A boundary value problem for a third-order equation (Q1363268)

The author deals with a third-order differential equation of the form \(x''' = f(t,x,x',x'')\), where \(f:[a,b]\times\mathbb{R}^{3} \rightarrow\mathbb{R}\) is a Carathéodory function, \(a\in\mathbb{R}\), \(b\in (a, +\infty)\). Under the assumption of the existence of two solutions \(\alpha\) and \(\beta\) of this equation with \(\alpha <\beta\), some sufficient conditions for the operators \(H_{1}, H_{2}, H_{3}\in C(S,\mathbb{R})\), being \(S\) the set of all solutions \(x\) of the third-order equation such that \(\alpha\leq x\leq\beta\), are given to guarantee the existence of a solution between \(\alpha\) and \(\beta\) of the boundary value problem \(x''' = f(t,x,x',x'')\), \(H_{1}(x)=0\), \(H_{2}(x)=0\), \(H_{3}(x)=0\). Finally, some existence results are derived for some particular cases of the operators \(H_{1}\), \(H_{2}\) and \(H_{3}\).