Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations

Positive solutions of systems of Hammerstein integral equations are studied by using the theory of the fixed-point index for compact maps defined on cones in Banach spaces. Criteria for the fixed-point index of the Hammerstein integral operators being 1 or 0 are given. These criteria are generalizations of previous results on a single Hammerstein integral operator. Some of criteria are new and involve the first eigenvalues of the corresponding systems of linear Hammerstein operators. The existence and estimates of the first eigenvalues are given. Applications are given to systems of fractional differential equations with two-point boundary conditions. The Green’s functions of the boundary value problems are derived and their useful properties are provided. As illustrations, the existence of nonzero positive solutions of two specific such boundary value problems is studied.