Existence of positive solutions to three-point phi-Laplacian BVPs via homotopic deformations

Under suitable conditions and via homotopic deformation, we provide existence results for a positive solution to the three-point $phi$-Laplacian boundary-value problem $$displaylines{ -( aphi(u'))'(x)=b(x) f(x,u(x)),quad xin ( 0,1), cr u(0)=alpha u(eta),quad u'(1)=0, }$$ where $phi:mathbb{R} omathbb{R}$ is an increasing homeomorphism with $phi(0) =0$, $b$ does not vanish identically, and $f$ is continuous.