This paper presents new fixed point theorems on lower and upper transversal spaces. The following main result is proved that if $T$ is a self-map on an orbitally DS-complete transversal lower space $(X, \rho)$, if there exists an upper semicontinuous function $G : X \to \mathbb{R}$ such that $\rho [x, Tx] \geq G(Tx) − G(x) \geq 0$ for every $x \in X$ and if $G(T^na) \rightarrow +\infty$ as $n \rightarrow \infty$ for some $a \in X$, then $T$ has a fixed point in $X$. For the lower transversal spaces are essential the mappings $T : X \to X$ which are unbounded variation, i.e., if there exists a function $A : X \times X \to \mathbb{R}^0_{+}$ such that $\sum_{n=0}^{\infty} A(T^nx, T^{n+1}x)= +\infty$ for arbitrary $x \in X$. On the other hand, for upper transversal spaces are essential the mappings $T : X \to X$ which are bounded variation.