Sign-changing solutions to second-order integral boundary value problems

In this paper, by using the fixed point index theory and Leray–Schauder degree theory, we consider the existence and multiplicity of sign-changing solutions to nonlinear second-order integral boundary value problem −u″(t)=f(u(t))−u″(t)=f(u(t)) for all t∈[0,1]t∈[0,1] subject to u(0)=0u(0)=0 and u(1)=g(∫01u(s)ds), where f,g∈C(R,R)f,g∈C(R,R). We obtain some new existence results concerning sign-changing solutions by computing hardly eigenvalues and the algebraic multiplicities of the associated linear problem. If ff and gg satisfy certain conditions, then this problem has at least six different nontrivial solutions: two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if ff and gg are also odd, then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.