Electron. J. Diff. Equ., Vol. 2016 (2016), No. 108, pp. 1-15.

Superlinear singular fractional boundary-value problems

Imed Bachar, Habib Maagli

In this article, we study the superlinear fractional boundary-value problem
 D^{\alpha }u(x) =u(x)g(x,u(x)),\quad 0<x<1, \cr
 u(0)=0,\quad \lim_{x\to0^{+}} D^{\alpha -3}u(x)=0,\cr
 \lim_{x\to0^{+}} D^{\alpha -2}u(x)=\xi ,\quad u''(1)=\zeta ,
where $3<\alpha \leq 4$, $D^{\alpha }$ is the Riemann-Liouville fractional derivative and $\xi ,\zeta \geq 0$ are such that $\xi +\zeta >0$. The function $g(x,u)\in C((0,1)\times [ 0,\infty ),[0,\infty))$ that may be singular at x=0 and x=1 is required to satisfy convenient hypotheses to be stated later. By means of a perturbation argument, we establish the existence, uniqueness and global asymptotic behavior of a positive continuous solution to the above problem.An example is given to illustrate our main results.

Submitted February 8, 2016. Published April 26, 2016.
Math Subject Classifications: 34A08, 34B15, 34B18, 34B27.
Key Words: Fractional differential equation; positive solution; Green's function; perturbation arguments.

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Imed Bachar
King Saud University, College of Science
Mathematics Department, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: abachar@ksu.edu.sa
Habib Mâagli
King Abdulaziz University, Rabigh Campus
College of Sciences and Arts, Department of Mathematics
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: abobaker@kau.edu.sa, habib.maagli@fst.rnu.tn

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