Existence and Stability Analysis of Sequential Coupled System of Hadamard-Type Fractional Differential Equations
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Authors: A. ZADA AND M. YAR
DOI: 10.46793/KgJMat2201.085Z
Abstract:
In this paper we study existence, uniqueness and Hyers-Ulam stability for a sequential coupled system consisting of fractional differential equations of Hadamard type, subject to nonlocal Hadamard fractional integral boundary conditions. The existence of solutions is derived from Leray-Schauder’s alternative, whereas the uniqueness of solution is established by Banach contraction principle. An example is also presented which illustrate our results.
Keywords:
Hadamard fractional derivative, sequential coupled system, fixed point theorem, Hyers-Ulam stability.
References:
[1] G. Adomian and G. E. Adomian, Cellular systems and aging models, Comput. Math. Appl. 11 (1985), 283–291.
[2] B. Ahmad, J. Juan, J. Nieto and A. Alseadi, A coupled system of Caputo-type sequential fractional differential equations with coupled (periodic/anti-periodic type) boundary conditions, Mediterr. J. Math. 2017 (2017), 227.
[3] B. Ahmad and S. K. Ntouayas, A fully Hadamard type integral boundary value problome of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal. 17 (2014), 348–360.
[4] Z. Ali, A. Zada and K. Shah, On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations, Bull. Malays. Math. Sci. Soc. 42 (2019), 2681–2699.
[5] R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology 27 (1983), 201–210.
[6] K. W. Blayneh, Analysis of age structured host-parasitoid model, Far East Journal of Dynamical Systems 4 (2002), 125–145.
[7] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2013.
[8] J. Hadamard, Essai sur l’etude des fonctions donnes par leur developpment de taylor, Journal des Mathématiques Pures et Appliquées 8 (1892), 86–101.
[9] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, SIngapore, 2000.
[10] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.
[11] S. Karthikeyan, C. Ravichandran and T. Gunasekar, Existence results for Hadamard type fractional functional integro-differential equations with integral boundary conditions, International Journal of Engineering Research 10 (2015), 6919–6932.
[12] H. Khan, Y. Li, W. Chen, D. Baleanu and A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Bound. Value Probl. 2017.
[13] H. Khan, Y. Li, H. Hongguang and A. Khan, Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with p-laplacian operator, J. Nonlinear Sci. Appl. 10 (2017), 5219–5229.
[14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Amsterdam, 2016.
[15] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in System Application 2 (1996), 963–968.
[16] T. Phollakrit, S. K. Ntouyas and T. Jessada, Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. 5 (2014), 1–9.
[17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[18] J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
[19] S. G. Samko, A. A. Kilbas and O. l. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
[20] K. Shah and R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions, Differ. Equ. Appl. 7(2) (2015), 245–262.
[21] R. Shah and A. Zada, A fixed point approach to the stability of a nonlinear Volterra integro-diferential equation with delay, Hacet. J. Math. Stat. 47 (2018), 615–623.
[22] S. O. Shah and A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales, Appl. Math. Comput. 359 (2019), 202–213.
[23] S. O. Shah, A. Zada and A. E. Hamza, Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales, Qual. Theory Dyn. Syst. (2019), DOI 10.1007/s12346–019–00315–x.
[24] S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh and T. Li, Stability of higher-order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl. 9 (2016), 4713–4721.
[25] J. Tariboon and W. Sudsutad, Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions, J. Nonlinear Sci. Appl. 9 (2016), 295–308.
[26] S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, USA, 1940.
[27] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theo. Diff. Equns. 63 (2011), 1–10.
[28] J. Wang, A. Zada and W. Ali, Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Numer. Funct. Anal. Optim. 33 (2012), 216–238.
[29] J. Wang, A. Zada and W. Ali, Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. 19 (2018), 553–560.
[30] A. Zada and S. Ali, Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul. 19 (2018), 763–774.
[31] A. Zada, S. Ali and Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Difference Equ. 2017.
[32] A. Zada, W. Ali and S. Farina, Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci. 40 (2017), 5502–5514.
[33] A. Zada, W. Ali and C. Park, Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type, Appl. Math. Comput. 350 (2019), 60–65.
[34] A. Zada, U. Riaz and F. U. Khan, Hyers-Ulam stability of impulsive integral equations, Boll. Unione Mat. Ital. 12 (2019), 453–467.
[35] A. Zada and S. O. Shah, Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat. 47 (2018), 1196–1205.
[36] A. Zada, S. O. Shah and R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problem, Appl. Math. Comput. 271 (2015), 512–518.
[37] A. Zada, S. Shaleena and T. Li, Stability analysis of higher order nonlinear differential equations in β-normed spaces, Math. Methods Appl. Sci. 42 (2019), 1151–1166.
[38] A. Zada, P. Wang, D. Lassoued and T. Li, Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems, Adv. Difference Equ. 2017.
[39] A. Zada, M. Yar and T. Li, Existence and stability analysis of nonlinearsequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math. 17 (2018), 103–125.
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