Positive solutions for some Riemann-Liouville fractional boundary value problems

Bachar, Imed¹ ; Mâagli, Habib²

¹ Mathematics Department, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
² Department of Mathematics, College of Sciences and Arts, Rabigh Campus, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia;Department of Mathematics, Faculte des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 7, p. 5093-5106.

Résumé

We study the existence and global asymptotic behavior of positive continuous solutions to the following nonlinear fractional boundary value problem

$ (p_\lambda) \begin{cases} D^\alpha u(t)=\lambda f(t,u(t)),\,\,\,\,\, t\in (0,1),\\ \lim_{t\rightarrow 0^+}t^{2-\alpha} u(t)=\mu, \quad u(1)=\nu, \end{cases} $

where $1 \alpha\leq 2; D^\alpha$ is the Riemann-Liouville fractional derivative, and $\lambda,\mu$ and $\nu$ are nonnegative constants such that $\mu + \nu > 0$. Our purpose is to give two existence results for the above problem, where $f(t; s)$ is a nonnegative continuous function on $(0; 1)\times[0;\infty)$; nondecreasing with respect to the second variable and satisfying some appropriate integrability condition. Some examples are given to illustrate our existence results.

Détail

DOI : 10.22436/jnsa.009.07.12

Classification : 34B27, 34A08, 34B18, 34B15, 47N20
Keywords: Fractional differential equation, positive solutions, Green's function, perturbation arguments, Schäuder fixed point theorem.

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