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

Thanin Sitthiwirattham and Jessada Tariboon

Journal of Applied Mathematics, 2012, vol. 2012, 1-15

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 ð ‘¢ ( ð ‘¡ − 1 ) + ð ‘Ž ( ð ‘¡ ) ð ‘“ ( ð ‘¢ ( ð ‘¡ ) ) = 0 , ð ‘¡ ∈ { 1 , 2 , … , 𠑇 } , ∑ ð ‘¢ ( 0 ) = ð ›½ 𠜂 ð ‘ = 1 ð ‘¢ ( ð ‘ ) , ∑ ð ‘¢ ( 𠑇 + 1 ) = ð ›¼ 𠜂 ð ‘ = 1 ð ‘¢ ( ð ‘ ) , where ð ‘“ is continuous, 𠑇 ≥ 3 is a fixed positive integer, 𠜂 ∈ { 1 , 2 , . . . , 𠑇 − 1 } , 0 < ð ›¼ < ( 2 𠑇 + 2 ) / 𠜂 ( 𠜂 + 1 ) , 0 < ð ›½ < ( 2 𠑇 + 2 − ð ›¼ 𠜂 ( 𠜂 + 1 ) ) / 𠜂 ( 2 𠑇 − 𠜂 + 1 ) , and Δ ð ‘¢ ( ð ‘¡ − 1 ) = ð ‘¢ ( ð ‘¡ ) − ð ‘¢ ( ð ‘¡ − 1 ) . We show the existence of at least one positive solution if ð ‘“ is either superlinear or sublinear.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:569313

DOI: 10.1155/2012/569313

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