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

Superlinear singular fractional boundary-value problems

Imed Bachar, Habib Maagli

Abstract:
In this article, we study the superlinear fractional boundary-value problem
$$\displaylines{ D^{\alpha }u(x) =u(x)g(x,u(x)),\quad 0<x<1, \cr u(0)=0,\quad \lim_{x\to0^{+}} D^{\alpha -3}u(x)=0,\cr \lim_{x\to0^{+}} D^{\alpha -2}u(x)=\xi ,\quad u''(1)=\zeta , }$$
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.

Show me the PDF file (267 KB), TEX file, and other files for this article.

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

Return to the EJDE web page