Electron. J. Diff. Equ., Vol. 2013 (2013), No. 64, pp. 1-8.

Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order

Ahmed M. A. El-Sayed, Ebtisam O. Bin-Taher

Abstract:
In this article we study the existence of positive solutions for the nonlocal multi-point boundary-value problem
$$\displaylines{ u''(t)+f(t, ^{c}D^{\alpha}u(t))=0, \quad \alpha \in(0, 1), \hbox{ a.e. } t\in(0, 1), \cr u(0)=0, \quad u(1)=\sum_{k=1}^m a_k u(\tau_k), \quad \tau_k\in(a, b)\subset (0, 1). }$$
We also consider the corresponding integral condition, and the two special cases $\alpha = 0 $ and $ \alpha = 1$.

Submitted March 19, 2012. Published March 5, 2013.
Math Subject Classifications: 34B10, 26A33
Key Words: Fractional calculus; boundary value problem; nonlocal condition; integral condition; positive solution.

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Ahmed M. A. El-Sayed
Faculty of Science, Alexandria University
Alexandria, Egypt
email: amasayed@hotmail.com
  Ebtisam O. Bin-Taher
Faculty of Science
Hadhramout Univeristy of Science and Technology
Hadhramout, Yemen
email: ebtsamsam@yahoo.com

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