• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.137-147
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• 2024

In the present paper, we introduce a new class of operators called p-demicompact operators between two lattice normed spaces X and Y. We study the basic properties of this class. Precisely, we give some conditions under which a p-bounded operator be p-demicompact. Also, a sufficient condition is given, under which each p-demicompact operator has a modulus which is p-demicompact. Further, we put in place some properties of this class of operators on lattice normed spaces.

• East Asian mathematical journal
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• v.28 no.1
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• pp.63-71
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• 2012

The aim of this article is to study an implicit iteration process with errors for a finite family of non-Lipschitzian asymptotically non expansive mappings in the intermediate sense in Banach spaces. Also we establish some strong convergence theorems and a weak convergence theorem for said scheme to converge to a common fixed point for non Lipschitzian asymptotically nonexpansive mappings in the intermediate sense. The results presented in this paper extend and improve the corresponding results of [1], [3]-[8], [10]-[11], [13]-[14], [16] and many others.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.439-442
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• 1995

Let ${X_n : n = 1,2,\cdots}$ be a sequence of pairwise independent identically distributed random vectors taking values in a separable Hilbert space H such that $E \Vert X_1 \Vert = \infty$. Let $S_n = X_1 + X_2 + \cdots + X_n$ and for any real $\alpha$ with $0 < \alpha < 1$ define a sequence ${\gamma_n(\alpha)}$ as $\gamma_n(\alpha) = inf {r : P(\Vert S_n \Vert \leq r) \geq \alpha}$. Then $$ lim_{n \to \infty} sup \Vert S_n \Vert/\gamma_n(\alpha) = \infty $$ holds. This is a generalization of Vvedenskaya[2].

• East Asian mathematical journal
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• v.26 no.5
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• pp.627-639
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• 2010

In this paper, we introduce a new iterative sequence finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a Banach space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly-monotone mapping, the fixed point problem and the classical variational inequality problem. Our results improve and extend the corresponding results announced by many others.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.303-315
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• 1994

Let X be a real Banach space with norm ∥ㆍ∥. Let T > 0, r ≥a be fixed constants. We denote by L/sup p/ the usual L/sup p/( -r, 0; X) with norm ∥ㆍ∥/sub p/ for 1 ≤p < ∞. Our object is to study the existence of solutions of nonlinear functional evolution equations of the type (FDE) x'(t) ＋ A(t)x(t) = G(t, x/sub t/), 0 ≤t ≤T, x/sub 0/ = ø.(omitted)

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.235-250
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• 2023

Let 1 ≤ p, q < ∞ be fixed, and let R = [r_jk] be an infinite scalar matrix such that 1 ≤ r_jk < ∞ and sup_j,k r_jk < ∞. Let 𝓑(𝑙_p, 𝑙_q) be the set of all bounded linear operator from 𝑙_p into 𝑙_q. For a fixed Banach algebra 𝐁 with identity, we define a new vector space S^R_p,q(𝐁) of infinite matrices over 𝐁 and a paranorm G on S^R_p,q(𝐁) as follows: let $$S^R_{p,q}({\mathbf{B}})=\{A:A^{[R]}{\in}{\mathcal{B}}(l_p,l_q)\}$$ and $G(A)={\parallel}A^{[R]}{\parallel}^{\frac{1}{M}}_{p,q}$, where $A^{[R]}=[{\parallel}a_{jk}{\parallel}^{r_{jk}}]$ and M = max{1, sup_j,k r_jk}. The existance of S^R_p,q(𝐁) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.65-69
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• 2001

For any closed subspace X of $\ell_p, \; 1<\kappa<\infty$, K(X) is proximinal in L(X), and if X is a Banach space with an unconditional shrinking basis, then K(X, c$_0$) is proximinal in L(X,$ \ell_\infty$).

• Communications of the Korean Mathematical Society
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• v.17 no.3
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• pp.467-474
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• 2002

Let A be C(D), A(D), P(T), C(T), or H$\^$$\infty$/(U). We evaluate the quotient group GL(A) by exp(A).

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.331-336
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• 1994

Let $P_{0}$ be a $\delta$-ring (a ring closed with respect to the forming of countable intersections) of subsets of a nonempty set $\Omega$. Let X and Y be Banach spaces and L(X, Y) the Banach space of all bounded linear operators from X to Y. A set function m : $P_{0}$ longrightarrow L(X, Y) is called an operator valued measure countably additive in the strong operator topology if for every x $\epsilon$ X the set function E longrightarrow m(E)x is a countably additive vector measure. From now on, m will denote an operator valued measure countably additive in the strong operator topology.(omitted)

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.251-266
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• 2014

In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.