We consider a Strang-type splitting method for an abstract semilinear evolution equation $${\partial }_{t}u=Au+F\left(u\right).$$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F.$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting...

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 $t=0,1$.

In this paper we deal with the four-point singular boundary value problem $$\left\{\begin{array}{cc}{\left({\phi }_p\left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+q\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,\hfill & t\in (0,1),\hfill \\ u^{\text{'}}\left(0\right)-\alpha u\left(\xi \right)=0,u^{\text{'}}\left(1\right)+\beta u\left(\eta \right)=0,\hfill \end{array}\right.$$ where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q\left(t\right)>0$, $t\in (0,1)$, and $f\in C\left(\right[0,1]\times (0,+\infty )\times ℝ,(0,+\infty \left)\right)$ may be singular at $u=0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.