Solvability of ( k , n - k )‎ Conjugate Boundary Value Problems with Integral Boundary Conditions at Resonance

Abstract EN

We investigate a ( k , n - k ) conjugate boundary value problem with integral boundary conditions.

By using Mawhin continuation theorem, we study the solvability of this boundary value problem at resonance.

It is shown that the boundary value problem ( - 1 ) n - k φ ( n ) ( x ) = f x , φ x , φ ′ x , … , φ ( n - 1 ) ( x ) , x ∈ [ 0,1 ] , φ ( i ) ( 0 ) = φ ( j ) ( 1 ) = 0 , 1 ≤ i ≤ k - 1 , 0 ≤ j ≤ n - k - 1 , φ ( 0 ) = ∫ 0 1 φ ( x ) d A ( x ) has at least one solution under some suitable conditions.