On eigenvalue intervals of higher-order boundary value problems with a sign-changing nonlinear term

In the paper, we shall consider the boundary value problem { u ( n ) + λ a ( t ) f ( t , u , u ′ , u ″ , u ( 3 ) , … , u ( n − 2 ) ) = 0 , n ≥ 2 , t ∈ ( 0 , 1 ) , u ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α u ( n − 2 ) ( 0 ) − β u ( n − 1 ) ( 0 ) = 0 , γ u ( n − 2 ) ( 1 ) + δ u ( n − 1 ) ( 1 ) = 0 where λ > 0 , α , β , γ and δ are constants satisfying α , γ > 0 and β , δ ≥ 0 . Intervals of λ are determined to ensure the existence of a positive solution of the boundary value problem according to the signs of a ( t ) and f .