Some results on the method of lower and upper solutions for non-compact domains

One of the best-known existence results on the method of lower and upper solutions in initial value problems for systems of O.D.E. is the classical Muller theorem. According to this fundamental result, for every pair $(u,v)$ of lower and upper solutions, such that $u(t)\leq v(t)$ in $[a,b]$, and every initial datum $x_0\in [u(a),v(a)]$, there exists at least one solution $\phi$ of the initial value problem with $$u(t)\leq \phi(t)\leq v(t)\ ,\ t\in [a,b]\ .$$ In a recent paper, we extended this result by replacing the assumption "$u(t)\leq v(t)$ in $[a,b]$" with the weaker one "$u(a)\leq v(a)$"$\ and \ $\max\{u,v\}$ and, for every initial datum $x_0\in [u(a),v(a)]$, we achieved the existence of at least one solution $\phi$ between $\min\{u,v\}$ and $\max\{u,v\}$. Moreover, we also proved an analogous result for mixed initial-terminal boundary value problems. The aim of this paper is to extend these theorems to non-compact domains. First, we prove the existence result for the initial value problem on an interval $[a,+\infty[$ under the same assumptions of the case of compact domains. Moreover, we also consider the case of upper and lower solutions in $AC_{loc}(]a,b])$. By means of a change of variable, we deduce from this last result a theorm regarding terminal value problems in domains of the type $[a,+\infty[$ with datum $y_0\in\erren$. Note that no hypothesis is assumed on the asymptotic behaviour of the lower and upper solutions, as, instead, was usual in this setting.