Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems. (19th August 2009) - Pei, Minghe; Chang, Sung Kag
Abstract : We are concerned with the higher-order nonlinear three-point boundary value problems:x ( n ) = f ( t, x, x ′, …, x ( n − 1 ) ), n ≥ 3, with the three point boundary conditionsg ( x ( a ), x ′ ( a ), …, x ( n − 1 ) ( a ) ) = 0 ;x ( i ) ( b ) = μ i, i = 0, 1, …, n − 3 ; h ( x ( c ), x ′ ( c ), …, x ( n − 1 ) ( c ) ) = 0, wherea < b < c, f : [ a, c ] × ℝ n → ℝ = ( − ∞, + ∞ ) is continuous, g, h : ℝ n → ℝ are continuous, andμ i ∈ ℝ, i = 0, 1, …, n − 3 are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.