Existence and uniqueness results for a third-order implicit differential equation (Q2389492)

The author derives sufficient conditions for existence and uniqueness of a solution to the boundary value problem \[ f(t,u(t),u'(t), u'''(t))=0, \quad 0\leq t \leq 1, \] \[ u(0)=u'(0)=u'(1)=0. \] Lower and upper solutions method, maximum principle and an iterative method are employed. The example \[ t(u(t))^{2}+(u'(t))^4+e^{-u'''(t)-3}-2 u'''(t)-1=0,\quad 0\leq t \leq 1, \] \[ u(0)=u'(0)=u'(1)=0 \] illustrates the application of the obtained results.