Asymptotics of solutions of a system of linear inhomogeneous singularly perturbed differential equations

Solutions with asymptotics in integral and fractional powers of the parameter ? are constructed for the vector differential equation $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$ in the case of resonance and multiple spectrum of the limit matrix. $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$