• Bulletin of the Korean Mathematical Society
• /
• v.59 no.1
• /
• pp.83-99
• /
• 2022

In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.

• Journal of the Korean Mathematical Society
• /
• v.53 no.4
• /
• pp.835-846
• /
• 2016

In this paper, we study entire solutions of some nonlinear difference equations and transcendental meromorphic solutons of some nonlinear differential equations. Our results generalize the results due to [11], [17].

• Journal of the Korean Operations Research and Management Science Society
• /
• v.32 no.1
• /
• pp.77-91
• /
• 2007

In this paper, we present a symbiotic evolutionary algorithm for multi-objective optimization. The goal in multi-objective evolutionary algorithms (MOEAs) is to find a set of well-distributed solutions close to the true Pareto optimal solutions. Most of the existing MOEAs operate one population that consists of individuals representing the entire solution to the problem. The proposed algorithm has a two-leveled structure. The structure is intended to improve the capability of searching diverse and food solutions. At the lower level there exist several populations, each of which represents a partial solution to the entire problem, and at the upper level there is one population whose individuals represent the entire solutions to the problem. The parallel search with partial solutions at the lower level and the Integrated search with entire solutions at the upper level are carried out simultaneously. The performance of the proposed algorithm is compared with those of the existing algorithms in terms of convergence and diversity. The optimization problems with continuous variables and discrete variables are used as test-bed problems. The experimental results confirm the effectiveness of the proposed algorithm.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1495-1506
• /
• 2021

The growth of solutions of second order complex differential equations f" + A(z)f' + B(z)f = 0 with transcendental entire coefficients is considered. Assuming that A(z) has a finite deficient value and that B(z) has either Fabry gaps or a multiply connected Fatou component, it follows that all solutions are of infinite order of growth.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.1281-1289
• /
• 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].

• Kyungpook Mathematical Journal
• /
• v.45 no.1
• /
• pp.105-114
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.1217-1227
• /
• 2018

In the paper, we study the growth and fixed point of solutions of high order linear differential equations with entire coefficients of [p, q]-order in the complex plane. We improve and extend some results due to T. B. Cao, J. F. Xu, Z. X. Chen, and J. Liu, J. Tu, L. Z. Shi.

• Communications of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.229-241
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.4
• /
• pp.983-1002
• /
• 2021

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]ⁿ + P²(z)f^m(z + 𝜂) = Q(z) and [f(z)f'(z)]ⁿ + P(z)[∆_𝜂f(z)]^m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P²(z) [f^(k)(z)]² + [αf(z + 𝜂) - 𝛽f(z)]² = e^r(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

• Kyungpook Mathematical Journal
• /
• v.49 no.4
• /
• pp.771-777
• /
• 2009

In this paper, we study the uniqueness problem on entire functions sharing fixed points and prove some theorems which are related to one famous problem of Hayman.