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

Abstract

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 ) ) for all t ∈ [ 0 , 1 ] subject to u ( 0 ) = 0 and u ( 1 ) = g ( ∫ 0 1 u ( s ) d s ) , where 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 f and g 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 f and g are also odd, then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.