• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.681-688
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• 1995

Fixed-point theory has an extension to coincidences. For a pair of maps $f,g:X_1 \to X_2$, a coincidence of f and g is a point $x \in X_1$ such that $f(x) = g(x)$, and $Coin(f,g) = {x \in X_1 $\mid$ f(x) = g(x)}$ is the coincidence set of f and g. The Nielsen coincidence number N(f,g) and the Lefschetz coincidence number L(f,g) are used to estimate the cardinality of Coin(f,g). The aspherical manifolds whose fundamental group has a normal solvable subgroup of finite index is called infrasolvmanifolds. We show that if $M_1,M_2$ are compact connected orientable infrasolvmanifolds, then $N(f,g) \geq $\mid$L(f,g)$\mid$$ for every $f,g : M_1 \to M_2$.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.171-181
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• 1995

Nielsen coincidence theory is concerned with the estimation of a lower bound for the number of coincidences of two maps $f,g: X \longrightarrow Y$. For this purpose the so-called Nielsen number N(f,g) is introduced, which is a lower bound for the number of coincidences ([1]). The relative Nielsen number N(f : X,A) in the fixed point theory is introduced in [3], which is a lower bound for the number of fixed points for all maps in the relative homotopy class of f:(X,A) $\longrightarrow$ (X,A), and its estimation is given in [5].

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.87-95
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• 1997

In general, the positive solution to biological reaction-diffusion equations is not unique. In this paper, we state the sufficient and necessary conditions of the existence of positive solutions, and give and the proof for the uniqueness of positive solutions for a certain elliptic interacting system.

• East Asian mathematical journal
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• v.38 no.1
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• pp.33-41
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• 2022

In this paper, we study singular Dirichlet boundary value problems involving ϕ-Laplacian. Using fixed point index theory, the existence of positive solutions is established under the assumption that the nonlinearity f = f(u) has a positive falling zero and is either superlinear or sublinear at u = 0.

• East Asian mathematical journal
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• v.38 no.5
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• pp.593-601
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• 2022

In this paper, we consider singular 𝜑-Laplacian problems with nonlocal boundary conditions. Using a fixed point index theorem on a suitable cone, the existence results for one or two positive solutions are established under the assumption that the nonlinearity may not satisfy the L¹-Carathéodory condition.

• East Asian mathematical journal
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• v.40 no.1
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• pp.85-94
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• 2024

In this paper we study the existence and multiplicity of positive solutions for p-Laplacian problems with a singular weight. Proofs mainly make use of Global Continuation Theorem and Fixed Point Index argument.

• Journal of Environmental Science International
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• v.24 no.4
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• pp.371-381
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• 2015

In the mountain streams in Jeju Island, strong turbulence and roughness usually made it nearly impossible to utilize most of intrusive instrumentation for streamflow discharge measurements. Instead, a non-intrusive fixed electro-magnetic wave surface velocimetry (fixed EWSV: Kalesto) became alternatively popular in many representative streams to measure stream discharge seamlessly. Currently, Kalesto has shown noteworthy performance with little loss in flood discharge measurements and also has successfully provided discharge for every minute. However, Kalesto has been operated to regard its measured one-point velocity as the representative mean velocity for the given cross-section. Therefore, it could be highly possible to potentially encompass discharge measurements errors. In this study, we analyzed the difference between such Kalesto discharge measurements and other alternative concurrent discharge measurements such as Acoustic Doppler Current Profiler (ADCP) and mobile EWSV which were able to measure velocity in multi-points in the cross-section. Consequently, Kalesto discharge deviated from ADCP discharge in amount of 48% for relatively low flow, and more than 20% difference for high flow compared with mobile EWSV discharge measurements. These results indicated that the one-point velocity measured by Kalesto should be used as a cross-sectional mean velocity, rather it should be accounted for as an index-velocity in conjunction with directly measured cross-sectional mean velocity by using more reliable instrumentations. After inducing Kalesto Discharge Correction Coefficient (KDCC) that actually means relationship between index velocity and cross-sectional mean velocity, the corrected discharge from Kalesto was significantly improved. Therefore, we found that index velocity method should be applied to obtain better accuracy of discharge measurement in case of Kalesto operation.

• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.271-279
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• 2009

Using an index theory for countably condensing maps, we show the existence of eigenvalues for countably k-set contractive maps and countably condensing maps in an infinite dimensional Banach space X, under certain condition that depends on the quantitative haracteristic, that is, the infimum of all $k\;{\geq}\;1$ for which there is a countably k-set-contractive retraction of the closed unit ball of X onto its boundary.

• Journal of the Korean GEO-environmental Society
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• v.10 no.3
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• pp.37-45
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• 2009

The experimental study was carried out to evaluate the characteristics of the point load index and the uniaxial compressive strength of inherently anisotropic shale in the laboratory. In the testing program the effects of size and the shape on the point load index were investigated both in the axial and diametral direction. In general, the point load index of the shale was constant when the length/diameter (L/D) ratio of the specimen is greater than 1.0 in the diametral direction. The point load index in axial direction shows slight decrease as the L/D ratio is increased and the corner breakage was observed when L/D ratio is greater than unity. The minimum point load index was observed in the bedding angle of $\beta=15^{\circ}{\sim}30^{\circ}$ in the axial point load tests and of $\beta=30^{\circ}$ in the uniaxial compression tests. The relationship between the point load index and the uniaxial compressive strength was linear to ${\sigma}_c=25.0 I_{s(50)}$ for the specimen with the bedding plane angle, $\beta$ at the range of $0^{\circ}{\sim}90^{\circ}$. On the other hand, this relationship was appeared linear to ${\sigma}_c=14.4 I_{s(50)}$ when the bedding angle, $\beta$ is fixed to 90${^{\circ}}$ and this correlation is much different from ${\sigma}c=22 I_{s(50)}, which is generally applied to the rock specimen with no bedding plane in ISRM (1985). The anisotropic strength with different $\beta$ angle shows the shoulder type and this can be suitably modelled by the corrected Ramamurthy'(1993)s equation with the index value of 'n' equal to 3.0.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.701-717
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• 2006

The positive coexistence of a simple food chain model with ratio-dependent functional response and cross-diffusion is discussed. Especially, when a cross-diffusion is small enough, the existence of positive solutions of the system concerned can be expected. The extinction conditions for all three interacting species and for one or two of three species are studied. Moreover, when a cross-diffusion is sufficiently large, the extinction of prey species with cross-diffusion interaction to predator occurs. The method employed is the comparison argument for elliptic problem and fixed point theory in a positive cone on a Banach space.