IJPAM: Volume 53, No. 3 (2009)

POSITIVE SOLUTION FOR THIRD ORDER THREE-POINT
BOUNDARY VALUE PROBLEMS ON TIME SCALES

K.R. Prasad$^1$, N.V.V.S. Suryanarayana$^2$
$^1$Department of Applied Mathematics
Andhra University
Visakhapatnam, 530 003, INDIA
e-mail: [email protected]
$^2$Department of Mathematics
VITAM College of Engineering
Visakhapatnam, 531 173, A.P. INDIA
e-mail: [email protected]


Abstract.We consider the three point third order boundary value problem on time scales

\begin{displaymath}y^{\Delta^{3}}(t)+ f(t, y(t), y^{\Delta}(t), y^{\Delta^{2}}(t))=0,~~~t\in[t_{1}, \sigma^{3}(t_{3})]\,,\end{displaymath}

subject to the general boundary conditions

\begin{displaymath}\begin{aligned} \alpha_{11}y(t_{1})+\alpha_{12}y^{\Delta}(t_... ...\alpha_{33}y^{\Delta^{2}}(\sigma(t_{3}))&=0\,, \end{aligned} \end{displaymath}

where $t_{1}<t_{2}<\sigma^{3}(t_{3})$ and $ \alpha_{ij},$ for $i,j=1,2,3$ are real constants. We establish a criterion for the existense of at least one positive solution by utilizing Krasnosel'skii Fixed Point Theorem for operators on a cone in a Banach space.

Received: May 13, 2009

AMS Subject Classification: 34B99, 39A99

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

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 3