IJPAM: Volume 74, No. 1 (2012)

POSITIVE SOLUTIONS TO A SECOND ORDER
$m$-POINT INTEGRAL BOUNDARY VALUE PROBLEM

Phollakrit Thiramanus$^1$, Jessada Tariboon$^2$
$^1$Department of Mathematics
Faculty of Applied Science
King Mongkut's University of Technology
North Bangkok, Bangkok, 10800, THAILAND
$^2$Centre of Excellence in Mathematics
CHE, Sri Ayutthaya Road
Bangkok 10400, THAILAND


Abstract. In this paper, by using the fixed-point index theorems, we study the existence of at least one or two positive solutions to the nonlinear second order $m$-point integral boundary value problem

\begin{displaymath}u''(t)+a(t)f(u(t))=0,\qquad 0<t<1,\end{displaymath}


\begin{displaymath}u(0)=0,\qquad u(1)=\sum_{i=1}^{m-1}\alpha_i\int_{\eta_{i-1}}^{\eta_i}u(s)ds,\end{displaymath}

where $0=\eta_0<\eta_1<\eta_2<\cdots<\eta_{m-2}<\eta_{m-1}=1$, $0<\sum_{i=1}^{m-1}\alpha_i(\eta_i^2-\eta_{i-1}^2)<2$, $\alpha_i\geq 0$ for $i\in\{1, \ldots, m-3\}\cup\{m-1\}$ and $\alpha_{m-2}>0$. $a\in C([0, 1],[0, \infty))$ and $f\in C([0,
\infty), [0, \infty))$. As an application, we give some examples that illustrate our results.

Received: November 11, 2011

AMS Subject Classification: 34B10, 34B15

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

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 74
Issue: 1