Third-order nonlinear singularly perturbed boundary value problem.
The authors study the following boundary value problems with the small parameter \(\varepsilon>0\) \[ \varepsilon x'''=f(t,x,x',\varepsilon), \] \[ \begin{cases} x(0)=A,\;g(x'(0),\;x'(1))=0,\\ h(x'(0),\;x'(1),\;x''(0),\;x''(1))=0,\end{cases} \] where \(g\) and \(h\) are continuous functions of their arguments under some additional restrictions. In particular, the following boundary conditions \[ x(0)=A,\;x'(0)=x'(1),\;x''(0)=x''(1), \] are considered. Specific upper and lower solutions are constructed, and the existence of solutions are shown and asymptotic estimates on solutions under suitable conditions are obtained. An example illustrating the applicability of the main results is given.
scientific article; zbMATH DE number 1880779