The Generalized Green's Function for Boundary Value Problem of Second Order Difference Equation. (23rd February 2015)

• Record Type: Journal Article
• Authors: Han, Xiaoling; Huang, Juanjuan
• Other Names: Sadarangani Kishin Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2015(2015)
• Publication Date: 2015-02-23
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2015/201946
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : Let b > 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 problem L 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 problem L 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 parameter λ is an eigenvalue of the linear problem L x = 0, U 1 ( x ) = 0, U 2 ( x ) = 0 or the problem L x = 0, U 3 ( x ) = 0, U 4 ( x ) = 0, and p : [ 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 have p ( a ) = p ( b + 1 ) .