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

Abstract

AbstractThe 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(∫01u(t)dβ(t)), where ∫01u(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