This paper gives sufficient conditions for new solutions of Peano's differential equation in the class of all lower continuous mappings. In this sense, this paper presents new fixed point theorems of Schauder type on lower transversal spaces. For the lower transversal space $(X,\rho)$ are essential the mappings $T: X\to X$ which are unbounded variation, i.e., if $\sum_{n=0}^{\infty} (T^{n}x, T^{n+1}x)=+\infty$ 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.