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Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order Cover

Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order

Open Access
|Feb 2017

References

  1. [1] S. Abbas and M. Benchohra, On the generalized Ulam-Hyers-Rassias stability for Darboux problem for partial fractional implicit differential equations. Appl. Math. E-Notes 14 (2014), 20-28.
  2. [2] S. Abbas, M. Benchohra and G M. N’Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.10.1007/978-1-4614-4036-9
  3. [3] S. Abbas, M. Benchohra and G M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  4. [4] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv Differ. Equat. 2009 (2009) Article ID 981728, 1-47.
  5. [5] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math, 109 (2010), 973-1033.
  6. [6] G.A. Anastassiou, Advances on Fractional Inequalities, Springer, New York, 2011.10.1007/978-1-4614-0703-4
  7. [7] C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2 (1998), 373-380.
  8. [8] T. Aoki, On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), G4-66.10.2969/jmsj/00210064
  9. [9] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculs Models and Numerical Methods, World Scientific Publishing, New York, 2012.10.1142/8180
  10. [10] D. Baleanu, Z.B. Güvenç and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.10.1007/978-90-481-3293-5
  11. [11] M. Benchohra and J.E. Lazreg, Nonlinear fractional implicit differential equations. Commun. Appl. Anal. 17 (2013), 471-482.
  12. [12] L. Byszewski, Theorem about existence and uniqueness of continuous solutions of nonlocal problem for nonlinear hyperbolic equation, Appl. Anal., 40 (1991), 173-180.10.1080/00036819108840001
  13. [13] Y.J. Cho, Th.M. Rassias and R. Saadati, Stability of Functional Equations in Random Normcd Spaces, Springer, New York, 2013.10.1007/978-1-4614-8477-6
  14. [14] P. Gavruta, A generalisation of the Hyers-Ulam-Rassias stability of approximately additive map¬pings, J. Math, Anal. Appl. 184 (1994), 431-436.10.1006/jmaa.1994.1211
  15. [15] D.H. Hyers, On the stability of the linear functional equation, Natl. Acad. Sci. U.S.A. 27 (1941), 222-224.10.1073/pnas.27.4.222107831016578012
  16. [16] R.W. Ibrahim, Stability for univalent solutions of complex fractional differential equations, Proc. Pakistan Acad. Sci. 49 (3) (2012), 227-232.10.1186/1687-1847-2012-192
  17. [17] S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137.10.1006/jmaa.1998.5916
  18. [18] S.M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19 (2006), 854-858.10.1016/j.aml.2005.11.004
  19. [19] K.W. Jun and H.M. Kim, On the stability of an n-dimensional quadratic and additive functional equation, Math. Inequal. Appl. 19 (9) (2006), 854-858.
  20. [20] S.M. Jung, K.S. Lee, Hyers-Ulam stability of first order linear partial differential equations with constant coefficients, Math. Inequal. Appl. 10 (2007), 261-266.
  21. [21] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  22. [22] G.H. Kim, On the stability of functional equations with square-symmetric operation, Math. Inequal. Appl 17 (4) (2001), 257-266.10.7153/mia-04-25
  23. [23] M. Obloza. Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Pracc Mat. 13 (1993), 259-270. 4037-4043.
  24. [24] M.D Otigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht, 2011.10.1007/978-94-007-0747-4
  25. [25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  26. [26] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.10.1090/S0002-9939-1978-0507327-1
  27. [27] J.M. Rassias, Functional Equations, Difference Inequalities and Ulam Stability Notions (F.U.N), Nova Science Publishers, Inc. New York, 2010.
  28. [28] Th.M. Rassias and J. Brzdek, Functional Equations in Mathematical Analysis, Springer, New York, 2012.10.1007/978-1-4614-0055-4
  29. [29] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26 (2010), 103-107.
  30. [30] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.
  31. [31] S.M. Ulam, Problems in Modem Mathematics, John Wiley and sons, New York, USA, 1940.
  32. [32] S.M. Ulam, A Collection of Mathematical Pmblems, Intcrscience, New York, 1960.
  33. [33] H.Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), 1075-1081.10.1016/j.jmaa.2006.05.061
  34. [34] J. Wang, M. Feckan and Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (20012), 258-264.10.1016/j.jmaa.2012.05.040
  35. [35] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equat. 63 (2011), 1-10.
  36. [36] J. Wang and Y. Zhang, Existence and stabihty of solutions to nonlinear impulsive differential equations in /3-normed spaces, Electron. J. Differential Equations (2014), No. 83, 1-10.
Language: English
Page range: 22 - 37
Submitted on: Jan 8, 2015
Accepted on: Jun 25, 2015
Published on: Feb 4, 2017
Published by: Sciendo
In partnership with: Paradigm Publishing Services
Publication frequency: 3 issues per year

© 2017 Mouffak Benchohra, Soufyane Bouriah, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.