The method of lower and upper solutions for 2nth-order multi-point boundary value problems

Abstract In this paper, we consider the following 2 n th-order multi-point boundary value problems ( − 1 ) n u ( 2 n ) ( t ) = f ( t , u ( t ) , − u ″ ( t ) , … , ( − 1 ) i u ( 2 i ) ( t ) , … , ( − 1 ) n − 1 u ( 2 n − 2 ) ( t ) ) , 0 t 1 , with the 2 ( n − m ) -point boundary conditions u ( 2 j ) ( 0 ) = u ( 2 j ) ( 1 ) = 0 , j = 0 , 1 , 2 , … , m , m ≤ n − 2 , u ( 2 j ) ( 0 ) − a j u ( 2 j ) ( ξ j ( 1 ) ) = 0 , u ( 2 j ) ( 1 ) − b j u ( 2 j ) ( ξ j ( 2 ) ) = 0 , j = m + 1 , … , n − 1 , where a j ≥ 0 , b j ≥ 0 , 0 ξ j ( 1 ) ξ j ( 2 ) ≤ 1 , j = m + 1 , … , n − 1 . The existence of iterative solutions is obtained by using the lower and upper solution method for the above 2 ( n − m ) -point boundary value problems.

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