Existence of positive solutions for \(p\)-Laplacian singular boundary value problems of functional differential equation (Q1775276)

The following boundary value problem is considered \[ (| y'|^{p-2}y')'+r(t)f(y_{t})=0,\quad t\in [0,1],\;p>1, \] \[ y'(0)=0,\quad y(t)=\Phi(t),\qquad t\in [1,1+\tau]. \] Here, \(y_{t}=y(t+\theta)\), \(\theta \in [0,\tau]\), \(0\leq \tau <1\); \(\Phi(t)\in [0,\infty)\) for \(t\in [1,1+\tau]\), \(\Phi\) is a continuous function; \(f\) is a nonnegative continuous function, \( r(t)\geq 0\) for \(t\in [0,1]\) is a measurable function and satisfies \[ 0< \int_{1}^{1+\tau}r(t)\,dt \leq \int_{0}^{1}r(t)<\infty, \] moreover \(r(t)\) is allowed to be zero on some subset of \([1,1+\tau]\) and has a singularity at the points \(t=0\) and \(t=1\). Existence theorems for positive solutions are proved.