IJPAM: Volume 56, No. 2 (2009)

MULTIPLE POSITIVE SOLUTIONS FOR NONLINEAR
THIRD ORDER GENERAL TWO-POINT BOUNDARY
VALUE PROBLEMS ON TIME SCALES

K.R. Prasad$^1$, P. Murali$^2$, S. Nageswara Rao$^3$
$^{1,2}$Department of Applied Mathematics
Andhra University
Visakhapatnam, 530 003, INDIA
$^1$e-mail: rajendra92@rediffmail.com
$^2$e-mail: murali$_{-}$uoh@yahoo.co.in
$^{3}$Department of Mathematics
Sri Prakash College of Engineering, Tuni, 533 401, A.P., INDIA
e-mail: sabbavarapu$_{-}$nag@yahoo.co.in


Abstract.We consider third order nonlinear general two-point boundary value problem on time scales,

\begin{displaymath} y^{\Delta\Delta\Delta}(t)+f(t,y(t))=0,\qquad t\in[a,b],\quad a<b, \end{displaymath}

subject to the general boundary conditions
\begin{align*} &\alpha_{11}y(a)-\alpha_{12}y(b)=0,\\ &\alpha_{21}y^{\Delta}(a... ..._{31}y^{\Delta\Delta}(a)-\alpha_{32}y^{\Delta\Delta}(\sigma(b))=0, \end{align*}
where the coefficients $\alpha_{1i}, \alpha_{2i},\alpha_{3i}$, $i=1,2$ are real positive constants. We establish the existence of at least one positive solution, and existence of at least three positive solutions for the two-point boundary value problem using Krasnosel'skii and Leggett-Williams fixed point theorems on a cone.

Received: August 25, 2009

AMS Subject Classification: 34B99, 39A99

Key Words and Phrases: time scales, boundary value problem, positive solution, cone, multiple positive solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 2