• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.559-577
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• 2020

We present uniform and L_p left Caputo-Bochner abstract iterated fractional Landau inequalities over ℝ₊. These estimate the size of second and third iterated left abstract fractional derivates of a Banach space valued function over ℝ₊. We give an application when the basic fractional order is ${\frac{1}{2}}$.

• Communications of the Korean Mathematical Society
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• v.18 no.2
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• pp.375-383
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• 2003

Under some conditions on an array of rowwise independent random variables, Hu et at. (1998) obtained a complete convergence result for law of large numbers with rate {a$\_$n/, n $\geq$ 1} which is bounded away from zero. We investigate the general situation for rate {a$\_$n/, n $\geq$ 1) under similar conditions.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.899-924
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• 2010

In the paper we introduce averages of each type and use these averages to construct examples of weakly compact operators on the space C ($\Omega$) for which the necessary and sufficient conditions that they be compact, absolutely summing or nuclear are distinct. A great number of concrete examples, in various situations, are given.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.895-908
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• 2016

Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $C_0$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $P_{\tau}(X)$ associated with the ${\tau}$-periodic evolutionary process on a Banach space X, where $P_{\tau}(X)$ stands for the space of all ${\tau}$-periodic continuous functions mapping ${\mathbb{R}}$ to X.

• Honam Mathematical Journal
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• v.7 no.1
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• pp.7-13
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• 1985

We show that a bounded linear qperator, T, on a Banach space, X, is $C^{n}$-scalar if the sepuence {$\frac{k!}{(k+n)!}{\phi}(T^{k+n}x)$}$_{k=0}^{\infty}$ is positive-definite, for sufficiently many $\phi$ in $X^{\ast}$, x in X. We use this to show that $(T_{n}f)(t){\equiv}tf(t)+nJf(t)$, where $If(t)=\int_{0}^{1}f(s)ds$, is $C^{n}$-scalar on $L^{p}([0,1],v)$, for $1{\leq}p{\leq}\infty$, for a large class of measures, v. Other corollaries include the spectral theorem for bounded symmetric operators on a Hilbert space.

• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1853-1878
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• 2017

Let $\{U(t,s)\}_{t{\geq}s}$ be a periodic evolutionary process with period ${\tau}$ > 0 on a Banach space X. Also, let L be the generator of the evolution semigroup associated with $\{U(t,s)\}_{t{\geq}s}$ on the phase space $P_{\tau}(X)$ of all ${\tau}$-periodic continuous X-valued functions. Some kind of variation-of-constants formula for the solution u of the equation $({\alpha}I-L)^nu=f$ will be given together with the conditions on $f{\in}P_{\tau}(X)$ for the existence of coefficients in the formula involving the monodromy operator $U(0,-{\tau})$. Also, examples of ODEs and PDEs are presented as its application.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.287-292
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• 2004

The operator $A \; {\in} \; L(H_{i})$, the Banach algebra of bounded linear operators on the complex infinite dimensional Hilbert space $\cal H_{i}$, is said to be p-hyponormal if $(A^\ast A)^P \geq (AA^\ast)^p$ for $p\; \in \; (0,1]$. Let (equation omitted) denote the completion of (equation omitted) with respect to some crossnorm. Let $I_{i}$ be the identity operator on $H_{i}$. Letting (equation omitted), where each $A_{i}$ is p-hyponormal, it is proved that the commuting n-tuple T = ($T_1$,..., $T_{n}$) satisfies Bishop's condition ($\beta$) and that if T is Weyl then there exists a non-singular commuting n-tuple S such that T = S + F for some n-tuple F of compact operators.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.147-160
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• 2018

In this article, we establish translation theorems for the analytic Fourier-Feynman transform of functionals in non-stationary Gaussian processes on Wiener space. We then proceed to show that these general translation theorems can be applied to two well-known classes of functionals; namely, the Banach algebra S introduced by Cameron and Storvick, and the space ${\mathcal{B}}^{(P)}_{\mathcal{A}}$ consisting of functionals of the form $F(x)=f({\langle}{\alpha}_1,x{\rangle},{\ldots},{\langle}{\alpha}_n,x{\rangle})$, where ${\langle}{\alpha},x{\rangle}$ denotes the Paley-Wiener-Zygmund stochastic integral ${\int_{0}^{T}}{\alpha}(t)dx(t)$.

• Communications of the Korean Mathematical Society
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• v.22 no.1
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• pp.75-90
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• 2007

Huffman, Park and Skoug introduced various results for the $L_p$ analytic Fourier-Feynman transform and the convolution for functionals on classical Wiener space which belong to some Banach algebra S introduced by Cameron and Strovic. Also Chang, Kim and Yoo extended the above results to an abstract Wiener space for functionals in the Fresnel class F(B) which corresponds to S. Recently Kim, Song and Yoo investigated more generalized relationships between the Fourier-Feynman transform and the convolution product for functionals in a generalized Fresnel class $F_{A_1,A'_2}$ containing F(B). In this paper, we establish various interesting relationships and expressions involving the first variation and one or two of the concepts of the Fourier-Feynman transform and the convolution product for functionals in $F_{A_1,A_2}$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.3
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• pp.203-215
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• 2009

In this article, we study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions u'(t) = Au(t) + f(t, u(t); Tu(t); Su(t)), $0{\leq}t{\leq}T_0$, $t{\neq}t_i$, u(0) + g(u) = $u_0$, ${\Delta}u(t_i)=I_i(u(t_i))$, i = 1,2,${\ldots}$p, 0<$t_1$<$t_2$<$\cdots$<$t_p$<$T_0$, in a Banach space X, where A is the infinitesimal generator of a $C_0$ semigroup, g constitutes a nonlocal conditions, and ${\Delta}u(t_i)=u(t_i^+)-u(t_i^-)$ represents an impulsive conditions.