the Growth of Meromorphic Solutions of \|Algebraic Differential Equations [J]. Acta mathematica scientia,Series A, 2013, 33(4): 759-765.
[5]	LV Feng. Some Further Results on Montel's Criterion [J]. Acta mathematica scientia,Series A, 2013, 33(3): 561-587.
[6]	SUN Dao-Chun. Random Meromorphic Functions in the Unit Disc [J]. Acta mathematica scientia,Series A, 2009, 29(3): 623-629.
[7]	YUAN Wen-Jun. An Oscillation Result for Higher Order Linear Differential Equations with One Dominant Coefficient [J]. Acta mathematica scientia,Series A, 2003, 23(2): 224-230.

Behaviour of Meromorphic Solutions of Complex Functional-Differential Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 26

Related Articles 7

Metrics

Comments

Recommended 10