Positive solutions of a 2n2nth-order boundary value problem involving all derivatives via the order reduction method

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the 2n2nth-order boundary value problem {(−1)nu(2n)=f(t,u,u′,…,(−1)[i2]u(i),…,(−1)n−1u(2n−1)),u(2i)(0)=u(2i+1)(1)=0(i=0,1,…,n−1), where n≥2n≥2 and f∈C([0,1]×R+2n,R+)(R+≔[0,∞)). 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.