On certain results of C. Bereanu and J. Mawhin (Q1932152)

This paper gives another proofs of well-known results obtained in [\textit{C. Bereanu} and \textit{J. Mawhin}, J. Differ. Equations 243, No. 2, 536--557 (2007; Zbl 1148.34013); J. Math. Anal. Appl. 352, No. 1, 218--233 (2009; Zbl 1170.34014); J. Fixed Point Theory Appl. 4, No. 1, 57--75 (2008; Zbl 1176.34019)]. The results of the cited papers guarantee the existence of a solution to the scalar differential equation \[ (\phi(u'))'=f(t,u,u'), \] where \(\phi:(-a,a)\to \mathbb R,\) \(\phi(0)=0,\) is an increasing homeomorphism and \(f:[0,T]\times\mathbb R^2\to\mathbb R\) is continuous, satisfying one of the following boundary conditions \[ \phi(u'(0))=g_0(u(0)),\;\phi(u'(T))=g_T(u(T)), \] where \(g_0,g_T:\mathbb R\to\mathbb R\) are continuous, \[ \begin{aligned} u(0)=A,&\;u(T)=B,\\u(0)=A,&\;u'(T)=C,\\u'(0)=C,&\;u(T)=A,\\u(0)=u(T),&\;u'(0)=u'(T).\end{aligned} \] The author replaces these boundary value problems by the corresponding boundary value problems for the first-order system \[ u'=\phi^{-1}(v),\;v'=f(t,u,\phi^{-1}(v)) \] and uses that, by the conditions imposed, if \(u(t)\) is a solution of the boundary value problems for the differential equation, then \((u(t),u'(t))\in[A-aT,A+aT]\times[-a,a]\) or \((u(t),u'(t))\in[-r,r]\times[-a,a]\) for \(t\in[0,T],\) where the constant \(r\) is defined in the assumptions.