A four-point boundary value problem with singular $$\phi $$-Laplacian

Abstract

We prove that the four-point boundary value problem $$\begin{aligned} -\left[ \phi (u') \right] ^{\prime }=f(t,u, u'), \quad u(0)=\alpha u(\xi ), \quad u(T)=\beta u(\eta ), \end{aligned}$$ where $$f:[0,T] \times \mathbb {R}^2 \rightarrow \mathbb {R}$$ is continuous, $$\alpha , \; \beta \in [0,1)$$ , $$0<\xi< \eta <T $$ , and $$\phi :(-a,a) \rightarrow \mathbb {R}$$ ( $$0<a<\infty $$ ) is an increasing homeomorphism, which is always solvable. When instead of f is some $$g:[0,T] \times [0, \infty ) \rightarrow [0, \infty )$$ , we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.