Singularly perturbed initial value problems for differential equations in a Banach space (Q1119789)

Consider the following initial value problem \((1)\quad dx/dt=u(t,x,y,\epsilon),\) \(x(a,\epsilon)=A(\epsilon)\), \(\epsilon dy/dt=v(t,x,y,\epsilon),\) \(y(a,\epsilon)=B(\epsilon)\), \(t\in (a,b]\). Assume that \(\epsilon\) is a positive small parameter and x, y, u, v, A, B are infinite dimensional vector functions. With help of the results on differential inequalities in Banach spaces (which are well-known, therefore I do not understand why the authors gave the proof of them) some asymptotic estimates for solutions of (1) are obtained. Those estimates yield the possibility of analysing of asymptotic behavior of solutions of (1) when \(\epsilon\) \(\to 0\).