Phase-space analysis of a non-oscillatory ''highly nonlinear'' boundary value problem (Q2367831)

The existence and the behaviour of solutions of the BVP (1) \(\varepsilon\ddot x=f(t,x)^ \omega\), (2) \(x(0)=\alpha\), \(x(1)=\beta\) is investigated under the assumptions stated in terms of the internal set theory that \(\varepsilon>0\) is an infinitesimal real number and \(\omega\in N\) is odd and unlimited. The function \(f\) is supposed to be increasing in the variable \(x\). Necessary and sufficient conditions for the existence of one or two boundary layers and the global behaviour of the solutions of (1), (2) and their asymptotic approximations are given. If \(\varepsilon^{1/\omega}\) is non-infinitesimal, then there exists a solution of (1), (2) which is not close to the curve \(f=0\), but to certain arcs of the curves \(f=\pm \varepsilon^{1/\omega}\) and certain straight lines. If \(\varepsilon^{1/\omega}\) is infinitesimal, then there is a solution of (1), (2) which approaches the curve \(f=0\).