• Title, Summary, Keyword: transcendental entire solutions

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EXISTENCE OF TRANSCENDENTAL MEROMORPHIC SOLUTIONS ON SOME TYPES OF NONLINEAR DIFFERENTIAL EQUATIONS

  • Hu, Peichu;Liu, Manli
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.991-1002
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    • 2020
  • We show that when n > m, the following delay differential equation fn(z)f'(z) + p(z)(f(z + c) - f(z))m = r(z)eq(z) of rational coefficients p, r doesn't admit any transcendental entire solutions f(z) of finite order. Furthermore, we study the conditions of α1, α2 that ensure existence of transcendental meromorphic solutions of the equation fn(z) + fn-2(z)f'(z) + Pd(z, f) = p1(z)eα1(z) + p2(z)eα2(z). These results have improved some known theorems obtained most recently by other authors.

ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS

  • Lu, Weiran;Li, Qiuying;Yang, Chungchun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1281-1289
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    • 2014
  • In this paper, we consider the differential equation $$F^{\prime}-Q_1=Re^{\alpha}(F-Q_2)$$, where $Q_1$ and $Q_2$ are polynomials with $Q_1Q_2{\neq}0$, R is a rational function and ${\alpha}$ is an entire function. We consider solutions of the form $F=f^n$, where f is an entire function and $n{\geq}2$ is an integer, and we prove that if f is a transcendental entire function, then $\frac{Q_1}{Q_2}$ is a polynomial and $f^{\prime}=\frac{Q_1}{nQ_2}f$. This theorem improves some known results and answers an open question raised in [16].

A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG

  • Majumder, Sujoy
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.411-421
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    • 2016
  • In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.

ON ZEROS AND GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS

  • Kumar, Sanjay;Saini, Manisha
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.229-241
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    • 2020
  • For a second order linear differential equation f" + A(z)f' + B(z)f = 0, with A(z) and B(z) being transcendental entire functions under some restrictions, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions, with a condition, have exponent of convergence of zeros equal to infinity. Also, we have extended these results to higher order linear differential equations.

ON GROWTH PROPERTIES OF TRANSCENDENTAL MEROMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH ENTIRE COEFFICIENTS OF HIGHER ORDER

  • Biswas, Nityagopal;Datta, Sanjib Kumar;Tamang, Samten
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1245-1259
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    • 2019
  • In the paper, we study the growth properties of meromorphic solutions of higher order linear differential equations with entire coefficients of [p, q] - ${\varphi}$ order, ${\varphi}$ being a non-decreasing unbounded function and establish some new results which are improvement and extension of some previous results due to Hamani-Belaidi, He-Zheng-Hu and others.

ENTIRE SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATION AND FERMAT TYPE q-DIFFERENCE DIFFERENTIAL EQUATIONS

  • CHEN, MIN FENG;GAO, ZONG SHENG
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.447-456
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    • 2015
  • In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

Non-homogeneous Linear Differential Equations with Solutions of Finite Order

  • Belaidi, Benharrat
    • Kyungpook Mathematical Journal
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    • v.45 no.1
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    • pp.105-114
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    • 2005
  • In this paper we investigate the growth of finite order solutions of the differential equation $f^{(k)}\;+\;A_{k-1}(Z)f^{(k-l)}\;+\;{\cdots}\;+\;A_1(z)f^{\prime}\;+\;A_0(z)f\;=\;F(z)$, where $A_0(z),\;{\cdots}\;,\;A_{k-1}(Z)\;and\;F(z)\;{\neq}\;0$ are entire functions. We find conditions on the coefficients which will guarantees the existence of an asymptotic value for a transcendental entire solution of finite order and its derivatives. We also estimate the lower bounds of order of solutions if one of the coefficient is dominant in the sense that has larger order than any other coefficients.

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RADIAL OSCILLATION OF LINEAR DIFFERENTIAL EQUATION

  • Wu, Zhaojun
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.911-921
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    • 2012
  • In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

Growth order of Meromorphic Solutions of Higher-order Linear Differential Equations

  • Xu, Junfeng;Zhang, Zhanliang
    • Kyungpook Mathematical Journal
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    • v.48 no.1
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    • pp.123-132
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    • 2008
  • In this paper, we investigate higher-order linear differential equations with entire coefficients of iterated order. We improve and extend the result of L. Z. Yang by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen and the extended Wiman-Valiron theory by Wang and Yi. We also consider the nonhomogeneous linear differential equations.