• 대한수학회지
• /
• 제55권3호
• /
• pp.531-551
• /
• 2018

In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.

• 대한수학회지
• /
• 제38권1호
• /
• pp.177-191
• /
• 2001

We consider a linear differential game described by the delay-differential equation in a Hilbert space H; (※Equations, See Full-text) U and V are Hilbert spaces, and B(t) and C(t) are families of bounded operators on U and V to H, respectively. A(sub)0 generates an analytic semigroup T(t) = e(sup)tA(sub)0 in H. The control variables g, and u and v are supposed to be restricted in the norm bounded sets (※Equations, See Full-text). For given x(sup)0 ∈ H and a given time t > 0, we study $\xi$-approximate controllability to determine x($.$) for a given g and v($.$) such that the corresponding solution x(t) satisfies ∥x(t) - x(sup)0∥ $\leq$ $\xi$($\xi$ > 0 : a given error).

• 대한수학회보
• /
• 제49권5호
• /
• pp.1015-1025
• /
• 2012

Let $[n]=\{1,2,{\cdots},n\}$, and let $PO_n$ be the partial order-preserving transformation semigroup on [n]. Let $$CPO_n=\{{\alpha}{\in}PO_n:({\forall}x,y{\in}dom{\alpha}),\;|x{\alpha}-y{\alpha}|{\leq}|x-y|\}$$ Then $CPO_n$ is a subsemigroup of $PO_n$. In this paper, we characterize Green's relations and the regularity of elements for $CPO_n$.

• 대한수학회보
• /
• 제50권3호
• /
• pp.801-809
• /
• 2013

In this paper, we construct nilpotent semigroups S such that $S^n=\{0\}$, $S^{n-1}{\neq}\{0\}$ and ${\Gamma}(S)$ is a refinement of the star graph $K_{1,n-3}$ with center $c$ together with finitely many or infinitely many end vertices adjacent to $c$, for each finite positive integer $n{\geq}5$. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpotent semigroups when $n=5$, 6 and in finite cases.

• 대한수학회보
• /
• 제55권2호
• /
• pp.379-403
• /
• 2018

In this paper we study the first initial boundary value problem for the 3D Navier-Stokes-Voigt equations with infinite delay. First, we prove the existence and uniqueness of weak solutions to the problem by combining the Galerkin method and the energy method. Then we prove the existence of a compact global attractor for the continuous semigroup associated to the problem. Finally, we study the existence and exponential stability of stationary solutions.

• Journal of applied mathematics & informatics
• /
• 제32권3_4호
• /
• pp.405-426
• /
• 2014

The concept of interval-valued (${\alpha},{\beta}$)-fuzzy ideals, interval-valued (${\alpha},{\beta}$)-fuzzy generalized bi-ideals are introduced in LA-semigroups, using the ideas of belonging and quasi-coincidence of an interval-valued fuzzy point with an interval-valued fuzzy set and some related properties are investigated. We define the lower and upper parts of interval-valued fuzzy subsets of an LA-semigroup. Regular LA-semigroups are characterized by the properties of the lower part of interval-valued (${\in},{\in}{\vee}q$)-fuzzy left ideals, interval-valued (${\in},{\in}{\vee}q$)-fuzzy quasi-ideals and interval-valued (${\in},{\in}{\vee}q$)-fuzzy generalized bi-ideals. Main Facts.

• Journal of applied mathematics & informatics
• /
• 제29권3_4호
• /
• pp.669-680
• /
• 2011

In this paper, some strong convergence theorems are obtained for hybrid method for modified Ishikawa iteration process of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in Hilbert spaces. The results presented in this article generalize and improve results of Tae-Hwa Kim and Hong-Kun Xu and others. The convergence rate of the iteration process presented in this article is faster than hybrid method of Tae-Hwa Kim and Hong-Kun Xu and others.

• 대한수학회보
• /
• 제44권4호
• /
• pp.731-742
• /
• 2007

Let X be a Banach space, $A:D(A){\subset}X{\rightarrow}X$ the generator of a compact $C_0-semigroup\;S(t):X{\rightarrow}X,\;t{\geq}0$, D a locally closed subset in X, and $f:(a,b){\times}C([-q,0];X){\rightarrow}X$ a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order that D be a viable domain of the semi linear differential equation of retarded type $$u#(t)=Au(t)+f(t,u_t),\;t{\in}[t_0,\;t_0+T],{u_t}_0={\phi}{\in}C([-q,0];X)$$ is the tangency condition $$\limits_{h{\downarrow}0}^{lim\;inf\;h^{-1}d(S(h)v(0)+hf(t,v);D)=0}$$ for almost every $t{\in}(a,b)$ and every $v{\in}C([-q,0];X)\;with\;v(0){\in}D$.

• 대한수학회보
• /
• 제57권3호
• /
• pp.623-647
• /
• 2020

The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers.

• 대한수학회지
• /
• 제36권4호
• /
• pp.747-756
• /
• 1999

In ring theory it is well-known that a ring R is (von Neumann) regular if and only if all right R-modules are flat. But the analogous statement for this result does not hold for a monoid S. Hence, in sense of S-acts, Liu (]10]) showed that, as a weak analogue of this result, a monoid S is regular if and only if all left S-acts satisfying condition (E) ([6]) are flat. Moreover, Bulmann-Fleming ([6]) showed that x is a regular element of a monoid S iff the cyclic right S-act S/p(x, x2) is flat. In this paper, we show that the analogue of this result can be held for automata and them characterize regular semigroups by flat automata.