On entire solutions of certain type of nonlinear differential equations

Fengrong Zhang; Linlin Wu; Jing Yang; Weiran Lü; Fengrong Zhang; Linlin Wu; Jing Yang; Weiran Lü

doi:10.3934/math.2020393

AIMS Mathematics

2020, Volume 5, Issue 6: 6124-6134. doi: 10.3934/math.2020393

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Research article

On entire solutions of certain type of nonlinear differential equations

Department of Mathematics, China University of Petroleum, Qingdao 266580, P. R. China

Received: 06 June 2020 Accepted: 15 July 2020 Published: 28 July 2020
MSC : 34M10, 30D05, 30D35

In this paper, we shall extend some results regarding the growth estimate of entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.
- entire solution,
- differential equation,
- mathieu equation,
- hyper-order,
- nevanlinna theory
Citation: Fengrong Zhang, Linlin Wu, Jing Yang, Weiran Lü. On entire solutions of certain type of nonlinear differential equations[J]. AIMS Mathematics, 2020, 5(6): 6124-6134. doi: 10.3934/math.2020393

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Abstract

In this paper, we shall extend some results regarding the growth estimate of entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.

References

[1]	S. Bank, Some results on analytic and meromorphic solutions of algebraic differential equations, Adv. Math., 15 (1975), 41-62. doi: 10.1016/0001-8708(75)90124-3
[2]	S. Bank, J. Langley, Oscillation theory for higher order linear differential equations with entire coeffients, Complex Var. Elliptic Eq., 16 (1991), 163-175.
[3]	Z. X. Chen, The growth of solutions of $f''+{\rm e}^{-z}f'+ Q(z)f = 0$ where the order $(Q) = 1$, Sci. China Ser. A, 45 (2002), 290-300.
[4]	Z. X. Chen, K. H. Shon, The hyper order of solution of second order differential equations and subnomal solutions of periodic equations, Taiwanese J. Math., 14 (2010), 611-628. doi: 10.11650/twjm/1500405809
[5]	Z. X. Chen, C. C. Yang, Some further results on the zeros and growths of entire solutions of second order linear differential equations, Kodai Math. J., 22 (1999), 273-285. doi: 10.2996/kmj/1138044047
[6]	A. E. Eremenko, Meromorphic solutions of algebraic differential equations, Russ. Math. Surv., 37 (1982), 61-95.
[7]	M. Frei, Über die subnormalen Löungen der Differentialgleichung $\omega''+{\rm e}^{-z}\omega' + (konst.)\omega = 0,$ Comment. Math. Helv. 36 (1962), 1-8.
[8]	F. Gackstatter, I. Laine, Zur Theorie der gewAhnlichen Differentialgleichungen im Komplexen, Ann. Polo. Math., 38 (1980), 259-287. doi: 10.4064/ap-38-3-259-287
[9]	A. A. Gol'dberg, On the single valued integrals of differential equations of the first order, Ukrainian Math. J., 8 (1956), 254-261 (Russian).
[10]	G. Gundersen, On the question of whether $f'' +{\rm e}^{-z}f' + B(z)f = 0$ can admit a solution $f\not\equiv 0$ of finite order, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 9-17. doi: 10.1017/S0308210500014451
[11]	W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964.
[12]	S. Hellerstein, J. Miles, J. Rossi, On the growth of solutions of certain linear differential equations, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 17 (1992), 343-365. doi: 10.5186/aasfm.1992.1723
[13]	I. Laine, Nevanlinna theory and complex differential equations, Walter de Gruyter, Berlin/New York, 1993.
[14]	P. Li, Entire solutions of certain type of differential equations II, J. Math. Anal. Appl., 375 (2011), 310-319. doi: 10.1016/j.jmaa.2010.09.026
[15]	P. Li, C. C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl., 320 (2006), 827-835. doi: 10.1016/j.jmaa.2005.07.066
[16]	L. W. Liao, C. C. Yang, J. J. Zhang, On meromorphic solutions of certain type of non-linear differential equations, Ann. Acad. Sci. Fenn. Math., 38 (2013), 581-593. doi: 10.5186/aasfm.2013.3840
[17]	M. Ozawa, On a solution of $\omega''+{\rm e}^{-z}\omega' + (az +b)\omega = 0,$ Kodai Math. J., 3 (1980), 295-309.
[18]	N. Steinmetz, Meromorphic solutions of second-order algebraic differential equations, Complex Var. Elliptic Eq., 13 (1989), 75-83.
[19]	N. Toda, On algebroid solutions of algebraic differential equations in the complex plane II, J. Math. Soc. Japan, 45 (1993), 705-717. doi: 10.2969/jmsj/04540705
[20]	J. Wang, W. R. Lü, The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients, Acta Math. Appl. Sinica, 27 (2004), 72-80 (Chinese).
[21]	E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge University Press, Cambridge, 1927.
[22]	H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Springer, Berlin Heidelberg, 1968.
[23]	C. C. Yang, On deficiencies of differential polynomials, Math. Z., 116 (1970), 197-204. doi: 10.1007/BF01110073
[24]	C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Science Press, Beijing/New York, 2003.
[25]	K. Yosida, A generalization of Malmquist's theorem, Japan. J. Math., 9 (1933), 253-256.
[26]	J. Zhang, L. W. Liao, On entire solutions of a certain type of nonlinear differential and difference equations, Taiwanese J. Math., 15 (2011), 2145-2157. doi: 10.11650/twjm/1500406427
[27]	J. J. Zhang, X. Q. Lu, L. W. Liao, On exact transcendental meromorphic solutions of nonlinear complex differential equations, Houston J. Math., 45 (2019), 439-453.

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