• Communications for Statistical Applications and Methods
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• v.1 no.1
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• pp.149-156
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• 1994

For an r > 2 and a finite B, $E\mid max \;1\leq k\leq n \;\sum\limits_{j=m+1}^{m+k}X_j\mid^r\leq Bn^ {\frac{r}{2}}$ (all $n\geq 1$) is obtained for a negatively associated sequence $\{X_j \;:\; j\in N\}$. We also derive the maximal inequelity for a negatively associated sequence. Stationarity is not required.

• East Asian mathematical journal
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• v.18 no.1
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• pp.95-109
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• 2002

In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.39-51
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• 2010

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

• East Asian mathematical journal
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• v.36 no.5
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• pp.571-581
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• 2020

In this paper, an equilibrium problems involving a finite family of maximal monotone operators and inverse-strongly monotone operators are introduced and investigated. A strong convergence theorem of common solutions is obtained in Hilbert spaces.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1147-1161
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• 2012

In this paper, a coincidence theorem for a compact ${\Re}\mathfrak{C}$-map is proved in an abstract convex space. Several more general coincidence theorems for noncompact ${\Re}\mathfrak{C}$-maps are derived in abstract convex spaces. Some examples are given to illustrate our coincidence theorems. As applications, an alternative theorem concerning the existence of maximal elements, an alternative theorem concerning equilibrium problems and a minimax inequality for three functions are proved in abstract convex spaces.

• Kyungpook Mathematical Journal
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• v.49 no.1
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• pp.31-39
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• 2009

Let $T_{\rho}$ be the generalized fractional integral operator associated to a function ${\rho}:(0,{\infty}){\rightarrow}(0,{\infty})$, as defined in [16]. For a function W on $\mathbb{R}^n$, we shall be interested in the boundedness of the multiplication operator $f{\mapsto}W{\cdot}T_{\rho}f$ on generalized Morrey spaces. Under some assumptions on ${\rho}$, we obtain an inequality for $W{\cdot}T_{\rho}$, which can be viewed as an extension of Olsen's and Kurata-Nishigaki-Sugano's results.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.465-478
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• 2006

In this paper, we introduce a new class of completely generalized strongly nonlinear quasivariational inequalities and establish its equivalence with a class of fixed point problems by using the resolvent operator technique. Utilizing this equivalence, we develop a three-step iterative scheme with errors, obtain a few existence theorems of solutions for the completely generalized non-linear strongly quasivariational inequality involving relaxed monotone, relaxed Lipschitz, strongly monotone and generalized pseudocontractive mappings and prove some convergence and stability results of the sequence generated by the three-step iterative scheme with errors. Our results include several previously known results as special cases.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.1-28
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• 2000

In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, We give a new proof of the Himmelberg fixed point theorem and G-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.141-148
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• 2022

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≤ 1, then Govil proved $$\max_{{\mid}z{\mid}=1}{\mid}p^{\prime}(z){\mid}{\leq}{\frac{n}{1+k^n}}\max_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$, provided |p'(z)| and |q'(z)| attain their maximal at the same point on the circle |z| = 1, where $$q(z)=z^n{\overline{p(\frac{1}{\overline{z}})}}$$. In this paper, we extend the above inequality to polar derivative of a polynomial. Further, we also prove an improved version of above inequality into polar derivative.