Electron. J. Differential Equations, Vol. 2017 (2017), No. 141, pp. 1-13.

Existence and asymptotic behavior of positive solutions for semilinear fractional Navier boundary-value problems

Habib Maagli, Abdelwaheb Dhifli

Abstract:
We study the existence, uniqueness, and asymptotic behavior of positive continuous solutions to the fractional Navier boundary-value problem
$$\displaylines{ D^{\beta }(D^{\alpha }u)(x)=-p(x)u^{\sigma },\quad \in (0,1), \cr \lim_{x\to 0}x^{1-\beta }D^{\alpha}u(x)=0,\quad u(1)=0, }$$
where $\alpha ,\beta \in (0,1]$ such that $\alpha +\beta >1$, $D^{\beta }$ and $D^{\alpha }$ stand for the standard Riemann-Liouville fractional derivatives, $\sigma \in (-1,1)$ and p being a nonnegative continuous function in (0,1) that may be singular at x=0 and satisfies some conditions related to the Karamata regular variation theory. Our approach is based on the Schauder fixed point theorem.

Submitted February 11, 2017. Published May 25, 2017.
Math Subject Classifications: 34A08, 34B15, 34B18, 34B27.
Key Words: Fractional Navier differential equations; Dirichlet problem; positive solution; asymptotic behavior; Schauder fixed point theorem.

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Habib Mâagli
King Abdulaziz University, Rabigh Campus
College of Sciences and Arts
Department of Mathematics, P.O. Box 344
Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn
Abdelwaheb Dhifli
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire
2092 Tunis, Tunisia
email: dhifli_waheb@yahoo.fr

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