Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays ✩

The paper is concerned with the delay differential equation u +λb(t)f (u(t −τ)) = 0 satisfying u(t) = 0 for −τ t 0 and u(1) = g( 1 0 u(t) dβ(t)), where 1 0 u(t) dβ(t) denotes the Riemann–Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient f (u) u (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t) = t , by using a property of the Riemann– Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established. © 2007 Elsevier Inc. All rights reserved.