Multiple sign-changing solutions for fourth order mm-point boundary value problems

In this paper, we study the existence and multiplicity of nontrivial solutions for the fourth order mm-point boundary value problems. Making use of the theory of the fixed point index in a cone and the Leray–Schauder degree, under general conditions on nonlinearity, we prove that there exist at least six different nontrivial solutions for the fourth order mm-point boundary value problems. Furthermore, if the nonlinearity is odd, we obtain that there exist at least eight different nontrivial solutions.