Abstract: We study the relationship λ between and the number of positive solutions of first order periodic boundary value problems{u'(t)+a(t)u(t)=λf(t,u(t)), t∈［0,T］,u(0)=u(T),where λ is a positive parameter, a∈C(R, ［0,+∞))且∫^T₀a(θ)dθ>0, f∈C(［0,T］×［0,+∞),(0,+∞))and f_∞=lim_u→∞ inf(f(t,u))/u=∞ uniformly for t∈［0,T］. By using the method of the upper and lower solutions and topological degree techniques, we obtain that the problem has no positive solution, exactly one positive solution and at least two positive solutions, when λ>λ^*, λ=λ^*, 0<λ<λ^*, respectively.

Key words: topological degree theory, existence, upper and lower solutions, multiplicity