Triangular System of Higher Order Singular Fractional Differential Equations
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Authors: A. TAIEB AND Z. DAHMANI
DOI: 10.46793/KgJMat2101.081T
Abstract:
In this paper, we introduce a high dimensional system of singular fractional differential equations. Using Schauder fixed point theorem, we prove an existence result. We also investigate the uniqueness of solution using the Banach contraction principle. Moreover, we study the Ulam-Hyers stability and the generalized-Ulam-Hyers stability of solutions. Some illustrative examples are also presented.
Keywords:
Caputo derivative, fixed point, singular fractional differential equation, existence, uniqueness, Ulam-Hyers stability.
References:
[1] M. A. Abdellaoui, Z. Dahmani and N. BedjaouiI, New existence results for a coupled system on nonlinear differential equations of arbitrary order, IJNAA 6(2) (2015), 65–75.
[2] R. P. Agarwal and D. O’Regan, Existence theory for singular initial and boundary value problems: a fixed point approach, Appl. Anal. 81 (2002), 391–434.
[3] R. P. Agarwal, D. O’Regan and S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57–68.
[4] C. Bai and J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150(3) (2004), 611–621.
[5] 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, Abstr. Appl. Anal. (2013), Article ID 368659, 15 pages.
[6] Z. Dahmani and A. Taïeb, Solvability of a coupled system of fractional differential equations with periodic and antiperiodic boundary conditions, Pure and Applied Mathematics Letters 1 (2015), 29–36.
[7] Z. Dahmani and A. Taïeb, New existence and uniqueness results for high dimensional fractional differential systems, Facta Univ. Ser. Math. Inform. 30(3) (2015), 281–293.
[8] Z. Dahmani and A. Taïeb, Solvability for high dimensional fractional differential systems with high arbitrary orders, Journal of Advanced Scientific Research in Dynamical and Control Systems 7(4) (2015), 51–64.
[9] Z. Dahmani and A. Taïeb, A coupled system of fractional differential equations involving two fractional orders, ROMAI J. 11(2) (2015), 141–177.
[10] Z. Dahmani, A. Taïeb and N. Bedjaoui, Solvability and stability for nonlinear fractional integro-differential systems of high fractional orders, Facta Univ. Ser. Math. Inform. 31(3) (2016), 629–644.
[11] M. Fečkan, J. Wang and Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (2012), 258–264.
[12] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, New Jersey, 2000.
[13] R. W. Ibrahim, Stability of a fractional differential equation, International Journal of Mathematical, Computational, Physical and Quantum Engineering 7(3) (2013), 300–305.
[14] R. W. Ibrahim, Ulam stability of boundary value oroblem, Kragujevac J. Math. 37(2) (2013), 287–297.
[15] S. M. Jung and T. M. Rassias, Generalized Hyers-Ulam stability of Riccati differential equation, Math. Inequal. Appl. 11 (2008), 777–782.
[16] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
[17] R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Adv. Difference Equ. 2014(292) (2014), 12 pages.
[18] Z. Lin, W. Wei and J. R. Wang, Existence and stability resullts for impulsive integro-differential equations, Ser. Math. Inform. 29(2) (2014), 119–130.
[19] F. Mainardi, Fractional calculus: some basic problems in continum and statistical mechanics, in: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 1997, 291–348.
[20] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[21] A. Taïeb and Z. Dahmani, A coupled system of nonlinear differential equations involving m nonlinear terms, Georgian Math. J. 23(3) (2016), 447–458.
[22] A. Taïeb and Z. Dahmani, The high order Lane-Emden fractional differential system: existence, uniqueness and Ulam stabilities, Kragujevac J. Math. 40(2) (2016), 238–259.
[23] A. Taïeb and Z. Dahmani, A new problem of singular fractional differential equations, J. Dyn. Syst. Geom. Theor. 14(2) (2016), 161–183.
[24] A. Taïeb and Z. Dahmani, On singular fractional differential systems and Ulam-Hyers stabilities, International Journal of Modern Mathematical Sciences 14(3) (2016), 262–282.
[25] A. Yang and W. Ge, Positive solutions for boundary value problems of n-dimension nonlinear fractional differential system, Bound. Value Probl. (2008), Article ID 437453, 15 pages.
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