• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1231-1252
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• 2006

In this paper we shall explicitly calculate the formula of the algebraic presentation of an embedding of the Teichmiiller space ${\Im}(M)$ into the Goldman space g(M). From this algebraic presentation, we shall show that the Goldman's length parameter on g(M) is an isometric extension of the Fenchel-Nielsen's length parameter on ${\Im}(M)$.

• East Asian mathematical journal
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• v.26 no.1
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• pp.81-93
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• 2010

In this paper, a strong convergence theorem of multi-step iterative schemes with errors for asymptotically quasi-nonexpansive type nonself mappings is established in a real uniformly convex Banach space. Our results extend the corresponding results of Wangkeeree [12], Xu and Noor [13], Kim et al.[1,6,7] and many others.

• East Asian mathematical journal
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• v.29 no.1
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• pp.23-37
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• 2013

The purpose of this paper is to study an implicit iteration process with errors and establish weak and strong convergence theorems to converge to common fixed points for a finite family of generalized asymptotically quasi-nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve and generalize some known results from the existing literature.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.215-231
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• 2008

Let E be a uniformly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm, C a nonempty closed convex subset of E, and $T:C{\rightarrow}{\mathcal{K}}(E)$ a multivalued nonself-mapping such that $P_T$ is nonexpansive, where $P_T(x)=\{u_x{\in}Tx:{\parallel}x-u_x{\parallel}=d(x,Tx)\}$. For $f:C{\rightarrow}C$ a contraction and $t{\in}(0,1)$, let $x_t$ be a fixed point of a contraction $S_t:C{\rightarrow}{\mathcal{K}}(E)$, defined by $S_tx:=tP_T(x)+(1-t)f(x)$, $x{\in}C$. It is proved that if C is a nonexpansive retract of E and $\{x_t\}$ is bounded, then the strong ${\lim}_{t{\rightarrow}1}x_t$ exists and belongs to the fixed point set of T. Moreover, we study the strong convergence of $\{x_t\}$ with the weak inwardness condition on T in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Our results provide a partial answer to Jung's question.

• The Pure and Applied Mathematics
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• v.22 no.2
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• pp.185-197
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• 2015

The author considers a Noor-type iterative scheme to approximate com- mon fixed points of an infinite family of uniformly quasi-sup(f_n)-Lipschitzian map- pings and an infinite family of g_n-expansive mappings in convex cone metric spaces. His results generalize, improve and unify some corresponding results in convex met- ric spaces [1, 3, 9, 16, 18, 19] and convex cone metric spaces [8].

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1203-1220
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• 2008

Let V be an arbitrary system of weights on an open connected subset G of ${\mathbb{C}}^N(N{\geq}1)$ and let B (E) be the Banach algebra of all bounded linear operators on a Banach space E. Let $HV_b$ (G, E) and $HV_0$ (G, E) be the weighted locally convex spaces of vector-valued analytic functions. In this paper, we characterize self-analytic mappings ${\phi}:G{\rightarrow}G$ and operator-valued analytic mappings ${\Psi}:G{\rightarrow}B(E)$ which generate weighted composition operators and invertible weighted composition operators on the spaces $HV_b$ (G, E) and $HV_0$ (G, E) for different systems of weights V on G. Also, we obtained compact weighted composition operators on these spaces for some nice classes of weights.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.269-331
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• 2015

For a fixed positive integer g, we let $\mathcal{P}_g=\{Y{\in}\mathbb{R}^{(g,g)}{\mid}Y=^tY>0\}$ be the open convex cone in the Euclidean space $\mathbb{R}^{g(g+1)/2}$. Then the general linear group GL(g, $\mathbb{R}$) acts naturally on $\mathcal{P}_g$ by $A{\star}Y=AY^tA(A{\in}GL(g,\mathbb{R}),\;Y{\in}\mathcal{P}_g)$. We introduce a notion of polarized real tori. We show that the open cone $\mathcal{P}_g$ parametrizes principally polarized real tori of dimension g and that the Minkowski modular space 𝔗g = $GL(g,\mathbb{Z}){\backslash}\mathcal{P}_g$ may be regarded as a moduli space of principally polarized real tori of dimension g. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.393-406
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• 2007

In the framework of generalized convex spaces, we show that generalized KKM maps can be regarded as ordinary KKM maps, and obtain some applications to equilibrium result, maximal element theorems, and almost fixed point theorems on multimaps of the Zima type.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.71-76
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• 1988

For each continuous convex function ${\phi}$ defined on an open convex subset $A_{\phi}$ of a Banach space X, if we define a positively homogeneous sublinear functional ${\sigma}_x$ on X by ${\sigma}_x(y)=\sup{\lbrace}f(y)\;:\;f{\in}{\partial}{\phi}(x){\rbrace}$, where ${\partial}{\phi}(x)$ is a subdifferential of ${\phi}$ at x, then we get the following characterization theorem of Gateaux differentiability (weak Asplund) sapce. THEOREM. For every ${\phi}$ above, $D_{\phi}={\lbrace}x{\in}A\;:\;\sup_{||u||=1}\;{\sigma}_x(u)+{\sigma}_x(-u)=0{\rbrace}$ contains dense (dense $G_{\delta}$) subset of $A_{\phi}$ if and only if X is a Gateaux differentiability (weak Asplund) space.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.905-913
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• 2022

We characterize metrizability and submetrizability for point-open, open-point and bi-point-open topologies on C(X, Y), where C(X, Y) denotes the set of all continuous functions from space X to Y ; X is a completely regular space and Y is a locally convex space.