A new extension of classical Muller's theorem

As it is well known, the classical Muller's theorem iis an important existence result in the theory of boundary value problems for ordinary differential equations. It finds several applications, like in the theory of monotone iterative techniques. Roughly speaking, considered a lower-solution $\gamma$ and an upper-solution $\Gamma$, this result asserts that if $\gamma(t)\leq\Gamma(t)$ in $[a,b]$, then for every $x_0\in [\gamma(a),\Gamma(a)]$ there exists at least one solution $\phi$ of initial value problem through $x_0$, such that $\gamma\leq\phi\leq\Gamma$ in $[a,b]$. The assumption $\gamma(t)\leq\Gamma(t)$ in $[a,b]$ reveals rather strong. In fact, in general, the lower-solutions are not always less than the upper-solutions, as simple examples show. The aim of this paper is to investigate on the existence of solutions even out of the interval where $\gamma\leq\Gamma$. More precisely, under the only assumption $\gamma(a)\leq\Gamma(a)$, we obtain the existence of at least one solution between $m=\min\{\gamma,\Gamma\}$ and $M=\max\{\gamma,\Gamma\}$, in the setting of single-valued differential equations under Carath\'eodory assumptions. We first prove a theorem in the scalar case which extends well-known analogous results. Subsequently, by using this result and by adapting a technique proposed by W. Walter in another setting, we achieve the extension to the vectorial case.