• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.361-376
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• 2000

Our purpose is to seek that the reachable set of the semilinear system $\frac{d}{dt}x(t){\;}={\;}Ax(t){\;}+{\;}f(t,x(t)){\;}+{\;}Bu(t)$ is equivalent to that of its corresponding to linear system (the case where f=0).Under the assumption that the system of generalized eigenspaces of A is complete, we will show that the reachable set corresponding to the linear system is independent of t in case A generates $C_0-semigroup$. An illustrative example for retarded system with time delay is given in the last section.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.559-575
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• 2008

A commutative hypercomplex system $L_1$(Q,m) is, roughly speaking, a space which is defined by a structure measure (c(A,B, r), (A,$B{\in}{\beta}$(Q)). Such space has bee studied by Berezanskii and Krein. Our main purpose is to establish a generalization of convolution semigroups and to discuss the role of the L$\'{e}$vy measure in the L$\'{e}$vy-Khinchin representation in terms of continuous negative definite functions on the dual hypercomplex system.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.2
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• pp.229-242
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• 2020

Functional fractional differential inclusions with impulse effects in general Banach spaces are studied. We discuss the situation when the semigroup generated by the linear part is equicontinuous and the multifunction is Caratheodory. First, we define the PC-mild solutions for functional fractional semilinear impulsive differential inclusions. We then prove the existence of PC-mild solutions for such inclusions by using the fixed point theorem, multivalued properties and applications of NCHM (noncompactness Hausdorff measure). Eventually, we enhance the acquired results by giving an example.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1217-1225
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• 2010

In this paper we define fuzzy interior $\Gamma$-ideals in ordered $\Gamma$-semigroups. We prove that in regular(resp. intra-regular) ordered $\Gamma$-semigroups the concepts of fuzzy interior $\Gamma$-ideals and fuzzy $\Gamma$-ideals coincide. We prove that an ordered $\Gamma$-semigroup is fuzzy simple if and only if every fuzzy interior $\Gamma$-ideal is a constant function. We characterize intra-regular ordered $\Gamma$-semigroups in terms of interior (resp. fuzzy interior) $\Gamma$-ideals.

• The Pure and Applied Mathematics
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• v.18 no.1
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• pp.31-39
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• 2011

Stone's theorem states that "A bounded linear operator A is infinitesimal generator of a $C_0$-group of unitary operators on a Hilbert space H if and only if iA is self adjoint". In this paper we establish a generalization of Stone's theorem in the framework of Hilbert $C^*$-modules.

• 전기의세계
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• v.31 no.3
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• pp.209-217
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• 1982

This paper considers a finite spectrum assignment Problem for a functional retarded linear differential system with delays in control only. In this problem, by generalizing from an abstract linear system characterized by Semigroups on a Hilbert space to a finite dimensional linear system, we unify the relationship between a control-delayed system and its non-delayed system, and then by using the spectrum of the generator-decomposition of Semigroup, we try to get a feedback law which yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned sets of n points in the complex plane. The comparative examinations between the standard spectrum assignment method and the method of spectral projection for the feedback law which consists of proportional and finite interval terms over present and past values of control variables are also considered. The analysis is carry down to the elementary spectral projection level because, in spite of all the research efforts, so far there has been no significant attempt to obtain the feedback implementation directly from the abstract representation forms in the case of multivariables.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1019-1036
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• 2016

In this paper, we study the existence of solutions and $L^2$-regularity for fractional order retarded neutral functional differential equations in Hilbert spaces. We no longer require the compactness of structural operators to prove the existence of continuous solutions of the non-linear differential system, but instead we investigate the relation between the regularity of solutions of fractional order retarded neutral functional differential systems with unbounded principal operators and that of its corresponding linear system excluded by the nonlinear term. Finally, we give a simple example to which our main result can be applied.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.183-195
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• 2019

Let R be a commutative ring with unity. In this paper, we characterize the unit elements, the regular elements, the ${\pi}$-regular elements and the clean elements of zero-symmetric near-ring of polynomials $R_0[x]$, when $nil(R)^2=0$. Moreover, it is shown that the set of ${\pi}$-regular elements of $R_0[x]$ forms a semigroup. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its "multiplication" operation.