




Volume 35 • Number 1 • 2012 

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On Lower SemiContinuity of IntervalValued Multihomomorphisms
S. Pianskool, P. Udomkavanich and P. Youngkhong
Abstract.
It is well known that if $f$ is a continuous homomorphism on $(\mathbb{R}, +)$, then there exists a constant $c \in \mathbb{R}$ such that $f(x) = cx$ for all $x \in \mathbb{R}$. Termwuttipong {\it et al.} extended this result to intervalvalued multifunctions on $\mathbb{R}$. They proved that an intervalvalued multifunction $f$ on $\mathbb{R}$ is an upper semicontinuous multihomomorphism on $(\mathbb{R}, +)$ if and only if $f$ has one of the following forms : $f(x) = \{cx\}, f(x) = \mathbb{R}, f(x) = (0, \infty), f(x) = (\infty, 0), f(x) = [\,cx, \infty), f(x) = (\infty, cx\,]$ where $c$ is a constant in $\mathbb{R}.$ In this paper, we extend the above well known result by considering lower semicontinuity. It is shown that an intervalvalued multifunction $f$ on $\mathbb{R}$ is a lower semicontinuous multihomomorphism on $(\mathbb{R}, +)$ if and only if $f$ is one of the following: $f(x) = \{cx\}, f(x) = \mathbb{R}, f(x) = (cx, \infty), f(x) = (\infty, cx), f(x) = [\,cx, \infty), f(x) = (\infty, cx\,]$ where $c$ is a constant in $\mathbb{R}$.
2010 Mathematics Subject Classification: 26A15, 26E25.
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