Abstract
Using the theory of coincidence degree, we establish
existence results of positive solutions for higher-order multi-point boundary value
problems at resonance for ordinary differential equation
u(n)(t)=f(t,u(t),u′(t),…,u(n−1)(t))+e(t),
t∈(0,1), with one of the following boundary conditions:
u(i)(0)=0,
i=1,2,…,
n−2,
u(n−1)(0)=u(n−1)(ξ), u(n−2)(1)=∑j=1m−2βju(n−2)(ηj), and
u(i)(0)=0,
i=1,2,…, n−1,
u(n−2)(1)=∑j=1m−2βju(n−2)(ηj),
where f:[0,1]×ℝn→ℝ=(−∞,+∞) is a continuous function, e(t)∈L1[0,1]βj∈ℝ (1≤j≤m−2, m≥4), 0<η1<η2<⋯<ηm−2<1, 0<ξ<1, all
the
β−j−s
have not the same sign. We also give some examples to demonstrate our
results.