Positive solutions of a 2nth-order boundary value problem involving all derivatives via the order reduction method

摘要

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the 2nth-order boundary value problem {(-l)~u~(2n) =∫(t, u, u',..., (-1)~(i/2)u~(i).....(-l)~(n-1)u~(2n-1), u~(2i)(0) = u~(2i+l)(1) = 0(i = 0,1,...,n- 1), where n ≥2 and f ∈ C((0,1| xR_+~(2n),R_+)(R_+= 10, ∞)). We first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first-order integro-differential equation and then use the fixed point index theory to prove the existence, multiplicity, and uniqueness of positive solutions for the resulting problem, based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators. Finally, as a byproduct, our main results are applied for establishing the existence, multiplicity and uniqueness of symmetric positive solutions for the Lidstone problem involving all derivatives.