Journal of Applied Mathematics

Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem

Thanin Sitthiwirattham and Jessada Tariboon

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Abstract

By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Δ 2 u ( t - 1 ) + a ( t ) f ( u ( t ) ) = 0 , t { 1,2 , , T } , u ( 0 ) = β s = 1 η u ( s ) , u ( T + 1 ) = α s = 1 η u ( s ) , where f is continuous, T 3 is a fixed positive integer, η { 1,2 , . . . , T - 1 } , 0 < α < ( 2 T + 2 ) / η ( η + 1 ) , 0 < β < ( 2 T + 2 - α η ( η + 1 ) ) / η ( 2 T - η + 1 ), and Δ u ( t - 1 ) = u ( t ) - u ( t - 1 ) . We show the existence of at least one positive solution if f is either superlinear or sublinear.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 569313, 15 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495127

Digital Object Identifier
doi:10.1155/2012/569313

Mathematical Reviews number (MathSciNet)
MR2923340

Zentralblatt MATH identifier
1244.39008

Citation

Sitthiwirattham, Thanin; Tariboon, Jessada. Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem. J. Appl. Math. 2012 (2012), Article ID 569313, 15 pages. doi:10.1155/2012/569313. https://projecteuclid.org/euclid.jam/1355495127


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