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Ulam-Hyers-Stability for nonlinear fractional neutral differential equations

Year 2019, Volume: 48 Issue: 1, 157 - 169, 01.02.2019
https://doi.org/10.15672/hujms.524435

Abstract

We discuss Ulam-Hyers stability, Ulam-Hyers-Rassias stability and Generalized Ulam-Hyers-Rassias stability for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative by using Picard operator. An example is also given to show the applicability of our results.

References

  • L. Chen, Y. He, Y. Chai and R. Wu, New results on stability and stabilization of a class of nonlinear fractional-order systems, Nonlinear Dyn. 75 (4), 633–641, 2014.
  • L. Chen, C. Liu, R. Wu, Y. He and Y. Chai, Finite-time stability criteria for a class of fractional-order neural networks with delay, Neural Comput. Appl. 27 (3), 549–556, 2016.
  • L. Chen, R. Wu, J. Cao and J.B. Liu, Stability and synchronization of memristorbased fractional-order delayed neural networks, Neural Netw., 71, 37–44, 2015.
  • L. Chen, R. Wu, Z. Chu and Y. He, Stabilization of fractional-order coupled systems with time delay on networks, Nonlinear Dyn. 88 (1), 521–528, 2017.
  • K. Diethelm and A.D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: Scientific Computing in Chemical Engineering II, 217–224, Springer, Berlin, Heidelberg, 1999.
  • A.M. El-Sayed, Fractional-order diffusion-wave equation, Internat. J. Theoret. Phys. 35 (2), 311–322, 1996.
  • W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68 (1), 46–53, (1995).
  • R. Hilfer, ed., Applications of fractional calculus in physics, World Scientific, 2000.
  • D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (4), 222–224, 1941.
  • D.H. Hyers, G. Isac and T.M. Rassias, Stability of functional equations in several variables, Birkhuser Boston. Inc., Boston, MA, 1998.
  • S.M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, 2001.
  • S.M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (10), 1135–1140, 2004.
  • S. Jung, A fixed point approach to the stability of differential equations $y'=F (x, y)$, Bull. Malays. Math. Sci. Soc. 33, 47–56, 2010.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006.
  • X. Li, S. Liu and W. Jiang, Positive solutions for boundary value problem of nonlinear fractional functional differential equations, Appl. Math. Comput. 217 (22), 9278– 9285, 2011.
  • S. Liu, X. Li, X.F. Zhou and W. Jiang, Synchronization analysis of singular dynamical networks with unbounded time-delays, Adv. Difference Equ. 193, 1–9, 2015.
  • K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Mathematics in Science and Engineering 198, 1993.
  • T. Miura, S. Miyajima and S.E. Takahasi, A characterization of Hyers–Ulam stability of first order linear differential operators, J. Math. Anal. Appl. 286 (1), 136–146, 2003.
  • T. Miura, S. Miyajima and S.E. Takahasi, Hyers–Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (1), 90–96, 2003.
  • M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13, 259–270, 1993.
  • M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14, 141–146, 1997.
  • D. Otrocol and V. Ilea, Ulam stability for a delay differential equation, Open Math. 11 (7), 1296–1303, 2013.
  • I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.
  • T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (2), 297–300, 1978.
  • M.U. Rehman, R.A. Khan and N.A. Asif, Three point boundary value problems for nonlinear fractional differential equations, Acta Math. Sci. 31 (4), 1337–1346, 2011.
  • I.A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babe(¸s)-Bolyai Math. 54, 125–133, 2009.
  • U. Saeed, CAS Picard method for fractional nonlinear differential equation, Appl. Math. Comput. 307, 102–112, 2017.
  • S.E. Takahasi, T. Miura and S. Miyajima, On the Hyers-Ulam stability of the Banach spacevalued differential equation $y'=\lambda y$, Bull. Korean Math. Soc. 39 (2), 309–315, 2002.
  • S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers. Inc., New York, 1968.
  • J. Wang, M. Fec and Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (1), 258–264, 2012.
  • 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. 63, 1–10, 2011.
  • J. Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (6), 2530–2538, 2012.
  • J. Wang and Y. Zhou, Mittag-Leffler–Ulam stabilities of fractional evolution equations, Appl. Math. Lett. 25 (4), 723–728, 2012.
  • J. Wang, Y. Zhou and M. Fec, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (10), 3389–3405, 2012.
  • W.Wei, X. Li and X. Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (10), 3468–3476, 2012.
Year 2019, Volume: 48 Issue: 1, 157 - 169, 01.02.2019
https://doi.org/10.15672/hujms.524435

Abstract

References

  • L. Chen, Y. He, Y. Chai and R. Wu, New results on stability and stabilization of a class of nonlinear fractional-order systems, Nonlinear Dyn. 75 (4), 633–641, 2014.
  • L. Chen, C. Liu, R. Wu, Y. He and Y. Chai, Finite-time stability criteria for a class of fractional-order neural networks with delay, Neural Comput. Appl. 27 (3), 549–556, 2016.
  • L. Chen, R. Wu, J. Cao and J.B. Liu, Stability and synchronization of memristorbased fractional-order delayed neural networks, Neural Netw., 71, 37–44, 2015.
  • L. Chen, R. Wu, Z. Chu and Y. He, Stabilization of fractional-order coupled systems with time delay on networks, Nonlinear Dyn. 88 (1), 521–528, 2017.
  • K. Diethelm and A.D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: Scientific Computing in Chemical Engineering II, 217–224, Springer, Berlin, Heidelberg, 1999.
  • A.M. El-Sayed, Fractional-order diffusion-wave equation, Internat. J. Theoret. Phys. 35 (2), 311–322, 1996.
  • W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68 (1), 46–53, (1995).
  • R. Hilfer, ed., Applications of fractional calculus in physics, World Scientific, 2000.
  • D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (4), 222–224, 1941.
  • D.H. Hyers, G. Isac and T.M. Rassias, Stability of functional equations in several variables, Birkhuser Boston. Inc., Boston, MA, 1998.
  • S.M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, 2001.
  • S.M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (10), 1135–1140, 2004.
  • S. Jung, A fixed point approach to the stability of differential equations $y'=F (x, y)$, Bull. Malays. Math. Sci. Soc. 33, 47–56, 2010.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006.
  • X. Li, S. Liu and W. Jiang, Positive solutions for boundary value problem of nonlinear fractional functional differential equations, Appl. Math. Comput. 217 (22), 9278– 9285, 2011.
  • S. Liu, X. Li, X.F. Zhou and W. Jiang, Synchronization analysis of singular dynamical networks with unbounded time-delays, Adv. Difference Equ. 193, 1–9, 2015.
  • K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Mathematics in Science and Engineering 198, 1993.
  • T. Miura, S. Miyajima and S.E. Takahasi, A characterization of Hyers–Ulam stability of first order linear differential operators, J. Math. Anal. Appl. 286 (1), 136–146, 2003.
  • T. Miura, S. Miyajima and S.E. Takahasi, Hyers–Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (1), 90–96, 2003.
  • M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13, 259–270, 1993.
  • M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14, 141–146, 1997.
  • D. Otrocol and V. Ilea, Ulam stability for a delay differential equation, Open Math. 11 (7), 1296–1303, 2013.
  • I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.
  • T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (2), 297–300, 1978.
  • M.U. Rehman, R.A. Khan and N.A. Asif, Three point boundary value problems for nonlinear fractional differential equations, Acta Math. Sci. 31 (4), 1337–1346, 2011.
  • I.A. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babe(¸s)-Bolyai Math. 54, 125–133, 2009.
  • U. Saeed, CAS Picard method for fractional nonlinear differential equation, Appl. Math. Comput. 307, 102–112, 2017.
  • S.E. Takahasi, T. Miura and S. Miyajima, On the Hyers-Ulam stability of the Banach spacevalued differential equation $y'=\lambda y$, Bull. Korean Math. Soc. 39 (2), 309–315, 2002.
  • S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers. Inc., New York, 1968.
  • J. Wang, M. Fec and Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (1), 258–264, 2012.
  • 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. 63, 1–10, 2011.
  • J. Wang, L. Lv and Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (6), 2530–2538, 2012.
  • J. Wang and Y. Zhou, Mittag-Leffler–Ulam stabilities of fractional evolution equations, Appl. Math. Lett. 25 (4), 723–728, 2012.
  • J. Wang, Y. Zhou and M. Fec, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (10), 3389–3405, 2012.
  • W.Wei, X. Li and X. Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (10), 3468–3476, 2012.
There are 35 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Azmat Ullah Khan Niazi This is me

Jiang Wei This is me

Mujeeb Ur Rehman This is me

Du Jun This is me

Publication Date February 1, 2019
Published in Issue Year 2019 Volume: 48 Issue: 1

Cite

APA Niazi, A. U. K., Wei, J., Rehman, M. U., Jun, D. (2019). Ulam-Hyers-Stability for nonlinear fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics, 48(1), 157-169. https://doi.org/10.15672/hujms.524435
AMA Niazi AUK, Wei J, Rehman MU, Jun D. Ulam-Hyers-Stability for nonlinear fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics. February 2019;48(1):157-169. doi:10.15672/hujms.524435
Chicago Niazi, Azmat Ullah Khan, Jiang Wei, Mujeeb Ur Rehman, and Du Jun. “Ulam-Hyers-Stability for Nonlinear Fractional Neutral Differential Equations”. Hacettepe Journal of Mathematics and Statistics 48, no. 1 (February 2019): 157-69. https://doi.org/10.15672/hujms.524435.
EndNote Niazi AUK, Wei J, Rehman MU, Jun D (February 1, 2019) Ulam-Hyers-Stability for nonlinear fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics 48 1 157–169.
IEEE A. U. K. Niazi, J. Wei, M. U. Rehman, and D. Jun, “Ulam-Hyers-Stability for nonlinear fractional neutral differential equations”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, pp. 157–169, 2019, doi: 10.15672/hujms.524435.
ISNAD Niazi, Azmat Ullah Khan et al. “Ulam-Hyers-Stability for Nonlinear Fractional Neutral Differential Equations”. Hacettepe Journal of Mathematics and Statistics 48/1 (February 2019), 157-169. https://doi.org/10.15672/hujms.524435.
JAMA Niazi AUK, Wei J, Rehman MU, Jun D. Ulam-Hyers-Stability for nonlinear fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics. 2019;48:157–169.
MLA Niazi, Azmat Ullah Khan et al. “Ulam-Hyers-Stability for Nonlinear Fractional Neutral Differential Equations”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, 2019, pp. 157-69, doi:10.15672/hujms.524435.
Vancouver Niazi AUK, Wei J, Rehman MU, Jun D. Ulam-Hyers-Stability for nonlinear fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics. 2019;48(1):157-69.