Several results for high dimensional singular fractional systems involving $n^2$ Caputo derivatives

Amele TAÏEB

doi:10.26637/MJM0603/0017

Abstract

In this paper, we introduce a high dimensional systems of singular fractional nonlinear differential equations involving $n^2$ Caputo derivatives. Using Schauder fixed point theorem and the contraction mapping principle, we investigate new existence and uniqueness results. Furthermore, we define and study the Ulam-Hyers stability and the generalized Ulam-Hyers stability for such systems. The application of the main results is illustrated by some examples.

Keywords:

fixed point, singular fractional differential equation, existence, uniqueness, Ulam-Hyers stability, Caputo derivative, generalized Ulam-Hyers stability

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Amele TAÏEB LPAM, Faculty ST, UMAB Mostaganem, Algeria.

Pages: 569-581
Date Published: 01-07-2018
Vol. 6 No. 03 (2018): Malaya Journal of Matematik (MJM)

M.A. Abdellaoui, Z. Dahmani and N. Bedjaoui, New Existence Results For A Coupled System On Nonlinear Differential Equations Of Arbitrary Order, IJNAA. $6(2015), 65-75$.

R.P. Agarwal, D. O'Regan and S. Staněk, Positive Solutions For Dirichlet Problems Of Singular Nonlinear Fractional Differential Equations, J. Math. Anal. Appl. $371(2010), 57-68$.

R. P. Agarwal, D. O'Regan and S. Staněk, Positive Solutions For Mixed Problems Of Singular Fractional Differential Equations, Mathematische Nachrichten. 285(2012), $27-41$

C. Bai, J. Fang, The Existence Of A Positive Solution For A Singular Coupled System Of Nonlinear Fractional Differential Equations Appl. Math. Comput. (2004), 611621 .

Z. Bai, W. Sun, Existence And Multiplicity Of Positive Solutions For Singular Fractional Boundary Value Problems, Computers & Mathematics with Applications. $63(2012), 1369-1381$

D. Baleanu, S.Z. Nazemi and S. Rezapour, The Existence Of Positive Solutions For A New Coupled System Of Multiterm Singular Fractional Integrodifferential Boundary Value Problems, Abstract And Applied Analysis. $(2013), 15 mathrm{pp}$.

Z. Dahmani, A. Taïeb, New Existence And Uniqueness Results For High Dimensional Fractional Differential Systems, Facta Nis Ser. Math. Inform. 30(2015), 281293.

${ }^{[8]}$ Z. Dahmani, A. Taïeb, Solvability For High Dimensional Fractional Differential Systems With High Arbitrary Orders, Journal Of Advanced Scientific Research In Dynamical And Control Systems. 7(2015), 51-64.

Z. Dahmani, A. Taïeb, A Coupled System Of Fractional Differential Equations Involing Two Fractional Orders, ROMAI Journal. 11 (2015), 141-177.

${ }^{[10]}$ Z. Dahmani, A. Taïeb, Solvability Of A Coupled System Of Fractional Differential Equations With Periodic And Antiperiodic Boundary Conditions, PALM Letters. (2015), $29-36$.

Z. Dahmani, A. Taïeb and N. Bedjaoui, Solvability And Stability For Nonlinear Fractional Integro-Differential Systems Of High Fractional Orders, Facta Nis Ser. Math. Inform. 31(2016), 629-644.

R. Hilfer, Applications Of Fractional Calculus In Physics, World Scientific, River Edge, New Jersey. (2000).

M. Houas, Z. Dahmani, On Existence Of Solutions For Fractional Differential Equations With Nonlocal Multi-Point Boundary Conditions, Lobachevskii. J. Math. $37(2016), 120-127$.

R.W. Ibrahim, Stability Of A Fractional Differential Equation, International Journal Of Mathematical, Computational, Physical And Quantum Engineering. 7 (2013), $300-305$.

R.W. Ibrahim, Ulam Stability Of Boundary Value Problem, Kragujevac Journal Of Mathematics. 37(2013), 287297.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory And Applications Of Fractional Differential Equations, Elsevier B.V., Amsterdam, The Netherlands. (2006).

R. Li, Existence Of Solutions For Nonlinear Singular Fractional Differential Equations With Fractional Derivative Condition, Advances In Difference Equations. (2014).

Z. Lin, W. Wei and J.R. Wang, Existence And Stability Resullts For Impulsive Integro-Differential Equations, Ser. Math. Inform. 29(2014), 119-130.

K.S. Miller, B. Ross, An Introduction To The Fractional Calculus And Fractional Differential Equations, Wiley, New York. (1993).

S. Staněk, The Existence Of Positive Solutions Of Singular Fractional Boundary Value problems, Computers & Mathematics With Applications. 62(2011), 1379-1388.

A. Taïeb, Z. Dahmani, A Coupled System Of Nonlinear Differential Equations Involving $mathrm{m}$ Nonlinear Terms, Georjian Math. Journal. 23(2016), 447-458.

A. Taïb, Z. Dahmani, The High Order Lane-Emden Fractional Differential System: Existence, Uniqueness And Ulam Stabilities, Kragujevac Journal of Mathematics. $40(2016), 238-259$.

A. Taïeb, Z. Dahmani, A New Problem Of Singular Fractional Differential Equations, Journal Of Dynamical Systems And Geometric Theory. 14(2016), 161-183.

A. Taïeb, Z. Dahmani, On Singular Fractional Differential Systems And Ulam-Hyers Stabilities, International Journal Of Modern Mathematical Sciences. 14(2016), 262-282.

A. Taïeb, Z. Dahmani, Fractional System of Nonlinear Integro-Differential Equations, To appear in Journal of Fractional Calculus and Applications, 2019