AIMS Mathematics, 2020, 5(6): 6124-6134. doi: 10.3934/math.2020393

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

On entire solutions of certain type of nonlinear differential equations

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

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.
  Article Metrics


1. S. Bank, Some results on analytic and meromorphic solutions of algebraic differential equations, Adv. Math., 15 (1975), 41-62.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved