Sharp estimates on the solutions to combined fractional boundary value problems on the half-line

Volume 9, Issue 5, pp 2331--2346 http://dx.doi.org/10.22436/jnsa.009.05.35

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Authors

Imed Bachar - College of Science, Mathematics Department, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia. Habib Maagli - College of Sciences and Arts, Rabigh Campus, Department of Mathematics, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia.

Abstract

We prove the existence and the uniqueness of a positive solution to the following combined fractional boundary value problem on the half-line \[ \begin{cases} D^\alpha u(t)+a_1(t)u^{\sigma_1}+ a_2(t)u^{\sigma_2}=0,\,\,\,\,\, t\in (0,\infty), 1<\alpha<2\\ \lim_{t\rightarrow 0}t^{2-\alpha}u(t)=0,\lim_{t\rightarrow \infty}t^{1-\alpha}u(t) =0, \end{cases} \] where \(D^\alpha\) is the standard Riemann{Liouville fractional derivative, \(\sigma_1; \sigma_2 \in (-1; 1)\), and \(a_1; a_2\) are non-negative continuous functions on (\(0,\infty\)), which may be singular at t = 0 and satisfying some convenient assumptions related to the Karamata regular variation theory. We also give sharp estimates on such solution.

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Keywords

• Riemann-Liouville fractional derivative
• Green's function
• Karamata regular variation theory
• positive solution
• fixed point theorem.

MSC

•  34B40
•  34B18
•  34B27

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