• Honam Mathematical Journal
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• v.30 no.1
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• pp.171-183
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• 2008

In a previous work [18], the authors investigated autocontinuity, converse-autocontinuity, uniformly autocontinuity, uniformly converse-autocontinuity, and fuzzy multiplicativity of monotone set function defined by Choquet integral([3,4,13,14,15]) instead of fuzzy integral([16,17]). We consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [18]. These integrals, which can be regarded as interval-valued aggregation operators, have been used in [10,11,12,19,20]. In this paper, we investigate some characterizations of monotone interval-valued set functions defined by the interval-valued Choquet integral such as autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.1
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• pp.83-90
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• 2013

Intervals can be used in the representation of uncertainty. In this regard, we consider monotone interval-valued set functions and the Choquet integral. This paper investigates characterizations of monotone interval-valued set functions and provides applications of the Choquet integral with respect to monotone interval-valued set functions, on the space of measurable functions with the Hausdorff metric.

• Journal of the Korean Institute of Intelligent Systems
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• v.18 no.3
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• pp.311-315
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• 2008

We introduce nonnegative interval-valued set functions and nonnegative measurable interval-valued Junctions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [17]. We also obtained absolutely continuity of them in [9]. In this paper, we investigate some characterizations of the monotone interval-valued set function defined by the interval-valued Choquet integral, and such as subadditivity, superadditivity, null-additivity, converse-null-additivity.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.227-234
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• 2007

At first, we consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. In this paper we investigate some properties and structural characteristics of the monotone interval-valued set function defined by an interval-valued Choquet integral.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2007.11a
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• pp.195-198
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• 2007

In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval-valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue's theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

• Journal of the Korean Institute of Intelligent Systems
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• v.17 no.6
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• pp.749-753
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• 2007

In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue's theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.91-97
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• 1993

In this article, we consider the case that S is a topologically complete subspace of $R^{k}$ , and that .GAMMA. is a set of monotone functions on S into S. It is obtained the sugficient condition for the existence of a unique invariant probability to which $P^{(n}$/(x,dy) converges exponentially fast in a metric stronger than the Kolmogorov's distance. This extends the earlier results of Bhattacharya and Lee (1988) who considered the case .GAMMA. a set of nondecreasing functions.tions.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2003.09a
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• pp.7-7
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• 2003

We studied closed set-valued Choquet integrals in two papers(1997, 2000) and convergence theorems under some sufficient conditions in two papers(2003), for examples : (i) convergence theorems for monotone convergent sequences of Choquet integrably bounded closed set-valued functions, (ii) covergence theorems for the upper limit and the lower limit of a sequence of Choquet integrably bounded closed set-valued functions. In this presentation, we consider fuzzy number-valued functions and define Choquet integrals of fuzzy number-valued functions. But these concepts of fuzzy number-valued Choquet inetgrals are all based on the corresponding results of interval-valued Choquet integrals. We also discuss their properties which are positively homogeneous and monotonicity of fuzzy number-valued Choquet integrals. Furthermore, we will prove convergence theorems for fuzzy number-valued Choquet integrals. They will be used in the following applications : (1) Subjectively probability and expectation utility without additivity associated with fuzzy events as in Choquet integrable fuzzy number-valued functions, (2) Capacity measure which are presented by comonotonically additive fuzzy number-valued functionals, and (3) Ambiguity measure related with fuzzy number-valued fuzzy inference.

• Korean Journal of Mathematics
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• v.25 no.2
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• pp.147-162
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• 2017

In this research works, we consider the general regularized penalty method for non-stationary set valued equilibrium problem in a Banach space. We define weak coercivity conditions and show that the weak and strong convergence problems of the regularized penalty method.

• Journal of the Korean Operations Research and Management Science Society
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• v.11 no.1
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• pp.36-43
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• 1986

In this study, the fuzzy programming is extended to handle various types of membership functions by transformation of the complicated fuzzy programming problems into the equivalent crisp linear programming problems with single objective. It is well-known that the fuzzy programming problem with linear membership functions (i.e., ramp type) can be easily transformed into a linear programming problem by introducing one dummy variable to minimize the worst unwanted deviation. However, until recently not many researches have been done to handle various general types of complicated linear membership functions which might be more realistic than ramp-or triangular-type functions. In order to handle these complicated membership functions, the goal dividing concept, which is based on the fuzzy set operation (i. e., intersection and union operations), has been prepared. The linear model obtained using the goal dividing concept is more efficient and single than the previous models [4, 8]. In addition, this result can be easily applied to any nonlinear membership functions by piecewise approximation since the membership function is continuous and monotone increasing or decreasing.