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.

An addendum was posted on December 4, 2013. It states that Theorem 3.2, the example, Corollary 3.3, and Corollary 3.4 are not correct. See the last page of this article.

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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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