Electron. J. Differential Equations, Vol. 2017 (2017), No. 240, pp. 1-16.

Positive solutions for superlinear Riemann-Liouville fractional boundary-value problems

Imed Bachar, Habib Maagli, Vicentiu D. Radulescu

Abstract:
Using a perturbation argument, we establish the existence and uniqueness of a positive continuous solution for the following superlinear Riemann-Liouville fractional boundary-value problem
$$\displaylines{ D^{\alpha }u( x) -u(x)\varphi (x,u(x))=0,\quad 0<x<1,\cr u(0)=u'(0)=\lim_{x\to 0^{+}} x^{4-\alpha}u''(x)=0,\quad u''(1)=a>0, }$$
where $3<\alpha \leq 4$ and $\varphi (x,t)$ satisfies a suitable integrability condition.

Submitted August 18, 2017. Published October 4, 2017.
Math Subject Classifications: 34A08, 34B15, 34B18, 34B27.
Key Words: Fractional differential equation; positive solution; Green's function; perturbation; Schauder fixed point theorem.

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Imed Bachar
King Saud University
Department of Mathematics, College of Science
P.O. Box 2455, Riyadh 11451, Saudi Arabia
email: abachar@ksu.edu.sa
Habib Mâagli
Department of Mathematics
College of Sciences and Arts
King Abdulaziz University
Rabigh Campus P.O. Box 344
Rabigh 21911, Saudi Arabia
email: abobaker@kau.edu.sa, habib.maagli@fst.rnu.tn
Vicentiu D. Radulescu
Department of Mathematics
Faculty of Sciences
King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia
email: vicentiu.radulescu@math.cnrs.fr

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