# Existence of Solutions and Ulam Stability for Caputo Type Sequential Fractional Differential Equations of Order α ∈ (2,3)

## Main Article Content

### Abstract

We study initial value problems of sequential fractional differential equations and inclusions involving a Caputo type differential operator of the form: $\left(^{C}D_{a+}^{\alpha }+\lambda _{1}~^{C}D_{a+}^{\alpha -1}+\lambda _{2}~^{C}D_{a+}^{\alpha -2}\right),$ where $\alpha \in (2,3)$ and $\lambda _{i} (i=1, 2) $ are nonzero constants. Several existence and uniqueness results are accomplished by means of fixed point theorems. Sufficient conditions for Ulam stability of the given problem are also presented. Examples are constructed for the illustration of obtained results. Then we investigate the inclusions case of the problem at hand. An initial value problem for coupled sequential fractional differential equations is also discussed.

## Article Details

### References

- F. Meral, T. Royston and R. Magin, Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 939-945.
- K. Oldham, Ractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010), 9-12.
- C. Lee and F. Chang, Fractional-order PID controller optimization via improved electromagnetism-like algorithm, Expert Syst. Appl. 37 (2010), 8871-8878.
- E. Ahmed, A. El-Sayed and H. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl. 325 (2007), 542-553.
- F. Liu and K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl. 62 (2011), 822-833.
- G. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl. 61 (2011), 68-78.
- J. Wang, Y. Zhou and W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Control Lett. 61 (2012), 472-476.
- R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes, J. Comput. Appl. Math. 229 (2009), 400-415.
- X. Jiang, M. Xu and H. Qi, The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes, Nonlinear Anal. Real World Appl. 11 (2010), 262-269.
- I. Sokolov, A. Chechkin and J. Klafter, Fractional diffusion equation for a power-law truncated Levy process, Physica A. 336 (2004), 245-251.
- R. Nigmatullin, T. Omay and D. Baleanu, On fractional filtering versus conventional filtering in economics, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 979-986.
- M. Faieghi, S. Kuntanapreeda, H. Delavari, D. Baleanu, LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynam. 72 (2013), 301-309.
- F. Zhang, G. Chen, C. Li, J. Kurths, Chaos synchronization in fractional differential systems, Phil. Trans. R. Soc. A 371 (2013), 20120155.
- K. Balachandran, M. Matar, J. J. Trujillo, Note on controllability of linear fractional dynamical systems, J. Control Decis. 3 (2016), 267-279.
- O.P. Agrawal, Generalized Variational Problems and Euler-Lagrange equations, Comput. Math. Appl. 59 (2010), 1852-1864.
- A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
- Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.
- B. Ahmad, M. M Matar, R. P. Agarwal, Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions, Bound. Value Probl. 2015:220 (2015), 13 pp.
- B. Ahmad, M. M. Matar, S. K. Ntouyas, On general fractional differential inclusions with nonlocal integral boundary conditions, Differ. Equ. Dyn. Syst. (2016), DOI:10.1007/s12591-016-0319-5.
- M. Matar, On Existence of positive solution for initial value problem of nonlinear fractional differential equations of order 1 < α â‰¤ 2, Acta Math. Univ. Comenianae, 84 (1) (2015), 51-57.
- L. Zhang, B. Ahmad, G. Wang, Explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions, Appl. Math. Comput. 268 (2015), 388-392.
- D. Qarout, B. Ahmad, A. Alsaedi, Existence theorems for semilinear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions, Fract. Calc. Appl. Anal. 19 (2016), 463-479.
- B. Ahmad, Sharp estimates for the unique solution of two-point fractional-order boundary value problems, Appl. Math. Lett. 65 (2017), 77-82.
- S. Aljoudi, B. Ahmad, J.J. Nieto, A. Alsaedi, A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos Solitons Fractals 91 (2016), 39-46.
- B. Ahmad, S.K. Ntouyas, A. Alsaedi, Fractional differential equations and inclusions with nonlocal generalized Riemann-Liouville integral boundary conditions, Int. J. Anal. Appl. 13 (2017), 231-247.
- H.M. Srivastava, Remarks on some families of fractional-order differential equations, Integral Transforms Spec. Funct. 28 (2017), 560-564.
- W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal. 72 (2010), 1768-1777.
- R. W. Ibrahim, Stability of a fractional differential equation, Miskolc Math. Notes 13 (2012), 39-52.
- R. Agarwal, S. Hristova and D. O'Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), 290-318.
- N. Brillou ¨ et-Belluot, J. Brzd ¸ ek, and K. Ciepli ´ nski, On some recent developments in Ulam's type stability, Abstr. Appl. Anal. (2012), Art. ID 716936, 41 pp.
- J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), Art. ID 63.
- S. Abbas, M. Benchohra and A. Petrusel, Ulam stability for partial fractional differential inclusions via Picard operators theory, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Art. ID 51.
- R.W. Ibrahim and H.A. Jalab, Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy 17 (2015), 3172-3181.
- D.R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London- New York, 1974.
- K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.
- S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory. Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997.
- H. F. Bohnenblust and S. Karlin, On a theorem of Ville, In Contributions to the Theory of Games. Vol. I, pp. 155-160, Princeton Univ. Press, 1950.
- A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser.Sci. Math. Astronom. Phys. 13 (1965), 781-786.
- A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.