The Generalized Green's Function for Boundary Value Problem of Second Order Difference Equation. (23rd February 2015) - Han, Xiaoling; Huang, Juanjuan
Abstract : Letb > a + 2 and[ a + 1, b + 1 ] = { a + 1, a + 2, …, b + 1 } . In this paper, by building the generalized Green's function for the problems, we study the solvability of the S-L problemL x = Δ [ p ( t - 1 ) Δ x ( t - 1 ) ] + [ q ( t ) + λ r ( t ) ] x ( t ) = - f ( t ), U 1 ( x ) = α 1 x ( a ) + α 2 Δ x ( a ) = 0, U 2 ( x ) = β 1 x ( b + 1 ) + β 2 Δ x ( b + 1 ) = 0, and the periodic S-L problemL x = Δ [ p ( t - 1 ) Δ x ( t - 1 ) ] + [ q ( t ) + λ r ( t ) ] x ( t ) = - f ( t ), U 3 ( x ) = x ( a ) - x ( b + 1 ) = 0, U 4 ( x ) = Δ x ( a ) - Δ x ( b + 1 ) = 0, where the parameterλ is an eigenvalue of the linear problemL x = 0, U 1 ( x ) = 0, U 2 ( x ) = 0 or the problemL x = 0, U 3 ( x ) = 0, U 4 ( x ) = 0, andp : [ a, b + 1 ] → ( 0, + ∞ ), r : [ a + 1, b + 1 ] → ( 0, + ∞ ), q ( t ) is defined and real valued on[ a + 1, b + 1 ], α 1 2 + α 2 2 ≠ 0, β 1 2 + β 2 2 ≠ 0, and in the periodic S-L problem we havep ( a ) = p ( b + 1 ) .