IJPAM: Volume 85, No. 2 (2013)

POSITIVE SOLUTIONS OF SUMMATION BOUNDARY VALUE
PROBLEM FOR A GENERALIZED SECOND-ORDER
DIFFERENCE EQUATION

Thanin Sitthiwirattham1, Jiraporn Reunsumrit2
1,2Department of Mathematics
Faculty of Applied Science
King Mongkut's University of Technology
North Bangkok, Bangkok, 10800, THAILAND
1Centre of Excellence in Mathematics
CHE, 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
\begin{align}
&\Delta^2y(t-1)+a(t)f(y(t))=0,~~~~~~t\in \{1,2,...,T\},\nonumber\...
...a y(s),~~~y(T+1)=\alpha\displaystyle \sum_{s=1}^\eta y(s),\nonumber
\end{align}
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 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 85
Issue: 2