Positive solutions of a2nth-order boundary value problem involving all derivatives via the order reduction method

Abstract

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