Let f : [ 0 , 1 ] × R 2 → R be function satisfying Caratheodory's conditions and e ( t ) ∈ L 1 [ 0 , 1 ] . Let η ∈ ( 0 , 1 ) , ξ i ∈ ( 0 , 1 ) , a i ≥ 0 , i = 1 , 2 , … , m − 2 , with ∑ i = 1 m − 2 a i = 1 , 0 < ξ 1 < ξ 2 < … < ξ m − 2 < 1 be given. This paper is concerned with the problem of existence of a solution for the following boundary value problems x ″ ( t ) = f ( t , x ( t ) , x ′ ( t ) ) + e ( t ) , 0 < t < 1 , x ′ ( 0 ) = 0 , x ( 1 ) = x ( η ) , x ″ ( t ) = f ( t , x ( t ) , x ′ ( t ) ) + e ( t ) , 0 < t < 1 , x ′ ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m − 2 a i x ( ξ i ) .

Conditions for the existence of a solution for the above boundary value problems are given using Leray Schauder Continuation theorem.