The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations (Q6048931)

The author considers singularly perturbed systems of differential equations of the form \[ \varepsilon ^{\sigma}(t^rd/dt)(\vec{u})=\vec{f}(t,\varepsilon ,\vec{u}), \] where \(\sigma\) is a positive integer and \(r\) is a non-negative integer. In the past such systems have been studied by different authors who have always assumed that \((\partial \vec{f}/\partial \vec{u})(0,0,\vec{0})\) is invertible. The author studies the situation when this assumption does not hold true. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey-type asymptotic expansion with respect to the parameter.