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An Equivalent Form of Picard’s Theorem and Beyond

Published online by Cambridge University Press:  20 November 2018

Bao Qin Li
Bao Qin Li*
Affiliation:
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199 USA, e-mail: libaoqin@fiu.edu
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Abstract

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This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation ${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

Keywords

30D2032A1535F20entire functionPicard’s Theoremfunctional equationpartial differential equation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 142 - 148
Copyright
Copyright © Canadian Mathematical Society 2018

References

[I] Ahlfors, L. V., Conformal invariants: topics in geometrical function theory. McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1973.Google Scholar
[2] Borel, E., Sur les zeros desfunctions entieres. Acta Math. 20 (1897), no. 1, 357–396. http://dx.doi.Org/10.1007/BF02418037.Google Scholar
[3] Courant, R. and Hubert, D., Methods of mathematicalphysics, II. Partial differential equations. Wiley Classics Library, John Wiley & Sons, 1991.Google Scholar
[4] Davis, B., Picard's theorem and Brownian motion. Trans. Amer. Math. Soc. 23 (1975), 353–362. http://dx.doi.Org/10.2307/1998050.Google Scholar
[5] Fuchs, W. H. J., Topics in the theory offunctions ofone complex variable. D. Van Nostrand, Princeton, NJ, 1967.Google Scholar
[6] Hayman, W. K., Meromorphic functions. Clarendon Press, Oxford, 1964.Google Scholar
[7] Hille, E., Ordinary differential equations in the complex domain. Dover, Mineola, NY, 1997.Google Scholar
[8] Lewis, J. L., Picard's theorem and Richman's theorem by way ofHarnack's inequality. Proc. Amer. Math. Soc. 122 (1994), 199–206. http://dx.doi.org/10.2307/2160861.Google Scholar
[9] Li, B. Q., On meromorphic Solutions off2 + g2 = 1. Math. Z. 258 (2008), 763–771. http://dx.doi.Org/10.1007/s00209-007-01 96-2.Google Scholar
[10] Li, B. Q., On Fermat-type functional and partial differential equations. Springer Proceedings in Mathematics, 16, Springer, Milan, 2012, pp. 209–222. http://dx.doi.org/10.1007/978-88-470-1947-8J3Google Scholar
[11] Li, B. Q., Estimates on derivatives and logarithmic derivatives of holomorphic functions and Picard's theorem. J. Math. Anal. Appl. 442 (2016), no. 2, 446–450. http://dx.doi.Org/10.1016/j.jmaa.2O16.04.060.Google Scholar
[12] Saleeby, E. G., Meromorphic Solutions of generalized inviscid Burgers’ equations and afamily of quadratic PDEs. J. Math. Anal. Appl. 425 (2015), 508–519. http://dx.doi.Org/10.1016/j.jmaa.2O14.12.046.Google Scholar
[13] Zhang, G. Y., Curves, domains and Picard's theorem. Bull. London Math. Soc. 34 (2002), 205–211. http://dx.doi.Org/10.1112/SOO24609301008712Google Scholar