The purpose of the present paper is to study the existence of solutions to initial value problems for nonlinear first order differential systems subject to nonlinear nonlocal initial conditions of functional type. The approach uses vector-valued metrics and matrices convergent to zero. Two existence results are given by means of Schauder and Leray-Schauder fixed point principles and the existence and uniqueness of the solution is obtained via a fixed point theorem due to Perov. Two examples are given to illustrate the theory.
In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form \[ y^{\prime }_i(t)-p_i(t)y_{i+1}(t)=0, \quad i=1,2,\dots , n-2, y^{\prime }_{n-1}(t)-p_{n-1}(t)|y_n(h_n(t))|^\alpha \mathop {\mathrm sgn}[y_n(h_n(t))]=0, y^{\prime }_n(t) \mathop {\mathrm sgn}[y_1(h_1(t))]+p_n(t)|y_1(h_1(t))|^\beta \, \le 0, \] where $ n\ge 3 $ is odd, $ \alpha >0$, $ \beta >0$.