• 대한수학회보
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• 제43권4호
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• pp.693-701
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• 2006

Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems $$(P)\{{\Delta}^2x(k-1)+g(k,y(k))=0,\;k{\in}[1,\;T],\;{\Delta}^2y(k-1)+f(k,\;x(k)=0,\;k{\in}[1,\;T],\;x(0)=y(0)=0,\;x(T+1)=y(T+1)=0$$ where T is a positive integer, [1, T] is the discrete interval {1, 2,..., T}, ${\Delat}x(k)=x(k+1)-x(k)$ is the forward difference operator and ${\Delta}^2x(k)={\Delta}({\Delta}x(k))$.

• 대한수학회보
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• 제58권3호
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• 한국수학교육학회지시리즈A:수학교육
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• 제26권1호
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• pp.41-45
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• 1987

In this paper, we give several fixed point theorems in a complete metric space for two multi-valued mappings commuting with two single-valued mappings. In fact, our main theorems show the existence of solutions of functional equations f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ and $\chi$=f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ under certain conditions. We also answer an open question proposed by Rhoades-Singh-Kulsherestha. Throughout this paper, let (X, d) be a complete metric space. We shall follow the following notations : CL(X) = {A; A is a nonempty closed subset of X}, CB(X)={A; A is a nonempty closed and founded subset of X}, C(X)={A; A is a nonempty compact subset of X}, For each A, B$\in$CL(X) and $\varepsilon$>0, N($\varepsilon$, A) = {$\chi$$\in$X; d($\chi$, ${\alpha}$) < $\varepsilon$ for some ${\alpha}$$\in$A}, E$\sub$A, B/={$\varepsilon$ > 0; A⊂N($\varepsilon$ B) and B⊂N($\varepsilon$, A)}, and (equation omitted). Then H is called the generalized Hausdorff distance function fot CL(X) induced by a metric d and H defined CB(X) is said to be the Hausdorff metric induced by d. D($\chi$, A) will denote the ordinary distance between $\chi$$\in$X and a nonempty subset A of X. Let R$\^$+/ and II$\^$+/ denote the sets of nonnegative real numbers and positive integers, respectively, and G the family of functions ${\Phi}$ from (R$\^$+/)$\^$s/ into R$\^$+/ satisfying the following conditions: (1) ${\Phi}$ is nondecreasing and upper semicontinuous in each coordinate variable, and (2) for each t>0, $\psi$(t)=max{$\psi$(t, 0, 0, t, t), ${\Phi}$(t, t, t, 2t, 0), ${\Phi}$(0, t, 0, 0, t)} $\psi$: R$\^$+/ \longrightarrow R$\^$+/ is a nondecreasing upper semicontinuous function from the right. Before sating and proving our main theorems, we give the following lemmas:

• 대순사상논총
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• 제25_2집
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• pp.1-43
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• 2015

Daesoonjinrihoe perceives that human suffering is caused not only by mental factors but also by the cosmic principle of Mutual Overcoming more fundamentally. The relationship influenced among all things in the universe is Mutual Beneficence and Mutual Overcoming. If these relations are properly cooperated in harmony, all things will reveal their energetic existence in a balanced state, however, if one side, particularly, Mutual Overcoming becomes excessive, indeed, it will cause trouble. That is, the sentient beings that are inclined to excessive Mutual Overcoming rather than indispensable Mutual Overcoming make grievances among themselves and cumulate them. In the end, heaven and earth lose its constant Way[常道], leading the sentient beings to all kinds of agonies. However, it may be a problem if every agony is depicted with Mutual Overcoming, that is, one single fixed frame, because Mutual Overcoming has to be interpreted diversely according to the circumstances. Thus, it is not possible to explain precisely about the agony if not understanding its spectrum. This study analyzed the ways how Daesoonjinrihoe explains the aspects of reasons and solutions to a variety of agonies. In the Supreme Scripture of Daesoonjinrihoe, the reasons for agony vary: first, it is caused by Mutual Overcoming of the cosmos itself, which is irrelevant to human beings. Secondly, man can be victimized because of Mutual Overcoming caused by man. Thirdly, there is a case of a victim who is damaged due to Mutual Overcoming caused by the other. Fourthly, man can be victimized because of Mutual Overcoming caused by someone else's unintended deeds. Fifthly, there is a case of man having agony in return when he, as a perpetrator, hurt others or resolve his inappropriate desires. Sixthly, man's deed unintendedly became a sin and have agony in return. Seventhly, man can be a self-perpetrator, conceiving vain desire, obsession and grievance. Eightly, there is a case of man having agony during enforcement of the law or practice of justice. Ninthly, man needs to endure agony during self-cultivation, enacting Reordering of the Universe(天地公事), ritual practice, promotion of fortune, and test. However, it is important to make sure that the reason for agony can interact with not only one but also diverse models in real situation. These nine models are included in the category of Mutual Overcoming; but, these cannot be classified in a single term, Mutual Overcoming because much information will be hidden, only to make it difficult to explain about agony more precisely. There are some characteristic observed in the concept of agony in Daesoonjinrihoe as follows: first, reasons for agony are caused by Mutual Overcoming although Mutual Overcoming can be interpreted differently according to the circumstances. Secondly, Daesoonjinrihoe makes a positive valuation for some agony. Thirdly, the solution to overcome a agony is brought to a conclusion, that is cultivation practice of Daesoonjinrihoe. Fourthly, the experience of agony and solution in Daesoonjinrihoe sometimes lie in the unit of family beyond individual. Fifthly, in the painful situation resulted from vain desire and obsession, Daesoonjinrihoe suggests a solution called Mutual Beneficence. That is, it emphasizes not only self-control but also active practice of considering others.