• 대한수학회지
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• 제34권3호
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• pp.581-598
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• 1997

This paper is concerned with the impulsive control problem $$ \dot{x}(t) = f(t, x) + g(t, x)\dot{u}(t), t \in [0, T], x(0) = \overline{x}, $$ where u is a possibly discontinuous control function of bounded variation, $f : R \times R^n \mapsto R^n$ is a bounded and Lipschitz continuous function, and $g : R \times R^n \mapsto R^n$ is continuously differentiable w.r.t. the variable x and satisfies $\mid$g(t,\cdot) - g(s,\cdot)$\mid$ \leq \phi(t) - \phi(s)$, for some increasing function $\phi$ and every s < t. We show that the map $u \mapsto x_u$ is Lipschitz continuous when u ranges in the set of step functions whose total variations are uniformly bounded, where $x_u$ is the solution of the impulsive control system corresponding to u. We also define the generalized solution of the impulsive control system corresponding to a measurable control functin of bounded variation.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.901-910
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• 2008

For some generalized N-function ${\Phi}$, some Holder type inequalities and bounded operators on spaces of type $W_M^{\Omega,\Phi}$ generalizing the $W^p$-spaces due to Pathak and Upadhyay are obtained.

• Journal of applied mathematics & informatics
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• 제14권1_2호
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• pp.365-376
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• 2004

The purpose of this study is to show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met. In particular, as a basis function, we are interested in using a conditionally positive definite function $\Phi$ whose generalized Fourier transform is of the form $\Phi(\theta)\;=\;F(\theta)$\mid$\theta$\mid$^{-2m}$ with a bounded function F > 0.

• 대한수학회보
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• 제37권2호
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• pp.229-247
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• 2000

This paper is concerned with the impulsive Cauchy problem where the control function u is a possibly discontinuous vector-valued function with finite total variation. We assume that the vector fields f, $g_i$(i=1,…, m) are dependent on the time variable. The impulsive Cauchy problem is of the form x(t)=f(t,x) +$\SUMg_i(t,x)u_i(t)$, $t\in$[0,T], x(0)=$\in\; R^n$, where the vector fields f, $g_i$ : $\mathbb{R}\; \times\; \mathbb{R}\; \longrightarrow\; \mathbb(R)^n$ are measurable in t and Lipschitz continuous in x, If $g_i's$ satisfy a condition that $\SUM{\mid}g_i(t_2,x){\mid}{\leq}{\phi}$ $\forallt_1\; <\; t-2,x\; {\epsilon}\;\mathbb{R}^n$ for some increasing function $\phi$, then the imput-output function can be continuously extended to measurable functions of bounded variation.

• 충청수학회지
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• 제2권1호
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• pp.55-64
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• 1989

In this paper, we study some properties of generalized function spaces of ${\kappa}$-, ${\phi}$-and ${\kappa}{\phi}$-bounded variations and general bounded variations.

• 대한수학회지
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• 제59권6호
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• pp.1139-1151
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• 2022

This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝ^N and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

• 대한수학회보
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• 제50권6호
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• pp.1981-1988
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• 2013

We say a positive integer n satisfies the Lehmer property if ${\phi}(n)$ divides n - 1, where ${\phi}(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime p and a positive integer n, or of the form ${\alpha}2^{\beta}+1$ for ${\alpha}{\leq}{\beta}$ does not satisfy the Lehmer property.

• 대한수학회논문집
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• 제30권3호
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• pp.239-252
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• 2015

Motivated mainly by certain interesting extensions of the ${\tau}$-hypergeometric function defined by Virchenko et al. [11] and some ${\tau}$-Appell's function introduced by Al-Shammery and Kalla [1], we introduce here the ${\tau}$-Lauricella functions $F_A^{(n),{\tau}_1,{\cdots},{\tau}_n}$, $F_B^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and $F_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and the confluent forms ${\Phi}_2^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and ${\Phi}_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ of n variables. We then systematically investigate their various integral representations of each of these ${\tau}$-Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.

• 대한수학회보
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• 제29권2호
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• pp.277-283
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• 1992

Let GF(q) be a finite field with q elements where q=p$^{n}$ for a prime number p and a positive integer n. Consider an arbitrary function .phi. from GF(q) into GF(q). By using the Largrange's Interpolation formula for the given function .phi., .phi. can be represented by a polynomial which is congruent (mod x$^{q}$ -x) to a unique polynomial over GF(q) with the degree < q. In [3], Wells characterized all polynomial over a finite field which commute with translations. Mullen [2] generalized the characterization to linear polynomials over the finite fields, i.e., he characterized all polynomials f(x) over GF(q) for which deg(f) < q and f(bx+a)=b.f(x) + a for fixed elements a and b of GF(q) with a.neq.0. From those papers, a natural question (though difficult to answer to ask is: what are the explicit form of f(x) with zero terms\ulcorner In this paper we obtain the exact form (together with zero terms) of a polynomial f(x) over GF(q) for which satisfies deg(f) < p$^{2}$ and (1) f(x+a)=f(x)+c for the fixed nonzero elements a and c in GF(q).

• Korean Journal of Mathematics
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• 제17권2호
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• pp.127-145
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• 2009

In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.