Electron. J. Diff. Equ., Vol. 2009(2009), No. 127, pp. 1-9.

Positive solutions for a system of nonlinear boundary-value problems on time scales

A. Kameswara Rao

Abstract:
We determine the values of a parameter $\lambda$ for which there exist positive solutions to the system of dynamic equations
$$\displaylines{ u^{\Delta \Delta}(t)+\lambda p(t)f(v(\sigma(t)))=0,\quad t\in[a, b]_\mathbb{T}, \cr v^{\Delta \Delta}(t)+\lambda q(t)g(u(\sigma(t)))=0, \quad t\in[a, b]_\mathbb{T}, }$$
with the boundary conditions, $\alpha u(a)-\beta u^{\Delta}(a)=0$, $\gamma u(\sigma^2(b))+\delta u^{\Delta}(\sigma(b))=0$, $\alpha v(a)-\beta v^{\Delta}(a)=0$, $\gamma v(\sigma^2(b))+\delta v^{\Delta}(\sigma(b))=0$, where $\mathbb{T}$ is a time scale. To this end we apply a Guo-Krasnosel'skii fixed point theorem.

Submitted July 6, 2009. Published October 4, 2009.
Math Subject Classifications: 39A10, 34B15, 34A40.
Key Words: Dynamic equations; eigenvalue intervals; positive solution; cone.

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A. Kameswara Rao
Department of Applied Mathematics, Andhra University
Visakhapatnam 530 003, India
email: kamesh_1724@yahoo.com

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