Abstract and Applied Analysis
Volume 2011 (2011), Article ID 391971, 14 pages
http://dx.doi.org/10.1155/2011/391971
Research Article

Existence of Positive Solutions for a Class of Delay Fractional Differential Equations with Generalization to N-Term

Azizollah Babakhani1 and Dumitru Baleanu2,3

1Faculty of Basic Science, Babol University of Technology, Babol 47148-71167, Iran
2Department of Mathematics and Computer Science, Faculty of Arts and Sciences Cankaya University, 06530 Balgat, Ankara, Turkey
3Institute of Space Sciences, P.O. Box, MG-23, RO 76900 Magurele-Bucharest, Romania

Received 25 July 2010; Accepted 17 March 2011

Academic Editor: Yoshikazu Giga

Copyright © 2011 Azizollah Babakhani and Dumitru Baleanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We established the existence of a positive solution of nonlinear fractional differential equations 𝔏 ( 𝐷 ) [ 𝑥 ( 𝑡 ) − 𝑥 ( 0 ) ] = 𝑓 ( 𝑡 , 𝑥 𝑡 ) , 𝑡 ∈ ( 0 , 𝑏 ] with finite delay 𝑥 ( 𝑡 ) = 𝜔 ( 𝑡 ) , 𝑡 ∈ [ − 𝜏 , 0 ] , where l i m 𝑡 → 0 𝑓 ( 𝑡 , 𝑥 𝑡 ) = + ∞ , that is, 𝑓 is singular at 𝑡 = 0 and 𝑥 𝑡 ∈ 𝐶 ( [ − 𝜏 , 0 ] , ℝ ≥ 0 ) . The operator of 𝔏 ( 𝐷 ) involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.