• Bulletin of the Korean Mathematical Society
• /
• v.50 no.2
• /
• pp.675-685
• /
• 2013

We prove, under some general assumptions, that a generator of any uniformly bounded Nemytskij operator, mapping a subset of space of functions of bounded variation in the sense of Wiener-Young into another space of this type, must be an affine function with respect to the second variable.

• Journal of applied mathematics & informatics
• /
• v.30 no.1_2
• /
• pp.83-92
• /
• 2012

In this paper, a generalized Shannon-McMillan theorem for the nonhomogeneous Markov chains indexed by an infinite tree which has a uniformly bounded degree is discussed by constructing a nonnegative martingale and analytical methods. As corollaries, some Shannon-Mcmillan theorems for the nonhomogeneous Markov chains indexed by a homogeneous tree and the nonhomogeneous Markov chain are obtained. Some results which have been obtained are extended.

• Journal of the Korean Mathematical Society
• /
• v.37 no.5
• /
• pp.707-720
• /
• 2000

Conley index is used study bifurcation from equilibria of full bounded solutions to parameter dependent families of ordinary differential equations of the form {{{{ {dx} over {dt} }}}} =$\varepsilon$F(x, t, $\mu$). It is assumed that F(x, t,$\mu$) is uniformly almost periodic in t.

• Journal of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.131-143
• /
• 2004

Let Ε and F be locally convex spaces over C. We assume that Ε is a nuclear space and F is a Banach space. Let f be a holomorphic mapping from Ε into F. Then we show that f is of uniformly bounded type if and only if, for an arbitrary locally convex space G containing Ε as a closed subspace, f can be extended to a holomorphic mapping from G into F.

• Honam Mathematical Journal
• /
• v.32 no.2
• /
• pp.227-237
• /
• 2010

For a discrete group G, the kernel of a homomorphism from bounded cohomology $\hat{H}^*(G)$ of G to the ordinary cohomology $H^*(G)$ of G is called the singular part of $\hat{H}^*(G)$. We give some results on the space of the singular part of the second bounded cohomology of G. Also some results on the second bounded cohomology of a uniformly perfect group are given.

• East Asian mathematical journal
• /
• v.24 no.3
• /
• pp.305-315
• /
• 2008

The purpose of this paper is to study the existence and uniqueness of the fixed point of uniformly pseudo-contractive operator and the solution of equation with uniformly accretive operator, and to approximate the fixed point and the solution by the Mann iterative sequence in an arbitrary Banach space or an uniformly smooth Banach space respectively. The results presented in this paper show that if X is a real Banach space and A : X $\rightarrow$ X is an uniformly accretive operator and (I-A)X is bounded then A is a mapping onto X when A is continuous or $X^*$ is uniformly convex and A is demicontinuous. Consequently, the corresponding results which depend on the assumptions that the fixed point of operator and solution of the equation are in existence are improved.

• Journal of applied mathematics & informatics
• /
• v.3 no.1
• /
• pp.47-54
• /
• 1996

We prove that if RUC(S) has a left invariant mean ${\rho}={T_{S} : s \;{\in}\; S}$ is a continuous repesentation of S as nonexpansive map-pings on a closed convex subset C of a p-uniformly convex and p-uniformly smooth Banach space and C contains an element of bounded orbit then C contains a common fixed point for ${\rho}$.

• Journal of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.581-598
• /
• 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.

• Honam Mathematical Journal
• /
• v.8 no.1
• /
• pp.1-19
• /
• 1986

We show that a (possibly unbounded) linear operator, T, is scalar on the real line (spectral operator of scalar type, with real spectrum) if and only if (iT) generates a uniformly bounded semigroup and $(1-iT)(1+iT)^{-1}$ is scalar on the unit circle. T is scalar on [0, $\infty$) if and only if T generates a uniformly bounded semigroup and $(1+T)^{-1}$ is scalar on [0,1). By analogy with these results, we define $C^0$-scalar, on the real line, or [0. $\infty$), for an unbounded operator. We show that a generator of a positive-definite group is $C^0$-scalar on the real line. and a generator of a completely monotone semigroup is $C^0$-scalar on [0, $\infty$). We give sufficient conditions for a closed operator, T, to generate a positive-definite group: the sequence < $\phi(T^{n}x)$ > $_{n=0}^{\infty}$ must equal the moments of a positive measure on the real line, for sufficiently many positive $\phi$ in $X^{*}$, x in X. If the measures are supported on [0, $\infty$), then T generates a completely monotone semigroup. On a reflexive Banach lattice, these conditions are also necessary, and are equivalent to T being scalar, with positive projection-valued measure. T generates a completely monotone semigroup if and only if T is positive and m-dispersive and generates a bounded holomorphic semigroup.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.2
• /
• pp.649-659
• /
• 2015

The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.