The author studies a boundary value problem for a system of nonlinear ordinary differential equations \[ (1)\quad \dot z=Z(z,t), \qquad (2) \quad lz= \varphi(z(\cdot)), \] where the nonlinear \(n\)-dimensional vector function \(Z(z,t)\) satisfies the conditions: \(Z( \cdot ,t)\in C^{1}[\|z-z_{0}\|\leq q]\) and \(Z(z, \cdot)\in C[a,b];\) \(l\) is a linear and \(\varphi\) is a nonlinear \(m\)-dimensional vector bounded functional; \(Z\) and \(\varphi\) have sufficiently small Lipschitz constants. In addition, the number of boundary conditions \( m \) does not coincides with the dimension \(n\) of the system of differential equations. The author proves a theorem on a necessary condition for the existence of a solution to problem (1), (2). He proposes to apply an iterative technique to study problem (1), (2). Two theorems on sufficient conditions for the existence of a unique solution to problem (1), (2) are demonstrated.