Existence of positive, negative and sign-changing periodic solutions for a class of integral equations

In this paper, the simultaneous existence of positive, negative and sign-changing periodic solutions for a class of integral equations of the form ϕ ( x ) = ∫ [ x , x + ω ] ∩ Ω K ( x , y ) f ( y , ϕ ( y − τ ( y ) ) ) d y , x ∈ Ω , is considered, where Ω is a closed subset of R N with a periodic structure. Our main result is different from most existing results since they provide three constant sign periodic solutions only.