Skip to main content

On Fermat-Type Functional and Partial Differential Equations

  • Conference paper
The Mathematical Legacy of Leon Ehrenpreis

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 16))

Abstract

This paper concerns entire and meromorphic solutions to functional and nonlinear partial differential equations of the form a 1 f m+a 2 g n=a 3 with function coefficients a k , k=1,2,3, where f and g are unknown functions or partial derivatives of an unknown function. We will discuss some recent results and also give, among other things, some new results on these equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baker, I.N.: On a class of meromorphic functions,. Proc. Am. Math. Soc. 17, 819–822 (1966)

    Article  MATH  Google Scholar 

  2. Chuang, C.: Sur la comparaison de la croissance d’une fonction méromorphe et de celle de sa dérivée. Bull. Sci. Math. 75, 171–190 (1951)

    MathSciNet  Google Scholar 

  3. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Partial Differential Equations, vol. II. Interscience, New York (1962)

    MATH  Google Scholar 

  4. Gundersen, G., Hayman, W.: The strength of Cartan’s version of Nevanlinna theory. Bull. Lond. Math. Soc. 36, 433–454 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gross, F.: On the equation f n+g n=1. Bull. Am. Math. Soc. 72, 86–88 (1966) (erratum, 72, 576 (1966))

    Article  MATH  Google Scholar 

  6. Granville, A., Tucker, T.J.: It’s as easy as abc. Not. Am. Math. Soc. 49, 1224–1231 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Han, Q.: On complex analytic solutions of the partial differential equation \(u_{z_{1}}^{m}+u_{z_{2}}^{m}=u^{m}\). Houst. J. Math. 35, 277–289 (2009)

    MATH  Google Scholar 

  8. Hemmati, J.: Entire solutions of first-order nonlinear partial differential equations. Proc. Am. Math. Soc. 125, 1483–1485 (1977)

    Article  MathSciNet  Google Scholar 

  9. Huber, A.: A novel class of solutions for a nonlinear third order wave equation generated by the Weierstrass transformation. Chaos Solitons Fractals 28, 972–978 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hurwitz, A., Courant, R.: Funktionentheorie, vol. 4. Springer, Berlin (1964)

    MATH  Google Scholar 

  11. Khavinson, D.: A note on entire solutions of the eikonal equation. Am. Math. Mon. 102, 159–161 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kujala, R.: Functions of finite λ-type in several complex variables. Trans. Am. Math. Soc. 161, 327–358 (1971)

    MathSciNet  MATH  Google Scholar 

  13. Jategaonkar, A.V.: Elementary proof of a theorem of P. Motel on entire functions. J. Lond. Math. Soc. 40, 166–170 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lang, S.: Old and new conjectured diophantine inequalities. Bull. Am. Math. Soc. 23, 37–75 (1990)

    Article  MATH  Google Scholar 

  15. Li, B.Q.: On certain functional and partial differential equations. Forum Math. 17, 77–86 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Li, B.Q.: Entire solutions of certain partial differential equations and factorization of partial derivatives. Trans. Am. Math. Soc. 357, 3169–3177 (2005)

    Article  MATH  Google Scholar 

  17. Li, B.Q.: On meromorphic solutions of f 2+g 2=1. Math. Z. 258, 763–771 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, B.Q.: Uniqueness of entire functions sharing four small functions. Am. J. Math. 119, 841–858 (1997)

    Article  MATH  Google Scholar 

  19. Montel, P.: Leçons sur les familles normales de functions analytiques et leurs applications. Gauthier-Villars, Paris (1927)

    Google Scholar 

  20. Saleeby, E.G.: Entire and meromorphic solutions of Fermat type partial differential equations. Analysis 19, 369–376 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Shabat, B.V.: Introduction to Complex Analysis, part II, Functions of Several Variables. Translation Mathematical Monographs, vol. 110. American Mathematical Society, Providence (1992)

    MATH  Google Scholar 

  22. Shabat, B.V.: Distribution of Values of Holomorphic Mappings. Translation Mathematical Monographs, vol. 61. American Mathematical Society, Providence (1985)

    MATH  Google Scholar 

  23. Stoll, W.: Introduction to the Value Distribution Theory of Meromorphic Functions. Springer, New York (1982)

    Google Scholar 

  24. Toda, N.: On the functional equation \(\sum_{i=0}^{p}a_{i}f_{i}^{n_{i}}=1\). Tohoku Math. J. 23, 289–299 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebra. Ann. Math. 141, 553–572 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, C.C.: A generalization of a theorem of P. Montel on entire functions. Proc. Am. Math. Soc. 26, 332–334 (1970)

    Article  MATH  Google Scholar 

  28. Yang, L.: Value Distribution Theory. Springer, Berlin (1993)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bao Qin Li .

Editor information

Editors and Affiliations

Additional information

Dedicated to the memory of the late Professor Leon Ehrenpreis.

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Italia

About this paper

Cite this paper

Li, B.Q. (2012). On Fermat-Type Functional and Partial Differential Equations. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_13

Download citation

Publish with us

Policies and ethics