Ulam stability results for the solutions of nonlinear implicit fractional order differential equations
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
___
- [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.