Ulam stability results for the solutions of nonlinear implicit fractional order differential equations

Year 2019, Volume: 48 Issue: 4, 1092 - 1109, 08.08.2019

Zeeshan Ali , Akbar Zada , Kamal Shah

Abstract

In this manuscript, we study the existence and uniqueness of solution for a class of fractional order boundary value problem (FBVP) for implicit fractional differential equations with Riemann-Liouville derivative. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the proposed problem. The concerned analysis is carried out through using classical technique of nonlinear functional analysis. The main results are illustrated by providing a couple of examples

Keywords

Riemann-Liouville fractional derivative, implicit fractional differential equations, existence, Green function, Ulam-Hyers stability

References

• [1] B. Ahmad and J.J. Nieto, Riemann–Laiouville fractional differential equations with fractional boundary conditions, Fixed Point Theor, 13 (2), 329–336, 2012.
• [2] B. Ahmad and S. Sivasundaram, On four–point nonlocal boundary value problems of nonlinear integro–differential equations of fractional order, Appl. Math. Comput. 217, 480–487, 2010.
• [3] N. Ahmad, Z. Ali, K. Shah, A. Zada and G. Ur Rahman, Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equations, Complexity, 2018, 1–15, 2018.
• [4] Z. Ali, A. Zada and K. Shah, Existence and stability analysis of three point boundary value problem, Int. J. Appl. Comput. Math. 2017, DOI:10.1007/s40819–017–0375–8.
• [5] G.A. Anastassiou, On right fractional calculus, Chaos Solitons Fractals, 42 (1), 365– 376, 2009.
• [6] D. Baleanu, Z.B. Güvenc and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
• [7] M. Benchohra and S. Bouriah, Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order, Moroccan J. Pure Appl. Anal. 1 (1), 22–37, 2015.
• [8] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71, 2391– 2396, 2009.
• [9] A. Browder, Mathematical Analysis: An Introduction, New York, Springer-Verlag, 1996.
• [10] M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi–point boundary value problem of fractional order, Comput. Math. Appl. 59, 3438–3443, 2010.
• [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [12] E. Hiffer (ed.), Application of Fractional Calculus in Physics, Word Scientific, Singapore, 2000.
• [13] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A, 27 (4), 222–224, 1941.
• [14] D.H. Hyers, G. Isac and T.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäiuser, Boston, 1998.
• [15] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations, in: Applied Mathematicals Sciences series, Springer–Verlag, New York, 99, 1993.
• [16] R.W. Ibrahim, Generalized Ulam–Hyers stability for fractional differential equations, Int. J. Math. 23 (5), 9 pages, 2012.
• [17] S.M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19, 854–858, 2006.
• [18] R.A. Khan and K. Shah, K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19, 515–526, 2015.
• [19] A.A. Kilbas, H.M. Srivasta and J.J. Trujilllo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 24, North-Holand, Amsterdam, 2006.
• [20] Y.H. Lee and K.W. Jun, A Generalization of the Hyers–Ulam–Rassias stability of Pexider equation, J. Math. Anal. Appl. 246, 627–638, 2000.
• [21] 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 (1), 1–8.
• [22] T. Li, A. Zada and S. Faisal, Hyers–Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl. 9, 2070–2075, 2016.
• [23] J.T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlin. Sci. Numer. Simul. 16 (3), 1140–1153, 2011.
• [24] R. Metzler and K. Joseph, Boundary value problems for fractional diffusion equations, Phys. A, 278 (1), 107–125, 2000.
• [25] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat. 13, 259–270, 1993.
• [26] K.B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41, 9–12, 2010.
• [27] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
• [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [29] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (2), 297–300, 1978.
• [30] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math. 62, 23–130, 2000.
• [31] F.A. Rihan, Numerical Modeling of Fractional-Order Biological Systems, Abstr. Appl. Anal. 2013, 11 pages, 2013.
• [32] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26, 103-107, 2010.
• [33] J. Sabatier, O.P. Agrawal and J.A.T. Machado, Advances in fractional calculus, Dordrecht, Springer, 2007.
• [34] 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), 245–262, 2015.
• [35] K. Shah, N. Ali and R.A. Khan, Existence of positive solution to a class of fractional differential equations with three point boundary conditions, Math. Sci. Lett. 5 (3), 291–296, 2016.
• [36] 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 Solitons Fractals, 77, 240–246, 2015.
• [37] K. Shah, H. Khalil and R.A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Prog. Fract. Differ. Appl. 1 (1), 1–8, 2016.
• [38] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. 22 (3), 291–298, 2007.
• [39] 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, 4713–4721, 2016.
• [40] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010.
• [41] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.
• [42] B.M. Vintagre, I. Podlybni, A. Hernandez and V. Feliu, Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal. 3 (3), 231–248, 2000.
• [43] J. Wang, L. Lv and W. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63, 1–10, 2011.
• [44] J.R. Wang, Y. Zhou and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (10), 3389–3405, 2012.
• [45] B. Xu, J. Brzdek and W. Zhang, Fixed point results and the Hyers–Ulam stability of linear equations of higher orders, Pacific J. Math. 273, 483–498, 2015.
• [46] A. Zada, S. Faisal and Y. Li, On the Hyers-Ulam Stability of First Order Impulsive Delay Differential Equations, J. Func. Spac. 2016, 6 pages, 2016.
• [47] 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, 512–518, 2015.