Positive solutions of nonlinear third-order mm-point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity

In this paper, several existence theorems of positive solutions are established for a nonlinear mm-point boundary value problem for the following third-order differential equations (ϕ(u″))′+a(t)f(t,u(t))=0,t∈(0,1),ϕ(u″(0))=∑i=1m−2aiϕ(u″(ξi)),u′(0)=0,u(1)=∑i=1m−2biu(ξi), where ϕ:R⟶Rϕ:R⟶R is an increasing homeomorphism and homomorphism and ϕ(0)=0ϕ(0)=0. The nonlinear term ff may change sign. As an application, an example to demonstrate our results is given.