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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Baker, I.N.: On a class of meromorphic functions,. Proc. Am. Math. Soc. 17, 819–822 (1966)
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)
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Partial Differential Equations, vol. II. Interscience, New York (1962)
Gundersen, G., Hayman, W.: The strength of Cartan’s version of Nevanlinna theory. Bull. Lond. Math. Soc. 36, 433–454 (2004)
Gross, F.: On the equation f n+g n=1. Bull. Am. Math. Soc. 72, 86–88 (1966) (erratum, 72, 576 (1966))
Granville, A., Tucker, T.J.: It’s as easy as abc. Not. Am. Math. Soc. 49, 1224–1231 (2002)
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)
Hemmati, J.: Entire solutions of first-order nonlinear partial differential equations. Proc. Am. Math. Soc. 125, 1483–1485 (1977)
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)
Hurwitz, A., Courant, R.: Funktionentheorie, vol. 4. Springer, Berlin (1964)
Khavinson, D.: A note on entire solutions of the eikonal equation. Am. Math. Mon. 102, 159–161 (1995)
Kujala, R.: Functions of finite λ-type in several complex variables. Trans. Am. Math. Soc. 161, 327–358 (1971)
Jategaonkar, A.V.: Elementary proof of a theorem of P. Motel on entire functions. J. Lond. Math. Soc. 40, 166–170 (1965)
Lang, S.: Old and new conjectured diophantine inequalities. Bull. Am. Math. Soc. 23, 37–75 (1990)
Li, B.Q.: On certain functional and partial differential equations. Forum Math. 17, 77–86 (2005)
Li, B.Q.: Entire solutions of certain partial differential equations and factorization of partial derivatives. Trans. Am. Math. Soc. 357, 3169–3177 (2005)
Li, B.Q.: On meromorphic solutions of f 2+g 2=1. Math. Z. 258, 763–771 (2008)
Li, B.Q.: Uniqueness of entire functions sharing four small functions. Am. J. Math. 119, 841–858 (1997)
Montel, P.: Leçons sur les familles normales de functions analytiques et leurs applications. Gauthier-Villars, Paris (1927)
Saleeby, E.G.: Entire and meromorphic solutions of Fermat type partial differential equations. Analysis 19, 369–376 (1999)
Shabat, B.V.: Introduction to Complex Analysis, part II, Functions of Several Variables. Translation Mathematical Monographs, vol. 110. American Mathematical Society, Providence (1992)
Shabat, B.V.: Distribution of Values of Holomorphic Mappings. Translation Mathematical Monographs, vol. 61. American Mathematical Society, Providence (1985)
Stoll, W.: Introduction to the Value Distribution Theory of Meromorphic Functions. Springer, New York (1982)
Toda, N.: On the functional equation \(\sum_{i=0}^{p}a_{i}f_{i}^{n_{i}}=1\). Tohoku Math. J. 23, 289–299 (1971)
Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebra. Ann. Math. 141, 553–572 (1995)
Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)
Yang, C.C.: A generalization of a theorem of P. Montel on entire functions. Proc. Am. Math. Soc. 26, 332–334 (1970)
Yang, L.: Value Distribution Theory. Springer, Berlin (1993)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of the late Professor Leon Ehrenpreis.
Rights 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
DOI: https://doi.org/10.1007/978-88-470-1947-8_13
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-1946-1
Online ISBN: 978-88-470-1947-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)