Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ENTIRE SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATION AND FERMAT TYPE q-DIFFERENCE DIFFERENTIAL EQUATIONS

  • CHEN, MIN FENG (LMIB and School of Mathematics and Systems Science Beihang University) ;
  • GAO, ZONG SHENG (LMIB and School of Mathematics and Systems Science Beihang University)
  • Received : 2015.05.12
  • Published : 2015.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

Keywords

References

  1. D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3, 457-474. https://doi.org/10.1017/S0308210506000102
  2. 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
  3. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f(z+${\eta}$) and diffdeence equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
  4. F. Gross, On the equation $f(z)^n+g(z)^n=1$, Bull. Amer. Math. Soc. 72 (1966), 86-88. https://doi.org/10.1090/S0002-9904-1966-11429-5
  5. F. Gross, On the equation $f(z)^n+g(z)^n=h(z)^n$, Amer. Math. Monthly 73 (1966), 1093-1096 https://doi.org/10.2307/2314644
  6. F. Gross, On the equation $f(z)^n+g(z)^n=h(z)^n$, J. Math. Anal. Appl. 314 (2006), 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
  7. W. K. Hayman, Meromorphic Function, Clarendon Press, Oxford, 1964.
  8. I. Laine, Nevanlinna Theory and Complex Differential Equations, Water de Gruyter, Berlin, 1993.
  9. S. Li and Z. S. Gao, Finite order meromorphic solutions of linear difference equations, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 5, 73-76. https://doi.org/10.3792/pjaa.87.73
  10. K. Liu and T. B. Cao, Entire solutions of Fermat type q-difference differential equations, Electron. J. Differential Equations 2013 (2013), no. 59, 1-10.
  11. K. Liu and L. Z. Yang, On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory 13 (2013), no. 3, 433-447. https://doi.org/10.1007/s40315-013-0030-2
  12. P. Montel, Lecons sur les familles de nomales fonctions analytiques et leurs applications, Gauthier-Viuars Paris (1927), 135-136.
  13. J. F. Tang and L. W. Liao, The transcendental meromorphic solutions of a certain type of nonlinear differential equations, J. Math. Anal. Appl. 334 (2007), no. 1, 517-527. https://doi.org/10.1016/j.jmaa.2006.12.075
  14. C. C. Yang, A generalization of a theorem of P. Montel on entire functions, Proc. Amer. Math. Sci. 26 (1970), 332-334. https://doi.org/10.1090/S0002-9939-1970-0264080-X
  15. C. C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan. Acad. Ser. A. Math. Sci. 86 (2010), 10-14. https://doi.org/10.3792/pjaa.86.10
  16. C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.

Cited by

  1. Existence of entire solutions of some non-linear differential-difference equations vol.2017, pp.1, 2017, https://doi.org/10.1186/s13660-017-1368-1