Existence of three positive solutions for quasi-linear boundary value problem (Q2508029)

Consider the boundary value problem \[ (\varphi_p(u'))'+a(t)f(u)=0 \text{ for }0<t<1, \] \[ u'(0)=0,\;u(1)+B_1(u'(\xi))=0,\tag{B} \] \[ u(0)-B_1(u'(\xi))=0,\;u'(1)=0, \] where \(\varphi_p(s)=|s|^{p-2}s\), \(p>1\), \(\xi\in(0,1)\), \(B_1\) is a nondecreasing continuous functions. The authors prove a new fixed-point theorem in a cone and apply this result to establish the existence of three positive solutions to (B).