Positive periodic solutions and nonlinear eigenvalue problems for functional differential equations

This paper is devoted to investigate the existence of positive periodic solution for a functional differential equation in the form of λLx = −b(t) f (x(t − τ(t))), where Lx = x ′ (t) − a(t)g(x(t))x(t). By using well-known fixed point index theory in a cone, values of λ are determined for which there exist positive periodic solutions for the above functional differential equation. The dependence of positive periodic solution xλ(t) on the parameter λ is also studied, i.e., lim λ→+∞ ‖xλ‖ = +∞ or lim λ→+∞ ‖xλ‖ = 0. 2000 Mathematics Subject Classification: 34K13, 92B05.