On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

R. Mandal; R. Biswas

doi:10.30970/ms.61.2.195-213

R. Mandal Raiganj University
R. Biswas Raiganj University

DOI: https://doi.org/10.30970/ms.61.2.195-213

Keywords: System, Fermat-type equation, Entire solution, Several complex variables, Partial differential-difference equation, Nevanlinna theory

Abstract

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Author Biographies

R. Mandal, Raiganj University

Assistant Professor,

Department of Mathematics, Raiganj University, Raiganj, West Bengal, India

R. Biswas, Raiganj University

Department of Mathematics, Raiganj University
Raiganj, West Bengal, India

References

A. Bandura, O. Skaskiv, Boundedness of the L-index in a direction of entire solutions of second order partial differential equation, Acta Comment. Univ. Tartu. Math., 22 (2018), №2, 223–234.

A. Bandura, O. Skaskiv, Linear directional differential equations in the unit ball: solutions of bounded L-index, Math. Slovaca, 69 (2019), №5, 1089–1098.

A. Bandura, O. Skaskiv, L. Smolovyk, Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction, Demonstr. Math., 52 (2019), №1, 482–489.

A. Bandura, O. Skaskiv, Analog of Hayman’s theorem and its application to some system of linear partial differential equations, J. Math. Phys. Anal. Geom., 15 (2019), №2, 170–191.

A. Bandura, O. Skaskiv, I. Tymkiv, Composition of entire and analytic functions in the unit ball, Carpathian Math. Publ., 14 (2022), 95–104.

C.A. Berenstein, D.C. Chang, B.Q. Li, On the shared values of entire functions and their partial differential polynomials in $mathbb{C}^n$, Forum Math., 8 (1996), 379–396.

R. Biswas, R. Mandal, Entire solutions for quadratic trinomial partial differential-difference functional equations in $mathbb{C}^n$, Novi Sad J. Math., https://doi.org/10.30755/NSJOM.15512.

J.F. Chen, S.Q. Lin, On the existence of solutions of Fermat-type differential-difference equations, Bull. Korean Math. Soc., 58 (2021), №4, 983–1002.

P.C. Hu, C.C. Yang, Malmquist type theorem and factorization of meromorphic solutions of partial differential equations, Complex Var., 27 (1995), 269–285.

P.C. Hu, C.C. Yang, Factorization of holomorphic mappings, Complex Var., 27 (1995), №3, 235–244.

P.C. Hu, C.C. Yang, Uniqueness of meromorphic functions on Cm, Complex Var., 30 (1996), 235–270.

R.J. Korhonen, A difference Picard theorem for meromorphic functions of several variables, Comput. Methods Funct. Theo., 12 (2012), №1, 343–361.

A.O. Kuryliak, O.B. Skaskiv, O.V. Zrum, Levy’s phenomenon for entire functions of several variables, Ufa Math. J., 6 (2014), №2, 111–120.

A.O. Kuryliak, I.Ye Ovchar, O.B. Skaskiv, Wiman’s inequality for the Laplace integrals, Int. J. Math. Anal.(N.S.), 8 (2014), №8, 381–385.

A.O. Kuryliak, S.I. Panchuk, O.B. Skaskiv, Bitlyan-Gol’dberg type inequality for entire functions and diagonal maximal term, Mat. Stud., 54 (2020), №2, 135–145.

A.O. Kuryliak, O.B. Skaskiv, S.R. Skaskiv, Levy’s phenomenon for analytic functions on a polydisc, Eur. J. Math., 6 (2020), №1, 138–152.

B.Q. Li, On entire solutions of Fermat-type partial differential equations, Int. J. Math., 15 (2004), 473–485.

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

H. Li, K. Zhang, H. Xu, Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables, AIMS Math., 6 (2021), №11, 11796–11814.

K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359 (2009), 384–393.

K. Liu, T. B. Cao, H. Z. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), №2, 147–155.

K. Liu, T.B. Cao, Entire solutions of Fermat type q-difference differential equations, Electron. J. Diff. Equ., 59 (2013), 1–10.

K. Liu, L. Yang, On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory, 13 (2013), 433–447.

K. Liu, L.Z. Yang, A note on meromorphic solutions of Fermat types equations, An. ¸Stiin¸t. Univ. Al. I. Cuza Iasi Mat. (N. S.), 1 (2016), 317–325.

R. Mandal, R. Biswas, On the transcendental entire functions satisfying some Fermat-type differentialdifference equations, Indian J. Math., 65 (2023), №2, 153–183.

L.I. Ronkin, Introduction to the theory of entire functions of several variables, Moscow: Nauka 1971 (Russian). American Mathematical Society, Providence, 1974.

E.G. Saleeby, On complex analytic solutions of certain trinomial functional and partial differential equations, Aequat. Math., 85 (2013), 553–562.

I.N. Sneddon, Elements of partial differential equations, Courier Corporation, 2006.

W. Stoll, Value distribution theory for meromorphic maps, Aspects of Mathematics, V.E7, Friedr. Vieweg & Sohn, Braunschweig, Wiesbaden, 1985.

H.Y. Xu, S.Y. Liu, Q.P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483 (2020), №2, https://doi.org/10.1016/j.jmaa.2019.123641.

C.C. Yang, P. Li, On the transcendental solutions of a certain type of non-linear differential equations, Arch. Math., 82 (2004), 442–448.

Z. Ye, A sharp form of Nevanlinna’s second main theorem for several complex variables, Math. Z., 222(1996), 81–95.