Qualitative properties of solutions of fractional order boundary value problems

Document Type : Research Paper


1 Faculty of Science, Alexandria University, Alexandria, Egypt

2 Department of Mathematics, College of Science, Qassim University, P.O. Box 6644 Buraidah 51452, Saudi Arabia

3 Department of Mathematics, Lebanese International University, Saida, Lebanon

4 Department of Mathematics, The International University of Beirut, Beirut, Lebanon


In this article, we discuss two boundary value problems for fractional-order differential equations. We show unique solutions exist and some data continuous dependence, with aim of proving some characteristics for these solutions of a coupled system of conjugate orders. These coupled systems are equivalent to coupled systems of second-order differential equations. Therefore, the analysis of the spectra of these problems is a consequence of that of second-order differential equations.


[1] T.S. Aleroev, Boundary-Value Problems for Differential Equations with Fractional Derivatives, Dissert. on Doctoral Degree Phys.-Math. Sci., MGU, Moscow, 2000.
[2] R.P. Agarwal, D. O’Regan and P.J.Y. Wong, Eigenvalues of a system of fredholm integral equations, Math.
Comput. Modell. 39 (2004) 1113–1150.
[3] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary
value conditions, J. Math. Anal. Appl. 389(1) (2012) 403–411 .
[4] D. Chalishajar and A. Kumar, Existence, uniqueness and Ulam’s stability of solutions for a coupled system of
fractional differential equations with integral boundary conditions, Math. 6(6) (2018) 96.
[5] A.M.A. El-Sayed and H.H.G. Hashem, Existence results for coupled systems of quadratic integral equations of
fractional orders, Optim. Lett. 7(6) (2013) 1251–1260.
[6] A.M.A. El-Sayed, H.H.G. Hashem and Sh M. Al-Issa, Characteristics of solutions of fractional hybrid integrodifferential equations in Banach algebra, Sahand Commun. Math. Anal. 18(3) (2020) 109–131.
[7] A.M.A. El-Sayed and Sh M. Al-Issa, On a set-valued functional integral equation of Volterra-Stieltjes type, J.
Math. Comput. Sci. 21(4) (2020) 273–285.
[8] A.M.A. El-Sayed, Sh M. Al-Issa and N.M. Mawed, Results on solvability of nonlinear quadratic integral equations
of fractional orders in Banach algebra, J. Nonlinear Sci. Appl. 14(4) (2021) 181–195.
[9] K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.
[10] J. Henderson and R. Luca, Positive solutions for a system of coupled fractional boundary value problems, Lith.
Math. J. 58(1) (2018) 15—32.
[11] M. Jleli, D. O’Regan and B. Samet, Lyapunov-type inequalities for coupled systems of nonlinear fractional differential equations via a fixed point approach, J. Fixed Point Theory Appl. 21(2) (2019) 1–15.
[12] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dovor Publ. Inc., 1975.
[13] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations,
Elsevier, North-Holland, 2006.
[14] W. Kumam, M.B. Zada, K. Shah and R. A. Khan, Investigating a coupled Hybrid system of nonlinear fractional
differential equations, Discrete Dyn. Nat. Soc. 2018 (2018) Article ID 5937572.
[15] K. Shah, J. Wang, H. Khalil and R.A. Khan, Existence and numerical solutions of a coupled system of integral
BVP for fractional differential equations, Adv. Difference Equ. 2018(1) (2018) 1–21.
[16] A. Shidfara and A. Molabahrami, Solving a system of integral equations by an analytic method, Math. Comput.
Modell. 54 (2011) 828-835.
[17] G.T. Wang, S.Y. Liu and L.H. Zhang, Eigenvalue problem for nonlinear fractional differential equations with
integral boundary conditions, Abstr. Appl. Anal. 2014 (2014) 916260.
[18] C. Yujun and S. Jingxian, On existence of positive solutions of coupled integral boundary value problems for a
nonlinear singular superlinear differential system, Electron. J. Qual. Theory Differential Equ. 41 (2012) 1–13.
[19] A.Y. Al-Hossain, Eigenvalues for iterative systems of nonlinear Caputo fractional order three point boundary value
problems, J. Appl. Math. Comput. 52 (2016) 157-172.
[20] He. Ying, The eigenvalue problem for a coupled system of singular p-Laplacian differential equations involving
fractional differential-integral conditions, Adv. Diff. Equ. 2016 (2016) 209.
[21] X.G. Zhang, L.S. Liu, B. Wiwatanapataphee and YH. Wu, The eigenvalue for a class of singular p-Laplacian
fractional differential equations involving the Riemann-Stieltjes integral boundary condition, Appl. Math. Comput.
235 (2014) 412–422 .
Volume 13, Issue 1
March 2022
Pages 3427-3440
  • Receive Date: 30 March 2021
  • Accept Date: 28 May 2021
  • First Publish Date: 25 January 2022