• 대한수학회보
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• 제39권4호
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• pp.527-534
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• 2002

The weighted shifts have been made a central role in the study of κ-hyponormality for hyponormal and subnormal operators in discrete notion. In this note we discuss the κ-hyponormality of weighted translation semigroups with a symbol function in continuous notion.

• 대한수학회보
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• 제51권5호
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• pp.1469-1484
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• 2014

In this paper we are concerned with a class of stochastic partial functional differential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined and employing the integrated semigroup and random dynamics theory, we present some appropriate conditions to guarantee the existence of a random attractor.

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.237-243
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• 2024

A similarity transformation of a solution of the Cauchy problem for the linear difference equation in Hilbert space has been studied. In this manuscript, we obtain necessary and sufficient conditions for linear equivalence of the discrete semigroup of operators, generated by the solution of the difference equation utilizing four Canonical semigroups.

• 대한수학회보
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• 제58권5호
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.

• 대한수학회보
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• 제45권4호
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• pp.717-728
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• 2008

Let H be a real Hilbert space and S = {T(s) : $0\;{\leq}\;s\;<\;{\infty}$} be a nonexpansive semigroup on H such that $F(S)\;{\neq}\;{\emptyset}$ For a contraction f with coefficient 0 < $\alpha$ < 1, a strongly positive bounded linear operator A with coefficient $\bar{\gamma}$ > 0. Let 0 < $\gamma$ < $\frac{\bar{\gamma}}{\alpha}$. It is proved that the sequences {$x_t$} and {$x_n$} generated by the iterative method $$x_t\;=\;t{\gamma}f(x_t)\;+\;(I\;-\;tA){\frac{1}{{\lambda}_t}}\;{\int_0}^{{\lambda}_t}\;T(s){x_t}ds,$$ and $$x_{n+1}\;=\;{\alpha}_n{\gamma}f(x_n)\;+\;(I\;-\;{\alpha}_nA)\frac{1}{t_n}\;{\int_0}^{t_n}\;T(s){x_n}ds,$$ where {t}, {${\alpha}_n$} $\subset$ (0, 1) and {${\lambda}_t$}, {$t_n$} are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\;{\in}\;F(S)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)\tilde{x},\;x\;-\;\tilde{x}{\rangle}\;{\leq}\;0$ for $x\;{\in}\;F(S)$.

• 대한수학회지
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• 제38권6호
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• pp.1191-1204
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• 2001

The paper is devoted to the study of fractional integration and differentiation on a finite interval [a, b] of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int_{a}^{x}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation ${\delta}+{\mu}({\delta}=xD,D=d/dx)$ with real $\mu$, being taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X^{p}_{c}$(a, b) of Lebesgue measurable functions f on $R_{+}=(0,{\infty})$ such that for c${\in}R=(-{\infty}{\infty})$, in particular in the space $L^{p}(0,{\infty})\;(1{\le}{\le}{\infty})$. The existence almost every where is established for the coorresponding Hadamard-type fractional derivative for a function g(x) such that $x^{p}$g(x) have $\delta$ derivatives up to order n-1 on [a, b] and ${\delta}^{n-1}[x^{\mu}$g(x)] is absolutely continuous on [a, b]. Semigroup and reciprocal properties for the above operators are proved.

• 대한수학회논문집
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• 제34권4호
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• pp.1365-1388
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• 2019

In this paper we consider a class of nonlinear nonlocal parabolic equations involving p-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on $L^p$-norm of the gradient and the nonlinear term is of polynomial type. We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method. Then we study the existence of global attractors in various spaces for the continuous semigroup generated by the problem. Finally, we investigate the existence and exponential stability of weak stationary solutions to the problem.