Solvability for Two Classes of Higher-Order Multi-Point Boundary Value Problems at Resonance. (21st February 2008) - Gao, Yunzhu; Pei, Minghe
Abstract : Using the theory of coincidence degree, we establish existence results of positive solutions for higher-order multi-point boundary value problems at resonance for ordinary differential equationu ( n ) ( t ) = f ( t, u ( t ), u ′ ( t ), …, u ( n − 1 ) ( t ) ) + e ( t ), t ∈ ( 0, 1 ), with one of the following boundary conditions:u ( i ) ( 0 ) = 0, i = 1, 2, …, n − 2, u ( n − 1 ) ( 0 ) = u ( n − 1 ) ( ξ ), u ( n − 2 ) ( 1 ) = ∑ j = 1 m − 2 β j u ( n − 2 ) ( η j ), andu ( i ) ( 0 ) = 0, i = 1, 2, …, n − 1, u ( n − 2 ) ( 1 ) = ∑ j = 1 m − 2 β j u ( n − 2 ) ( η j ), wheref : [ 0, 1 ] × ℝ n → ℝ = ( − ∞, + ∞ ) is a continuous function, e ( t ) ∈ L 1 [ 0, 1 ] β j ∈ ℝ ( 1 ≤ j ≤ m − 2, m ≥ 4 ), 0 < η 1 < η 2 < ⋯ < η m − 2 < 1, 0 < ξ < 1, all theβ − j − s have not the same sign. We also give some examples to demonstrate our results.