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In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ${\left({\phi }_p\left(u^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\right)=0$, $0<t<1$, where ${\phi }_p\left(s\right):={\left|s\right|}^{p-2}s$, $p>1$, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient $q\left(t\right)$ may be singular at $t=0,1$.