On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class $L_q$

We consider a system of first-order ordinary linear differential equations with coefficients depending on an arbitrary parameter $\lambda$. For large $\lambda$, if the coefficients are smooth with respect to $x$, then there are known classical exponentially asymptotic (with respect to $\lambda$) formulas for the solution of the system. We generalize such formulas to the case in which the coefficients belong to the class $L_q$, $q>1$. We use a new method for the reduction of problems to an integral system of special form.