Nonexistence of solutions to a fractional differential boundary value problem

Volume 9, Issue 5, pp 2233--2243 http://dx.doi.org/10.22436/jnsa.009.05.27

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Authors

Maysaa Al-Qurashi - College of Sciences, Mathematics department, King Saud University, P. O. Box 45 180, Riyadh 11 551, Saudi Arabia. Lakhdar Ragoub - Mathematics Department, College of Computers and Information Systems, Al Yamamah University, P. O. Box 45 180, Riyadh 11 512, Saudi Arabia.

Abstract

We investigate new results about Lyapunov-type inequality by considering a fractional boundary value problem subject to mixed boundary conditions. We give a necessary condition for nonexistence of solutions for a class of boundary value problems involving Riemann-Liouville fractional order. The order considered here is \(3 < \alpha\leq 4\). The investigation is based on a construction of Green's function and on finding its corresponding maximum value. In order to illustrate the result, we provide an application of Lyapunov-type inequality for an eigenvalue problem and we show how the necessary condition of existence can be employed to determine intervals for the real zeros of the Mittag-Leffler function.

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Keywords

• Lyapunov's inequality
• Green's function
• Riemann-Liouville derivative
• mixed boundary conditions.

MSC

•  34A08
•  34A40
•  26D10
•  33E12

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