The authors study the initial value problems \[ \left\{\begin{aligned} x^{\prime }\left( t\right)& = f\left( t,x^{\prime }\left( t\right) ,x^{\prime }\left( \beta \left( t\right) \right) ,x\left( t\right) ,x\left( \alpha \left( t\right) \right) \right) ,\quad t\in J=[0,T], \\ x\left( 0\right) &= 0 \end{aligned}\right.\tag{1} \] and \[ \left\{\begin{aligned} x^{\prime }\left( t\right)& = g\left( t,x^{\prime }\left( t\right) ,x^{\prime }\left( \beta \left( t\right) \right) ,x\left( t\right) ,x\left( \alpha \left( t\right) \right) \right) ,\quad t\in J=[0,T], \\ x\left( T\right)& = 0. \end{aligned}\right.\tag{2} \] Applying the monotone iterative method to the integral equations equivalent to (1) and (2), they prove some existence results for extremal solutions and some existence and uniqueness results for the solution to (1) and (2).