Eigenvalues of boundary value problems for higher order di erential equations

We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n − 1 ) = 0 , where 0,\alpha ,\beta ,\gamma$ " xmlns:mml="http://www.w3.org/1998/Math/MathML"> λ > 0 , α , β , γ and δ are constants satisfying 0,\beta , \delta \ge 0, \beta + \alpha > 0$ " xmlns:mml="http://www.w3.org/1998/Math/MathML"> α γ + α δ + β γ > 0 , β , δ ≥ 0 , β + α > 0 and 0$ " xmlns:mml="http://www.w3.org/1998/Math/MathML"> δ + γ > 0 to characterize the values of λ so that it has a positive solution. For the special case λ = 1 , sufficient conditions are also established for the existence of positive solutions.