Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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EFFECT OF PERTURBATION IN THE SOLUTION OF FRACTIONAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

  • ABDO, MOHAMMED. S. (RESEARCH SCHOLAR AT DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY) ;
  • PANCHAL, SATISH. K. (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY)
  • Received : 2018.01.07
  • Accepted : 2018.03.12
  • Published : 2018.03.25
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Abstract

In this paper, we study the initial value problem for neutral functional differential equations involving Caputo fractional derivative of order ${\alpha}{\in}(0,1)$ with infinite delay. Some sufficient conditions for the uniqueness and continuous dependence of solutions are established by virtue of fractional calculus and Banach fixed point theorem. Some results obtained showed that the solution was closely related to the conditions of delays and minor changes in the problem. An example is provided to illustrate the main results.

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References

  1. S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Differential Equations, 9 (2011) 1-11.
  2. M. S. Abdo and S. K. Panchal, Existence and continuous dependence for fractional neutral functional differential equations, Journal of Mathematical Modeling(JMM), 5 (2017) 153-170.
  3. M. S. Abdo and S. K. Panchal, Some New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations, Annals of Pure and Applied Mathematics, 16 (2018), 345-352. https://doi.org/10.22457/apam.v16n2a11
  4. R. Agarwal, Y. Zhou and Y. He, Existence of fractional neutral functional differential equations, Computers and Mathematics with Applications, 59 (2010) 1095-1100. https://doi.org/10.1016/j.camwa.2009.05.010
  5. M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338 (2008) 1340-1350. https://doi.org/10.1016/j.jmaa.2007.06.021
  6. J. Cao, H. Chen and W. Yang, Existence and continuous dependence of mild solutions for fractional neutral abstract evolution equations, Advances in Difference Equations, 2015 (2015) 1-6.
  7. D. Delboso and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204 (1996) 609-625. https://doi.org/10.1006/jmaa.1996.0456
  8. K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002) 229-248. https://doi.org/10.1006/jmaa.2000.7194
  9. K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes inMathematics 2004, Springer, Berlin, 2010.
  10. Q. Dong, Existence and continuous dependence for weighted fractional differential equations with infinite delay, Advance Difference Equations, 2014 (2014) 1-13.
  11. A. M. A. El-Sayed, F. Gaafar and M. El-Gendy, Continuous dependence of the solution of Ito stochastic differential equation with nonlocal conditions, Applied Mathematical Sciences, 10(2016) 1971-1982. https://doi.org/10.12988/ams.2016.64135
  12. J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial Ekvac, 21 (1978) 11-41.
  13. E. Hernandez, Existence results for partial neutral functional integrodifferential equations with unbounded delay, Journal of Mathematical Analysis and Applications, 292 (2004). 194-210. https://doi.org/10.1016/j.jmaa.2003.11.052
  14. Y. Hino, S. Murakami and T. Naito, Functional differential equations with infinite delay, Lecture Notes in Math 1473, Springer-Verlag, Berlin, 1991.
  15. A. A. Kilbas, H.M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006.
  16. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008) 3337-3343. https://doi.org/10.1016/j.na.2007.09.025
  17. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  18. GM. N.Mophou, GM. Guerekata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Applied Mathematics and Computation, 216 (2010) 61-69. https://doi.org/10.1016/j.amc.2009.12.062
  19. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  20. H. A. H. Salem, On the existence of continuous solutions for a singular system of non-linear fractional differential equations, Applied Mathematics and Computation, 198 (2008) 445-452. https://doi.org/10.1016/j.amc.2007.08.063
  21. S. G. Samko, A. A. Kilbas and O. I.Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
  22. M. Sharma, and S. Dubey, Analysis of Fractional Functional Differential Equations of Neutral Type with Nonlocal Conditions, Differential Equations and Dynamical Systems, 25 (2017) 499-517. https://doi.org/10.1007/s12591-016-0290-1
  23. Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations,Computers &Mathematics with Applications, 59 (2010) 1063-1077. https://doi.org/10.1016/j.camwa.2009.06.026
  24. Y. Zhou, Basic theory of fractional differential equations, Singapore, World Scientific, 2014.
  25. Y. Zhou and F. Jiao and J. Li, Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009) 2724-2733. https://doi.org/10.1016/j.na.2009.01.105