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SOME RESULTS ON MEROMORPHIC SOLUTIONS OF Q-DIFFERENCE DIFFERENTIAL EQUATIONS

  • Lingyun Gao (Department of mathematics Jinan University) ;
  • Zhenguang Gao (Department of mathematics Jinan University) ;
  • Manli Liu (Department of mathematics South China Agricultural University)
  • Received : 2021.10.24
  • Accepted : 2023.02.17
  • Published : 2023.05.31

Abstract

In view of Nevanlinna theory, we investigate the meromorphic solutions of q-difference differential equations and our results give the estimates about counting function and proximity function of meromorphic solutions to these equations. In addition, some interesting results are obtained for two general equations and a class of system of q-difference differential equations.

Keywords

Acknowledgement

The authors would like to thank the referees for their thorough comments and helpful suggestions.

References

  1. G. E. Andrews, R. A. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge Univ. Press, Cambridge, 1999. https://doi.org/10.1017/CBO9781107325937 
  2. 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 
  3. M. F. Chen, Y. Y. Jiang, and Z. S. Gao, Growth of meromorphic solutions of certain types of q-difference differential equations, Adv. Difference Equ. 2017 (2017), Paper No. 37, 16 pp. https://doi.org/10.1186/s13662-017-1072-y 
  4. L. Y. Gao, On m-admissible components of solution, Complex Variables Theory Appl. 35 (1998), no. 4, 297-306. https://doi.org/10.1080/17476939808815088 
  5. W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. 
  6. Y. Z. He and X. Z. Xiao, Algebroid Functions and Ordinary Differential Equations, Science Press, Beijing 1988. 
  7. P.-C. Hu and M. L. Liu, A Malmquist type theorem for a class of delay differential equations, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 1, 131-145. https://doi.org/10.1007/s40840-020-00941-8 
  8. P.-C. Hu, W. Wang, and L. Wu, Entire solutions of differential-difference equations of Fermat type, Bull. Korean Math. Soc. 59 (2022), no. 1, 83-99. https://doi.org/10.4134/BKMS.b210099 
  9. Z. B. Huang, Value distribution and uniqueness on q-differences of meromorphic functions, Bull. Korean Math. Soc. 50 (2013), no. 4, 1157-1171. https://doi.org/10.4134/BKMS.2013.50.4.1157 
  10. I. Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, 15, de Gruyter, Berlin, 1993. https://doi.org/10.1515/9783110863147 
  11. I. Laine and Z. Latreuch, Remarks on meromorphic solutions of some delay-differential equations, Anal. Math. 48 (2022), no. 4, 1081-1104. https://doi.org/10.1007/s10476-022-0172-4 
  12. I. Laine and C.-C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556-566. https://doi.org/10.1112/jlms/jdm073 
  13. A. Z. Mokhon'ko and V. D. Mokhon'ko, Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations, Sib. Math. J. 15 (1974), no. 6, 921-934.  https://doi.org/10.1007/BF00966560
  14. Z. H. Tu and X. Z. Xiao, On the meromorphic solutions of system of higher-order algebraic differential equations, Complex Variables Theory Appl. 15 (1990), no. 3, 197-209. https://doi.org/10.1080/17476939008814451 
  15. Y. Wang, Solutions of complex difference and q-difference equations, Adv. Difference Equ. 2016 (2016), no. 1, 1-22. https://doi.org/10.1186/s13662-016-0790-x 
  16. X. L. Wang, H. Wang, and H. Y. Xu, The existence of solutions of q-difference differential equations, SpringerPlus. 5 (2016), no. 1, 1-9. https://doi.org/10.1186/s40064-016-2179-4 
  17. L. Xu and T.-B. Cao, Meromorphic solutions of delay differential equations related to logistic type and generalizations, Bull. Sci. Math. 172 (2021), Paper No. 103040, 25 pp. https://doi.org/10.1016/j.bulsci.2021.103040 
  18. J. F. Xu and L. Luo, Some q-shift difference results on Hayman conjecture and uniqueness theorems, Bull. Iranian Math. Soc. 48 (2022), no. 3, 1193-1204. https://doi.org/10.1007/s41980-021-00574-y 
  19. H. Y. Xu, L. Yang, and H. Wang, Growth of the solutions of some q-difference differential equations, Adv. Difference Equ. 2015 (2015), 172, 12 pp. 
  20. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing/New York, 2003. 
  21. J. Zhang and R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 369 (2010), no. 2, 537-544. https://doi.org/10.1016/j.jmaa.2010.03.038 
  22. X.-M. Zheng and Z. X. Chen, Some properties of meromorphic solutions of q-difference equations, J. Math. Anal. Appl. 361 (2010), no. 2, 472-480. https://doi.org/10.1016/j.jmaa.2009.07.009