On existence of multiple positive solutions for ϕϕ-Laplacian multipoint boundary value

In this paper, we consider the following multipoint boundary value problem with one-dimensional ϕϕ-Laplacian View the MathML source(ϕ(x′(t)))′+q(t)f(t,x(t),x′(t))=0,t∈(0,1),x(0)=∑i=1n−2αix(ξi),ϕ(x′(1))=∑i=1n−2βiϕ(x′(ξi)), Turn MathJax on where ϕ(⋅)ϕ(⋅) is an odd and increasing homeomorphism, ξi∈(0,1)ξi∈(0,1) with 0<ξ1<ξ2<⋯<ξn−2<10<ξ1<ξ2<⋯<ξn−2<1, αiαi and βiβi are nonnegative constants and f(t,x(t),x′(t))f(t,x(t),x′(t)) is continuous and allowed to change sign. By fixed point theorems, we obtain new results on the existence of at least three positive solutions of this boundary value problem, which includes and improves some related results in the relevant literature.