On Lower Semi-Continuity of Interval-Valued Multihomomorphisms

S. Pianskool ; P. Udomkavanich ; P. Youngkhong

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 1

Résumé

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 interval-valued multifunctions on $\mathbb{R}$. They proved that an interval-valued multifunction $f$ on $\mathbb{R}$ is an upper semi-continuous 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 semi-continuity. It is shown that an interval-valued multifunction $f$ on $\mathbb{R}$ is a lower semi-continuous 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 $\mathbb{R}$.

Détail

Classification : 26A15, 26E25.