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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Local Existence and Ulam Stability Results for Nonlinear Fractional Differential Equations

Journal of Applied Nonlinear Dynamics 9(4) (2020) 655--666 | DOI:10.5890/JAND.2020.12.009

Houssem Eddine Khochemane$^{1}$, Abdelouaheb Ardjouni$^{2}$ , Amin Guerouah$^{3}$, Salah Zitouni$^{2}$

$^1$ Ecole normale sup'{e}rieure d'enseignement technologique, Azzaba-Skikda, Algeria

$^2$ Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria

$^3$ Mustapha Ben Boulaid University, Batna 2-fesdis, 05001 Batna, Algeria

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The aim of this paper is to study the existence and uniqueness for nonlinear fractional differential equations involving Caputo's fractional derivative using the Krasnoselskii and Banach fixed point theorems on one hand and to establish the Ulam stability on the other hand. Finally, An example is given to substantiate the usefulness of the obtained results.


\bibitem {4}Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), \textit{Theory and applications of fractional differential equations}, Amsterdam, the Netherlands, North-Holland.


  1. [1] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Amsterdam, the Netherlands, North-Holland.
  2. [2] Miller, K.S. and Ross, B. (1993), An Introduction to the fractional calculus and differential equations, John Wiley, New York.
  3. [3] Podlubny, I. (1999), Fractional differential equations, San Diego, CA, USA, Academic Press.
  4. [4] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993), Fractional integrals and derivatives: Theory and applications, Yverdon, Switzerland, Gordon and Breach Science.
  5. [5] Agarwal, R.P., Benchohra, M. and Hamani, S. (2010), A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109, 973-1033.
  6. [6] Agarwal, R.P. and O'Regan, D. (2002), Existence theory for singular initial and boundary value problems: A fixed point approach, Appl. Anal., 81, 391-434.
  7. [7] Ahmad, B. and Otero-Espinar, V. (2009), Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions, Bound. Value Probl., 2009.
  8. [8] Anastassiou, G.A. (2011), Advances on fractional inequalities, Berlin, Germany, Springer.
  9. [9] Babakhani, A. and Gejji, V.D. (2003), Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 434-442.
  10. [10] Bai, Z.B. and Qiu, T.T. (2009), Existence of positive solution for singular fractional differential equation, Appl. Math. Comput., 215, 2761-2767.
  11. [11] Balachandran, K. and Park, JY. (2009), Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal., 71, 4471-4475.
  12. [12] Bashir, A. and Sivasundaram, S. (2008), Some existence results for fractional integro-differential equations with nonlocal conditions, Communications in Applied Analysis, 12, 107-112.
  13. [13] Benchohra, M. and Lazreg, J.E. (2013), Nonlinear fractional implicit differential equations, Commun. Appl. Anal., 17, 471-482.
  14. [14] Buckwar, E. and Luchko, Y. (1998), Invariance of a partial differential equation of fractional order under lie group of scaling traps formations, J. Math. Anal. Appl., 227, 81-97.
  15. [15] Delbosco, D. and Rodino, L. (1996), Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625.
  16. [16] Kou, C.H., Liu, J. and Ye, Y. (2010), Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations, Discr. Dyn. Nat. Soc., 2010, 1-15.
  17. [17] Ulam, S.M. (1940), Problems in modern mathematics, John Wiley and Sons, New York, U.S.A..
  18. [18] Yu, C. and Gao, G. (2005), Existence of fractional differential equations, J. Math. Anal. Appl., 310, 26-29.
  19. [19] Zhang, S. (2000), The existence of a positive solution for a fractional differential equation, J. Math. Anal. Appl., 252, 804-812.
  20. [20] Arara, A., Benchohra, M., Hamidi, N. and Nieto, J.J. (2010), Fractional order differential equations on an unbounded domain, Nonlin. Anal., 72, 580-586.
  21. [21] Baleanu, D. and Mustafa, O.G. (2010), On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 59, 1835-1841.
  22. [22] Lakshmikantham, V. and Vatsala, A.S. (2008), Basic theory of fractional differential equations, Nonlinear Anal., 69, 2677-2682.
  23. [23] Hyers, D.H. (1941), On the stability of the linear functional equation, Natl. Acad. Sci. U.S.A., 27, 222-224.
  24. [24] Rassias, Th.M. (1978), On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300.
  25. [25] Abdo, M.S. and Panchal, S.K. (2018), Uniqueness and stability results of fractional neutral differential equations with infinite delay, Int. J. Math. And Appl., 6(2-A), 403-410.
  26. [26] Atmania, R. and Bouzitouna, S. (2019), Existence and Ulam stability results for two-orders fractional differential equation, Acta Math. Univ. Comenianae, LXXXVIII(1), 1-12.
  27. [27] Hyers, D.H., Isac, G. and Rassias, Th.M. (1998), Stability of functional equations in several variables, Birkhauser.
  28. [28] Jung, S.M. (2001), Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor.
  29. [29] Moghaddam, B.P. and Machado, J.A.T. (2017), A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels, Fractioanl calculus and applied analysis, 20(4), 1023-1042.
  30. [30] Moghaddam, B.P., Machado, J.A.T. and Behforooz, H. (2017), An integro quadratic spline approach for a class of variable-order fractional initial value problems, Chaos, Solitons and Fractals, 102, 354-360.
  31. [31] Moghaddam, B.P. and Machado, J.A.T. (2017), SM-algorithms for approximating the variable-order fractional derivative of high order, Fundamenta Informaticae, 151, 293-311.
  32. [32] Moghaddam, B.P. and Machado, J.A.T. (2017), Time analysis of forced variable-order fractional Van der Pol oscillator, Eur. Phys. J. Special Topics, 226, 3803-3810.
  33. [33] Moghaddam, B.P., Zhang, L., Lopes, A.M., Tenreiro Machado, J.A. and Mostaghim, Z.S. (2019), Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, Stochastics,
  34. [34] Changpin, L. and Shahzad, S. (2016), Existence and continuation of solutions for Caputo type fractional differential equations, Electronic Journal of Differential Equations, 2016(207), 1-14.
  35. [35] Babakhani, A. (2012), Existence and uniqueness of solution for class of fractional order differential equations on an unbounded domain, Advances in Difference Equations, 2012(41), 1-8.
  36. [36] Smart, D.R. (1980), Fixed point theorems, Cambridge Uni. Press., Cambridge.