On the existence of solution for a singular Riemann-Liouville fractional differential system by using measure of non-compactness


1 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

2 Department of Mathematics, Islamic Azad University, Ilam Branch, Ilam, Iran


We investigate the existence of solution for a singular fractional differential system with Riemann-Liouville
integral boundary conditions by using the measure of non-compactness.


[1] Z. Bai, T. Qui, Existence of positive solution for singular fractional differential equation, Appl. Math. Comput.,
215 (2009), 2761–2767. 

[2] G. Darbo, Punti uniti in transformazioni a condominio non compatto, Rend. Sem. Math. Univ. Padova., 24
(1955), 84–92. 

[3] Y. Liu, H. Shi, Existence of unbounded positive solutions for BVPs of singular fractional differential equations,
J. Nonlinear Sci. App., 5 (2012), 281–293. 

[4] E. Malkowsky, V. Rakocevi, An introduction into the theory of sequence spaces and measure of non-compactness,
Zb. rad. Beogr., 17 (2000), 143–234. 

[5] I. Podlubny, Fractional differential equations, Academic Press, (1999).

[6] Sh. Rezapour, M. Shabibi, A singular fractional differential equation with Riemann-Liouville integral boundary
condition, J. Adv. Math. Stud., 8 (2015), 80–88. 

[7] M. Shabibi, Sh. Rezapour, S. M. Vaezpour, A singular fractional integro-differential equation, to appear in Univ.
Politech. Buch. Sci. Bull. Ser. A. 

[8] S. Stanek, The existence of positive solutions of singular fractional boundary value problems, Comput. Math.
Appl., 62 (2011), 1379–1388.

[9] N. E. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, J. Inequal. Appl., 2006
(2006), 12 pages. 

[10] Y. Wang, Positive solution for a system of fractional integral boundary value problem, Bound. Value probl., 2013
(2013), 2013, 14 pages. 

[11] R. Yan, Sh. Sun, H. Lu, Y. Zhao, Existence of solutions for fractional differential equations with integral boundary
conditions, Adv. Difference Equ., 2014 (2014), 13 pages.