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On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient

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Abstract

The aim of this study is to investigate the existence and other properties of solution of nonlinear fractional integro–differential equations with constant coefficient. Also with the help of Pachpatte’s inequality, we prove the continuous dependence of the solutions.

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References

  1. Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations, vol. 27. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-4036-9

    Book  MATH  Google Scholar 

  2. Aghajani, A., Pourhadi, E., Trujillo, J.: Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 16(4), 962–977 (2013). https://doi.org/10.2478/s13540-013-0059-y

    Article  MathSciNet  MATH  Google Scholar 

  3. Anastassiou, G.: Advances on Fractional Inequalities. Springer, New York (2011)

    Book  Google Scholar 

  4. Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. 71(10), 4471–4475 (2009). https://doi.org/10.1016/j.na.2009.03.005

    Article  MathSciNet  MATH  Google Scholar 

  5. Baleanu, D., Güvenç, Z., Machado, J.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2000). https://doi.org/10.1007/978-90-481-3293-5

    Book  Google Scholar 

  6. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus. Models and Numerical Methods. World Scientific, New York (2012)

    Book  Google Scholar 

  7. Cabrera, I., Harjani, J., Sadarangani, K.: Existence and uniqueness of solutions for a boundary value problem of fractional type with nonlocal integral boundary conditions in Hölder spaces. Mediterr. J. Math. 15, 1–15 (2018). https://doi.org/10.1007/s00009-018-1142-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, New York (2010)

    Book  Google Scholar 

  9. Diethelm, K., Ford, N.J.: Analysis of Fractional Differential Equations. J. Math. Anal. Appl. 265(2), 229–248 (2002). https://doi.org/10.1006/jmaa.2000.7194

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong, X., Wang, J., Zhou, Y.: On nonlocal problems for fractional differential equations in Banach spaces. Opuscula Math. 31(3), 341–357 (2011)

    Article  MathSciNet  Google Scholar 

  11. Furati, K.M., Tatar, N.: Long time behavior for a nonlinear fractional model. J. Math. Anal. Appl. 332(1), 441–454 (2007). https://doi.org/10.1016/j.jmaa.2006.10.027

    Article  MathSciNet  MATH  Google Scholar 

  12. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, New York (2000)

    Book  Google Scholar 

  13. Jagtap, T.B., Kharat, V.V.: On existence of solution to nonlinear fractional integrodifferential system. J. Trajectory 22(1), 40–46 (2014)

    Google Scholar 

  14. Kharat, V.V.: On existence and uniqueness of fractional integrodifferential equations with an integral fractional boundary condition. Malaya J. Mat. 6(3), 485–491 (2018)

    Article  MathSciNet  Google Scholar 

  15. Kendre, S.D., Jagtap, T.B., Kharat, V.V.: On nonlinear fractional integrodifferential equations with nonlocal condition in Banach spaces. Nonlinear Anal. Differ. Equat. 1(3), 129–141 (2013)

    Article  Google Scholar 

  16. Kendre, S.D., Kharat, V.V., Jagtap, T.B.: On abstract nonlinear fractional integrodifferential equations with integral boundary condition. Comm. Appl. Nonlinear Anal. 22(3), 93–108 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Kendre, S.D., Kharat, V.V., Jagtap, T.B.: On fractional integrodifferential equations with fractional non-separated boundary conditions. Int. J. Appl. Math. Sci. 13(3), 169–181 (2013)

    Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, 204th edn. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  19. Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)

    MATH  Google Scholar 

  20. Liang, J., Liu, Z., Wang, X.: Solvability for a couple system of nonlinear fractional differential equations in a Banach space. Fract. Calc. Appl. Anal. 16(1), 51–63 (2013). https://doi.org/10.2478/s13540-013-0004-0

    Article  MathSciNet  MATH  Google Scholar 

  21. Luchko, Y.U.R.I.I., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24(2), 207–233 (1999)

    MathSciNet  MATH  Google Scholar 

  22. N’Guérékata, G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70(5), 1873–1876 (2009). https://doi.org/10.1016/j.na.2008.02.087

    Article  MathSciNet  MATH  Google Scholar 

  23. N’Guérékata, G.M.: Corrigendum: A Cauchy Problem for some Fractional Differential Equations. Commun. Math. Anal. 7(1), 11 (2009). http://math-res-pub.org/cma/7

  24. Nieto, J., Ouahab, A., Venktesh, V.: Implicit fractional differential equations via the Liouville–Caputo derivative. Mathematics 3(2), 398–411 (2015). https://doi.org/10.3390/math3020398

    Article  MATH  Google Scholar 

  25. Pachpatte, B.: Inequalities for Differential and Integral Equations. Academic Press, New York (1998)

    MATH  Google Scholar 

  26. Pierri, M., O’Regan, D.: On non-autonomous abstract nonlinear fractional differential equations. Appl. Anal. 94(5), 879–890 (2015). https://doi.org/10.1080/00036811.2014.905679

    Article  MathSciNet  MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  28. Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integr. Equ. Oper. Theory 16(1), 33–49 (2010). https://doi.org/10.1007/s00020-010-1767-x

    Article  MathSciNet  MATH  Google Scholar 

  29. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  30. Tarasov, V.E.: Fractional dynamics: applications of fractional calculus to dynamics of particles. Higher Education Press, Heidelberg (2010)

    Book  Google Scholar 

  31. Tate, S., Dinde, H.T.: Some theorems on Cauchy problem for nonlinear fractional differential equations with positive constant coefficient. Mediterr. J. Math. 14(2), 1–17 (2017). https://doi.org/10.1007/s00009-017-0886-x

    Article  MathSciNet  MATH  Google Scholar 

  32. Tidke, H.L.: Some theorems on fractional semilinear evolution equations. J. Appl. Anal. 18(2), 209–224 (2012). https://doi.org/10.1515/jaa-2012-0014

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, J., Li, X.: A uniform method to UlamHyers stability for some linear fractional equations. Mediterr. J. Math. 13(2), 625–635 (2016). https://doi.org/10.1007/s00009-015-0523-5

    Article  MathSciNet  Google Scholar 

  34. Zhou, Y., Shen, X.H., Zhang, L.: Cauchy problem for fractional evolution equations with Caputo derivative. Eur. Phys. J. Spec. Top. 222(8), 1749–1765 (2013). https://doi.org/10.1140/epjst/e2013-01961-5

    Article  Google Scholar 

  35. Zhou, Y., Jiao, F., Pečarić, J.: Abstract Cauchy problem for fractional functional differential equations. Topol Methods Nonlinear anal. 42(1), 119–136 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11(5), 4465–4475 (2010). https://doi.org/10.1016/j.nonrwa.2010.05.029

    Article  MathSciNet  MATH  Google Scholar 

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Tate, S., Kharat, V.V. & Dinde, H.T. On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient. Mediterr. J. Math. 16, 41 (2019). https://doi.org/10.1007/s00009-019-1325-y

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  • DOI: https://doi.org/10.1007/s00009-019-1325-y

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