Bulletin of the Korean Mathematical Society (대한수학회보)

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SOME RESULTS RELATED TO COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS OF CERTAIN TYPES

  • Liu, Kai (Department of Mathematics Nanchang University) ;
  • Dong, Xianjing (Department of Mathematics Nanchang University)
  • Received : 2013.06.26
  • Published : 2014.09.30
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Abstract

In this paper, we consider the growth and existence of solutions of differential-difference equations of certain types. We also consider the differential-difference analogues of Br$\ddot{u}$ck conjecture and give a short proof on a theorem given by Li, Yang and Yi [18]. Our additional purpose is to explore the similarity or difference on some problems in differential, difference and differential-difference fields.

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References

  1. W. Bergweiler and J. K. Langley, Zeros of difference of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 133-147. https://doi.org/10.1017/S0305004106009777
  2. R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.
  3. R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1-2, 21-24. https://doi.org/10.1007/BF03322176
  4. Z. X. Chen, Growth and zeros of meromorphic solution of some linear difference equations, J. Math. Anal. Appl. 373 (2011), no. 1, 235-241. https://doi.org/10.1016/j.jmaa.2010.06.049
  5. Z. X. Chen, Zeros of entire solutions to complex linear difference equations, Acta Math. Sci. Ser. B Engl. Ed. 32 (2012), no. 3, 1141-1148.
  6. Z. X. Chen and K. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), no. 2, 235-244. https://doi.org/10.11650/twjm/1500407625
  7. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f($z+{\eta}$) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129.
  8. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223, (1998), no. 1, 88-95. https://doi.org/10.1006/jmaa.1998.5959
  9. R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
  10. R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
  11. R. G. Halburd and R. J. Korhonen, Meromorphic solutions of difference equations, integrability and the discrete Painleve equations, J. Phys. A 40 (2007), no. 6, 1-38. https://doi.org/10.1088/1751-8113/40/1/001
  12. W. K. Hayman, Meromorphic Functions, Oxford at the Clarendon Press, 1964.
  13. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory 1 (2001), no. 1, 27-39. https://doi.org/10.1007/BF03320974
  14. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. L. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), no. 1, 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
  15. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin-New York, 1993.
  16. S. Li and Z. S. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Arch. Math. (Basel) 97 (2011), no. 5, 475-483. https://doi.org/10.1007/s00013-011-0324-4
  17. B. Q. Li and E. G. Saleeby, On solutions of functional differential equations f'(x)=a(x)f(g(x))+b(x)f(x)+c(x) in the large, Israel J. Math. 162 (2007), 335-348. https://doi.org/10.1007/s11856-007-0101-z
  18. X. M. Li, X. Yang, and H. X. Yi, Entire functions sharing an entire function of smaller order with their shifts, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 2, 34-39. https://doi.org/10.3792/pjaa.89.34
  19. X. M. Li, H. X. Yi, and C. Y. Kang, Notes on entire functions sharing an entire function of a smaller order with their difference operators, Arch. Math. (Basel) 99 (2012), 261-270. https://doi.org/10.1007/s00013-012-0425-8
  20. K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math. (Basel) 92 (2009), no. 3, 270-278. https://doi.org/10.1007/s00013-009-2895-x
  21. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2003.

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