# IJPAM: Volume 85, No. 2 (2013)

**POSITIVE SOLUTIONS OF SUMMATION BOUNDARY VALUE**

PROBLEM FOR A GENERALIZED SECOND-ORDER

DIFFERENCE EQUATION

PROBLEM FOR A GENERALIZED SECOND-ORDER

DIFFERENCE EQUATION

Thanin Sitthiwirattham

Faculty of Applied Science

King Mongkut's University of Technology

North Bangkok, Bangkok, 10800, THAILAND

CHE, Sri Ayutthaya Road, Bangkok, 10400, THAILAND

^{1}, Jiraporn Reunsumrit^{2}^{1,2}Department of MathematicsFaculty of Applied Science

King Mongkut's University of Technology

North Bangkok, Bangkok, 10800, THAILAND

^{1}Centre of Excellence in MathematicsCHE, Sri Ayutthaya Road, Bangkok, 10400, THAILAND

**Abstract. **In this paper, by using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem

where f is continuous, T≥3 is a fixed positive integer, η∈{1,2,...,T-1}, 0<α<(2T+2)/(η(η+1), 0<β<(2T+2-αη(η+1))/(η(2T-η+1)) and Δ y(t-1)=y(t)-y(t-1) is the forward difference operator. We show the existence of at least one positive solution if f is neither superlinear and sublinear by applying the fixed point theorem in cones.

**Received: **December 18, 2012

**AMS Subject Classification: **39A10

**Key Words and Phrases: **positive solution, boundary value problem, fixed point theorem, cone

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**DOI: 10.12732/ijpam.v85i2.12**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2013

**Volume:**85

**Issue:**2