• Kyungpook Mathematical Journal
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• v.13 no.2
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• Honam Mathematical Journal
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• v.31 no.3
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• pp.381-398
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• 2009

Let $G\;=\;O(k){\times}O(k){\times}O(q)$ and let $M^n$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^{n+1}$. Then we obtain a theorem: If $M^n$ has 2 distinct principal curvatures at some point p, then the square norm of the second fundamental form of $M^n$, S = n.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.877-894
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• 2013

In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green's function, a nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem in a cone. Some examples are included to show the applicability of our results.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.2
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• pp.79-91
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• 2010

In this paper, a class of fractional integrodifferential equations of mixed type with nonlocal conditions is considered. First, using contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. Second, the existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with nonlocal conditions is also presented.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.437-442
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• 2014

In this paper we prove Ekeland's variational principle in the setting of locally complete spaces for lower semi continuous functions from above and bounded below. We use this theorem to prove Caristi's fixed point theorem in the same setting and also for lower semi continuous functions.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.233-246
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• 2016

In this paper, we introduce the class of analytic extensions of M-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an M-hyponormal operator T is subscalar of order 2k + 2. Finally we obtain that an analytic extension of an M-hyponormal operator satisfies Weyl's theorem.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1229-1236
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• 2009

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator S$_K$ on a vector-valued Hardy space H$^2$(${\Omega}$, K) is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions and quasi-inner divisors.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.167-176
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• 2009

The aim of this paper is to obtain three interesting results for reducibility of Kamp$\'{e}$ de $\'{e}$riet function. The results are derived with the help of contiguous Gauss's second summation formulas obtained earlier by Lavoie et al. The results obtained by Bailey, Rathie and Nagar follow special cases of our main findings.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.295-300
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• 1992

In [5] it was shown that if T .mem. L(X) is regular on a Banach space X, with finite dimensional intersection T$^{-1}$ (0).cap.T(X) and if S .mem. L(X) is invertible, commute with T and has sufficiently small norm then T - S in upper semi-Fredholm, and hence essentially one-one, in the sense that the null space of T - S is finite dimensional ([4] Theorem 2; [5] Theorem 2). In this note we extend this result to incomplete normed space.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.63-101
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• 2021

In this research paper, an attempt has been made to provide an interesting extension of the well-known and useful Kummer's second theorem. Several applications have also been given.