Electron. J. Diff. Equ., Vol. 2014 (2014), No. 37, pp. 1-10.

Existence of positive solutions for p-Laplacian an m-point boundary value problem involving the derivative on time scales

Abdulkadir Dogan

Abstract:
We are interested in the existence of positive solutions for the -Laplacian dynamic equation on time scales,
$$ (\phi_p(u^\Delta(t)))^\nabla+a(t)f(t,u(t),u^\Delta(t))=0,\quad t\in(0,T)_{\mathbb{T}}, $$
subject to the multipoint boundary condition,
$$ u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i), \quad u^\Delta(T)=0, $$
where $\phi_p(s)=|s|^{p-2} s$, $p>1$, $\xi_i\in [0,T]_{\mathbb{T}}$, $ 0<\xi_1<\xi_2<\dots<\xi_{m-2}<\rho(T)$. By using fixed point theorems, we prove the existence of at least three non-negatvie solutions, two of them positive, to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first order derivative explicitly. An example is given to illustrate the main result.

Submitted December 3, 2013. Published January 30, 2014.
Math Subject Classifications: 34B15, 34B16, 34B18, 39A10.
Key Words: Time scales; boundary value problem; p-Laplacian; positive solution; fixed point theorem.

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Abdulkadir Dogan
Department of Applied Mathematics
Faculty of Computer Sciences
Abdullah Gul University, Kayseri, 38039 Turkey
Tel: +90 352 224 88 00 Fax: +90 352 338 88 28
email: abdulkadir.dogan@agu.edu.tr

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