Positive solutions for some Riemann-Liouville fractional boundary value problems

Volume 9, Issue 7, pp 5093--5106 http://dx.doi.org/10.22436/jnsa.009.07.12

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Authors

Imed Bachar - Mathematics Department, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia. Habib Mâagli - 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.

Abstract

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.

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Keywords

• Fractional differential equation
• positive solutions
• Green's function
• perturbation arguments
• Schäuder fixed point theorem.

MSC

•  34B27
•  34A08
•  34B18
•  34B15
•  47N20

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