Positive periodic solutions of second-order nonlinear differential systems with two parameters

By employing the Deimling fixed point index theory, we consider a class of second-order nonlinear differential systems with two parameters (λ,μ)∈R+2∖{(0,0)}. We show that there exist three nonempty subsets of R+2∖{(0,0)}: ΓΓ, Δ1Δ1 and Δ2Δ2 such that R+2∖{(0,0)}=Γ∪Δ1∪Δ2 and the system has at least two positive periodic solutions for (λ,μ)∈Δ1(λ,μ)∈Δ1, one positive periodic solution for (λ,μ)∈Γ(λ,μ)∈Γ and no positive periodic solutions for (λ,μ)∈Δ2(λ,μ)∈Δ2. Meanwhile, we find two straight lines L1L1 and L2L2 such that ΓΓ lies between L1L1 and L2L2.