• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.161-171
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• 2012

We consider the fourth-order differential equation with one-dimensional $p$-Laplacian (${\phi}_p(x^{\prime\prime}(t)))^{\prime\prime}=f(t,x(t),x^{\prime}(t),x^{\prime\prime}(t)$) a.e. $t{\in}[0,1]$, subject to the boundary conditions $x^{\prime\prime}}(0)=0$, $({\phi}_p(x^{\prime\prime}(t)))^{\prime}{\mid}_{t=0}=0$, $x(0)={\sum}_{i=1}^n{\mu}_ix({\xi}_i)$, $x(t)=x(1-t)$, $t{\in}[0,1]$, where ${\phi}_p(s)={\mid}s{\mid}^{p-2}s$, $p$ > 1, 0 < ${\xi}_1$ < ${\xi}_2$ < ${\cdots}$ < ${\xi}_n$ < $\frac{1}{2}$, ${\mu}_i{\in}\mathbb{R}$, $i=1$, 2, ${\cdots}$, $n$, ${\sum}_{i=1}^n{\mu}_i=1$ and $f:[0,1]{\times}\mathbb{R}^3{\rightarrow}\mathbb{R}$ is a $L^1$-Carath$\acute{e}$odory function with $f(t,u,v,w)=f(1-t,u,-v,w)$ for $(t,u,v,w){\in}[0,1]{\times}\mathbb{R}^3$. We obtain the existence of at least one nonconstant symmetric solution by applying an extension of Mawhin's continuation theorem due to Ge. Furthermore, an example is given to illustrate the results.

• Kyungpook Mathematical Journal
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• 제47권1호
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• pp.1-20
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• 2007

In this paper, oscillation and nonoscillation criteria are established for nonlinear neutral delay differential equations with several positive and negative coefficients $$[x(t)-R(t)x(t-r)]^{\prime}+\sum_{i=1}^{m}Pi(t)H_i(x(t-{\tau}_i))-\sum_{j=1}^{n}Q_j(t)H_j(x(t-{\sigma}_j))=0$$. Our criteria improve and extend many results known in the literature. In addition we prove that under appropriate hypotheses, if every solution of the associated linear equation with constant coefficients, $$y^{\prime}(t)+\sum_{i=1}^{m}(p_i-\sum_{k{\in}J_i}qk)y(t-{\tau}_i)=0$$, oscillates, then every solution of the nonlinear equation also oscillates.

• 대한수학회보
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• 제54권4호
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• pp.1141-1158
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• 2017

In this paper, we study the value distribution of the derivative of a Dirichlet L-function $L^{\prime}(s,{\chi})$ at the a-points ${\rho}_{a,{\chi}}={\beta}_{a,{\chi}}+i{\gamma}_{a,{\chi}}$ of $L^{\prime}(s,{\chi})$. We give an asymptotic formula for the sum $${\sum_{{\rho}_{a,{\chi}};0<{\gamma}_{a,{\chi}}{\leq}T}\;L^{\prime}({\rho}_{a,{\chi}},{\chi})X^{{\rho}_{a,{\chi}}}\;as\;T{\rightarrow}{\infty}$$, where X is a fixed positive number and ${\chi}$ is a primitive character mod q. This work continues the investigations of Fujii [4-6], $Garunk{\check{s}}tis$ & Steuding [8] and the authors [12].

• 대한수학회보
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• 제61권3호
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• pp.611-619
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• 2024

For a finite group G, let 𝜓(G) denote the sum of element orders of G. If ${\psi}^{{\prime}{\prime}}(G)\,=\,{\frac{\psi(G)}{{\mid}G{\mid}^2}}$, we show here that the image of 𝜓'' on the class of all Dihedral groups whose order is twice a composite number greater than 4 is dense in $[0,\,{\frac{1}{4}}]$. We also derive some properties of 𝜓'' on the class of all dihedral groups whose order is twice a prime number.

• 대한수학회지
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• 제59권3호
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• pp.439-448
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• 2022

For a natural number n, let R(n) denote the number of representations of n as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most $O(N^{{\frac{4}{9}+{\varepsilon}})$ positive integers not exceeding N.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.27-34
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• 2013

In this paper, we consider the Duffing type p-Laplacian equation with multiple deviating arguments of the form $$({\varphi}_p(x^{\prime}(t)))^{\prime}+Cx^{\prime}(t)+go(t,x(t))+\sum_{k=1}^ngk(t,x(t-{\tau}_k(t)))=e(t)$$. By using the coincidence degree theory, we establish new results on the existence and uniqueness of periodic solutions for the above equation. Moreover, an example is given to illustrate the effectiveness of our results.

• 대한수학회보
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• 제55권3호
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• pp.763-776
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• 2018

In this paper we consider the element-wise (Hadamard) product (or sum) of elliptic divisibility sequences and study the periodic structure of these sequences. We obtain that the element-wise product (or sum) of elliptic divisibility sequences are periodic modulo a prime p like linear recurrence sequences. Then we study periodicity properties of product sequences. We generalize our results to the case of modulo $p^l$ for some prime p > 3 and positive integer l. Finally we consider the p-adic behavior of product sequences and give a generalization of [9, Theorem 4].

• 대한수학회지
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• 제54권4호
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• pp.1175-1187
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• 2017

The spectral problem $$-y^{{\prime}{\prime}}+q(x)y={\lambda}y,\;0 < x < 1, \atop y(0)cos{\beta}=y^{\prime}(0)sin{\beta},\;0{\leq}{\beta}<{\pi};\;{\frac{y^{\prime}(1)}{y(1)}}=h({\lambda})$$ is considered, where ${\lambda}$ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and $$h({\lambda})=a{\lambda}+b-\sum\limits_{k=1}^{N}{\frac{b_k}{{\lambda}-c_k}},$$ with the real coefficients and $a{\geq}0$, $b_k$ > 0, $c_1$ < $c_2$ < ${\cdots}$ < $c_N$, $N{\geq}0$. The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권3호
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• pp.337-345
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• 2023

Different versions of the boundary Schwarz lemma for the 𝒩 (𝜌) class are discussed in this study. Also, for the function g(z) = z+b₂z²+b₃z³+... defined in the unit disc D such that g ∈ 𝒩 (𝜌), we estimate a modulus of the angular derivative of g(z) function at the boundary point 1 ∈ 𝜕D with g'(1) = 1 + 𝜎 (1 - 𝜌), where ${\rho}={\frac{1}{n}}{\sum\limits_{i=1}^{n}}g(c_i)={\frac{g^{\prime}(c_1)+g^{\prime}(c_2)+{\ldots}+g^{\prime}(c_n)}{n}}{\in}g^{\prime}(D)$ and 𝜌≠1, 𝜎 > 1 and c₁, c₂, ..., c_n ∈ 𝜕D. That is, we shall give an estimate below |g"(1)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z ≠ 0. Estimating is made by using the arithmetic average of n different derivatives g'(c₁), g'(c₂), ..., g'(c_n).

• 대한수학회논문집
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• 제32권4호
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• pp.959-970
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• 2017

In this paper, we investigate the transcendental meromorphic solutions with finite order of two different types of difference equations $${\sum\limits_{j=1}^{n}}a_jf(z+c_j)={\frac{P(z,f)}{Q(z,f)}}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ and $${\prod\limits_{j=1}^{n}}f(z+c_j)={\frac{P(z,f)}{Q(z,f)}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ that share three distinct values with another meromorphic function. Here $a_j$, $b_k$, $d_l$ are small functions of f and $a_j{\not{\equiv}}(j=1,2,{\ldots},n)$, $b_p{\not{\equiv}}0$, $d_q{\not{\equiv}}0$. $c_j{\neq}0$ are pairwise distinct constants. p, q, n are non-negative integers. P(z, f) and Q(z, f) are two mutually prime polynomials in f.