• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.297-304
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• 2011

In this paper, we introduce closure operators [${\cdot}$]c and [${\cdot}$]a on a space and study some relations among [${\cdot}$]c, [${\cdot}$]a and countable tightness. We introduce the concepts of a strongly sequentially closed set and a strongly sequentially open set and show that a space X has countable tightness if and only if every strongly sequentially closed set is closed if and only if every strongly sequentially open set is open. Finally we find a generalization of the weak Fr$\'{e}$chet-Urysohn property which is equivalent to countable tightness.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.199-208
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• 2012

In this paper, we study some properties of spaces having countable tightness and spaces having weakly countable tightness. We obtain some necessary and sufficient conditions for a space to have countable tightness. And we introduce a new concept of weakly countable tightness which is a generalization of countable tightness and show some properties of spaces having weakly countable tightness.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1151-1164
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• 2009

Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\rightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $\rightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${\neq}{\emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, ($n{\geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\lambda_n$} $\subset$ (0, 1) satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n$ = 0 and $\sum_{n=0}^{\infty}{\lambda}_n={\infty}$, {$\beta_n$} $\subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.