• 대한수학회지
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• 제45권6호
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• pp.1561-1576
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• 2008

In this paper we shall prove some weighted norm inequalities of the form $${\int}_{R^n}\;|Tf(x)|^pu(x)dx\;{\leq}\;C_p\;{\int}_{R^n}\;|f(x)|^pNu(x)dx$$ for certain rough singular integral T and maximal singular integral $T^*$. Here u is a nonnegative measurable function on $R^n$ and N denotes some maximal operator. As a consequence, some vector valued inequalities for both T and $T^*$ are obtained. We shall also get a boundedness result of T on the Triebel-Lizorkin spaces.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권2호
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• pp.155-166
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• 2004

In this paper, we introduce new monotone concepts for set-valued mappings, called generalized C(x)-L-pseudomonotonicity and weakly C(x)-L-pseudomonotonicity. And we obtain generalized Minty-type lemma and the existence of solutions to vector variational inequality problems for weakly C(x)-L-pseudomonotone set-valued mappings, which generalizes and extends some results of Konnov & Yao [11], Yu & Yao [20], and others Chen & Yang [6], Lai & Yao [12], Lee, Kim, Lee & Cho [16] and Lin, Yang & Yao [18].

• 대한수학회보
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• 제50권3호
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• pp.777-786
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• 2013

The purpose of this paper is to establish some relationships between proper efficiency of set-valued optimization problems and proper efficiency of vector variational-like inequalities under the assumptions of generalized cone-preinvexity. Our results extend and improve the corresponding results in the literature.

• 호남수학학술지
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• 제31권2호
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• pp.247-258
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• 2009

This paper introduces new mixed vector FQ-implicit variational inequality problems and corresponding mixed vector FQ-implicit complementarity problems for set-valued mappings, and studies the equivalence between them under certain assumptions in Banach spaces. It also derives some new existence theorems of solutions for them with examples under suitable assumptions without monotonicity. This paper generalizes and extends many results in [8, 10, 19-22].

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제6권1호
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• pp.70-76
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• 2006

Some Stampacchia type of generalized weak vector quasivariational-like inequalities for fuzzy mappings was introduced and the existence of solutions to them under non-compact assumption was considered using the particular form of the generalized Ky Fan's section theorem due to Park [15]. As a corollary, Stampacchia type of generalized vector quasivariational-like inequalities for fuzzy mappings was studied under compact assumption using Ky Fan's section theorem [7].

• 충청수학회지
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• 제25권4호
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• pp.703-709
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• 2012

Recently, Kazmi and Khan [7] introduced a kind of equilibrium problem called generalized system (GS) with a single-valued bi-operator F. Next, in [10], the first author considered a generalization of (GS) into a multi-valued circumstance called the multi-valued quasi-generalized system (in short, MQGS). In the current work, we provide an extension of (MQGS) into a system of (MQGS) in general settings. This system is called the generalized multi-valued quasi-generalized system (in short, GMQGS). Using the existence theorem for abstract economy by Kim [8], we prove the existence of solutions for (GMQGS) in the framework of Hausdorff topological vector spaces. As an application, an existence result of a system of generalized vector quasi-variational inequalities is derived.

• 대한수학회지
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• 제55권3호
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• pp.671-694
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• 2018

In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\;{\mathbb{R}}^n,\;w_1){\times}{\cdots}{\times}L^{p_m}(l^{q_m};\;{\mathbb{R}}^n,\;w_m)$ to $L^p(l^q;\;{\mathbb{R}}^n,\;{\nu}_{\vec{w}})$, with $p_1,{\cdots},p_m$, $q_1,{\cdots},q_m{\in}(1,\;{\infty})$, $1/p=1/p_1+{\cdots}+1/p_m$, $1/q=1/q_1+{\cdots}+1/q_m$ and ${\vec{w}}=(w_1,{\cdots},w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the $Calder{\acute{o}}n$ commutator.

• 대한수학회보
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• 제58권2호
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• pp.403-417
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• 2021

This note is devoted to establishing the boundedness for some classes of Littlewood-Paley square operators defined by the kernels without any regularity on the mixed radial-angular spaces. The corresponding vector-valued versions are also presented. As applications, the corresponding results for the Littlewood-Paley g∗λ function and the Littlewood-Paley function related to the area integrals are also obtained.

• 충청수학회지
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• 제7권1호
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• pp.59-67
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• 1994

Let (${\Omega},{\Sigma},{\mu}$) be a measure space. A function $f:{\Omega}{\rightarrow}X$ is said to be equioscillated if for each set $A{\in}{\Sigma}$ of positive measure and for each ${\epsilon}$ > 0, there is a measurable subset B of A of positive measure such that the inequality s$sup_{{\omega}{\in}B}x^*f({\omega})-inf_{{\omega}{\in}B}x^*f({\omega})$ < ${\epsilon}$ holds for every $x^*$ with $||x^*||{\leq}1$. Strong measurability of a vector valued function is characterized using equioscillation.

• 대한수학회지
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• 제60권1호
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• pp.33-69
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• 2023

Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝⁿ → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.