Existence, Uniqueness and Stability of Implicit Switched Coupled Fractional Differential Equations of ψ-Hilfer Type

References

[1] R. P. Agarwal, Y. Zhou and Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59(3) (2010), 1095–1100.10.1016/j.camwa.2009.05.010Search in Google Scholar

[2] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006.Search in Google Scholar

[3] R. Magin, Fractional calculus in bioengineering, Crit. Rev. Biom. Eng. 32 (2004), 1–104.10.1615/CritRevBiomedEng.v32.10Search in Google Scholar

[4] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Soft. 41 (2010), 9–12.10.1016/j.advengsoft.2008.12.012Search in Google Scholar

[5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.Search in Google Scholar

[6] K. Shah, H. Khalil and R. A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos. Solit. Fract. 77 (2015), 240–246.10.1016/j.chaos.2015.06.008Search in Google Scholar

[7] J. Wang, K. Shah and A. Ali, Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Meth. Appl. Sci. 41(6) (2018), 1–11.10.1002/mma.4748Search in Google Scholar

[8] A. Khan, T. S. Khan, M. I. Syam and H. Khan, Analytical solutions of time-fractional wave equation by double Laplace transform method, Eur. Phys. J. Plus. 134 (2019), 163.10.1140/epjp/i2019-12499-ySearch in Google Scholar

[9] H. Khan, Y. Li, A. Khan and A. Khan, Existence of solution for a fractional-order Lotka-Volterra reaction-diffusion model with Mittag-Leffler kernel, Math. Meth. Appl. Sci. 42(9)(2019), 3377–3387.10.1002/mma.5590Search in Google Scholar

[10] H. Khan, J. F. GÓmez–Aguilar, A. Khan and T. S. Khan, Stability analysis for fractional order advection–reaction diffusion system, Physica A 521 (2019), 737–751.10.1016/j.physa.2019.01.102Search in Google Scholar

[11] H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Adv. Difference Equ. 2019 (2019), 18.10.1186/s13662-019-1965-zSearch in Google Scholar

[12] H. Khan, T. Abdeljawad, C. Tunc and A. Alkhazzan, A. Khan Minkowski’s inequality for the AB–fractional integral operator, J. Inequal Appl. 2019 (2019), 96.10.1186/s13660-019-2045-3Search in Google Scholar

[13] R. A. Khan and K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19 (2015), 515–526.Search in Google Scholar

[14] X. Liu, M. Jia and W. Ge, Multiple solutions of a p - Laplacian model involving a fractional derivative, Adv. Difference Equ. 2013 (2013), 126.10.1186/1687-1847-2013-126Search in Google Scholar

[15] G. Wang, B. Ahmad and L. Zhang, Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal. Theory, Methods and Appl. 74 (2011), 792–804.10.1016/j.na.2010.09.030Search in Google Scholar

[16] L. Zhang, B. Ahmad, G. Wang and R. P. Agarwal, Nonlinear fractional integro–differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51–56.10.1016/j.cam.2013.02.010Search in Google Scholar

[17] K. M. Furati and M. D. Kassim, Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differ. Equ. 235 (2013), 1–10.Search in Google Scholar

[18] K. M Furati, M. D Kassim and N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. 64(6) (2012), 1616–1626.10.1016/j.camwa.2012.01.009Search in Google Scholar

[19] R. Hilfer, Threefold Introduction to Fractional Derivatives, in Anomalous Transport: Foundations and Applications, edited by R. Klages, G. Radons, I.M. Sokolov, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008, doi: 10.1002/9783527622979.ch2.Search in Google Scholar

[20] R. Kamocki and C. Obcznnski, On fractional Cauchy-type problems containing Hilfer derivative, Electron. J. Qual. Theory Differ. Equ. 50 (2016), 1–12.10.14232/ejqtde.2016.1.50Search in Google Scholar

[21] Ž.Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct. 21(11) (2010), 797–814.10.1080/10652461003675737Search in Google Scholar

[22] J. R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850–859.10.1016/j.amc.2015.05.144Search in Google Scholar

[23] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York, 1960.Search in Google Scholar

[24] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.10.1073/pnas.27.4.222Search in Google Scholar PubMed PubMed Central

[25] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.10.1090/S0002-9939-1978-0507327-1Search in Google Scholar

[26] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat. 13 (1993), 259–270.Search in Google Scholar

[27] T. Li and A. Zada, Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ. 2016 (2016), 153.10.1186/s13662-016-0881-8Search in Google Scholar

[28] A. Zada, W. Ali and S. Farina, Ulam–Hyers stability of nonlinear differential equations with fractional integrable impulsis, Math. Meth. Appl. Sci. 40 (2017), 5502–5514.10.1002/mma.4405Search in Google Scholar

[29] 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 (2017), 317.10.1186/s13662-017-1376-ySearch in Google Scholar

[30] A. Zada, O. Shah and R. Shah, Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput. 271 (2015), 512–518.10.1016/j.amc.2015.09.040Search in Google Scholar

[31] S. Harikrishnan, K. Shah, D. Baleanu and K. Kanagarajan, Note on the solution of random differential equations via ψ - Hilfer fractional derivative, Adv. Difference. Equ. 2018 (2018), 224, 1–10.10.1186/s13662-018-1678-8Search in Google Scholar

[32] J. V. C. Sousa and E. C. de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simulat 60 (2018), 72–91.10.1016/j.cnsns.2018.01.005Search in Google Scholar

[33] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 460–481.10.1016/j.cnsns.2016.09.006Search in Google Scholar

[34] J. V. C. Sousa and E. C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using ψ-Hilfer operator, (2017), arXiv:1711.07339.10.1007/s11784-018-0587-5Search in Google Scholar

[35] A. Granas and J. Dugundji, Fixed Point Theory, Springer–Verlag, New York, 2003.10.1007/978-0-387-21593-8Search in Google Scholar

[36] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.Search in Google Scholar

[37] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math. 26 (2010), 103–107.Search in Google Scholar