Existence of unbounded positive solutions for BVPs of singular fractional differential equations

1123
Downloads

1633
Views
Authors
Yuji Liu
 Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, P. R. China.
Haiping Shi
 Basic Courses Department, Guangdong Construction Vocational Technology Institute, Guangzhou 510450, P. R. China.
Abstract
In this article, we establish the existence of multiple unbounded positive solutions to the boundary value
problem of the nonlinear singular fractional differential equation
\[
\begin{cases}
D^\alpha_{ 0^+}u(t) + f(t; u(t)) = 0; t \in (0; 1); 1 < \alpha < 2,\\
[I^{2\alpha}_{ 0^+} u(t)]'_{t=0} = 0\\
u(1) = 0.
\end{cases}
\]
Our analysis relies on the well known fixed point theorems in the cones in Banach spaces. Here \(f\) is singular
at \(t = 0\) and \(t = 1\).
Share and Cite
ISRP Style
Yuji Liu, Haiping Shi, Existence of unbounded positive solutions for BVPs of singular fractional differential equations, Journal of Nonlinear Sciences and Applications, 5 (2012), no. 4, 281293
AMA Style
Liu Yuji, Shi Haiping, Existence of unbounded positive solutions for BVPs of singular fractional differential equations. J. Nonlinear Sci. Appl. (2012); 5(4):281293
Chicago/Turabian Style
Liu, Yuji, Shi, Haiping. "Existence of unbounded positive solutions for BVPs of singular fractional differential equations." Journal of Nonlinear Sciences and Applications, 5, no. 4 (2012): 281293
Keywords
 Singular fractional differential equation
 boundary value problem
 unbounded positive solution
 Fixed Point Theorem.
MSC
References

[1]
K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation , Wiley, New York (1993)

[2]
S. G. Samko, A. A. Kilbas, O. I. Marichev , Fractional Integral and Derivative, Theory and Applications, Gordon and Breach (1993)

[3]
Z. Bai, H. Lv , Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl. , 311 (2005), 495505.

[4]
A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problemsI, Applicable Analysis, 78 (2001), 153192.

[5]
A. Arara, M. Benchohra, N. Hamidi, J. J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis TMA, 72 (2010), 580586

[6]
Z. Bai, On positive solutions of a nonlocal fractional boundary value problem , Nonlinear Analysis, 72 (2010), 916924.

[7]
R. Dehghant, K. Ghanbari, Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bulletin of the Iranian Mathematical Society, 33 (2007), 114.

[8]
S. Z. Rida, H. M. ElSherbiny, A. A. M. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Physics Letters A, 372 (2008), 553558.

[9]
X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis TMA, 71 (2009), 46764688.

[10]
F. Zhang , Existence results of positive solutions to boundary value problem for fractional differential equation, Positivity, 13 (2008), 583599.

[11]
R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28 (1979), 673688.