IJPAM: Volume 52, No. 5 (2009)

POSITIVE SOLUTIONS FOR STURM-LIOUVILLE
BOUNDARY VALUE PROBLEMS ON A MEASURE CHAIN

Sheng-Quan Liang$^1$, Jian-Ping Sun$^2$
$^1$Gansu Polytechnic College of
Animal Husbandry and Engineering
Huangyang Town, Wuwei, Gansu, 733006, P.R. CHINA
$^2$Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, P.R. CHINA
e-mail: [email protected]


Abstract.In this paper we consider the following differential equation on a measure chain

\begin{displaymath} u^{\Delta \Delta }(t)+f(t,u(\sigma (t)))=0,t\in [a,b], \end{displaymath}

satisfying Sturm-Liouville boundary value condition
\begin{align*} &\alpha u(a)-\beta u^\Delta (a) =0, \\ &\gamma u(\sigma (b))+\delta u^\Delta (\sigma (b)) = 0. \end{align*}
Some results of the existence and multiplicity are obtained by using Krasnoselskii's Fixed Point Theorem in a cone. In particular, it is proved that the above boundary value problem has $N$ positive solutions under suitable conditions, where $N$ is an arbitrary positive integer.

Received: February 27, 2008

AMS Subject Classification: 34B15, 39A10

Key Words and Phrases: measure chain, positive solution, cone, fixed point

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 5