The authors present sufficient conditions for the existence of extremal solutions (the greatest and the smallest Carathéodory solutions) between some given lower and upper solutions for the initial value problem $$x'=f(t,x),\,\,\,t\in I=[t_0,t_0+L],\,\,\,\,x(t_0)=x_0,$$ where the function $f$ is bounded by an integrable function on the sector delimited by the graphs of the lower and upper solutions, and also verifies some additional assumptions. Then they show that the existence of a pair of well-ordered piecewise continuous lower and upper solutions implies the existence of a better pair of continuous lower and upper solutions. Two existence results for weak solutions in the case $f$ does not have a strong bound (the singular case) are also proved. Some considerations for singular quasimonotone systems and several examples which illustrate the obtained results are finally addressed.