• Korean Journal of Mathematics
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• v.30 no.4
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• pp.571-592
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• 2022

By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.635-645
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• 2015

We will show the general solution of the functional equation $$\begin{eqnarray}f(x+ay)+f(x-ay)+2(a^2-1)f(x)\\=a^2f(x+y)+a^2f(x-y)+2a^2(a^2-1)f(y)\end{eqnarray}$$ and investigate the Hyers-Ulam stability of the quartic set-valued functional equation.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.493-507
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• 1995

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.