The optimal behavior of solutions to fractional impulsive stochastic integro-differential equations and its control problems

  • Zuomao YanEmail author
  • Xingxue Yan


In this paper, we study a new class of fractional nonlinear impulsive stochastic integro-differential equations with infinite delay in separable Hilbert spaces. Firstly, by using the measure of noncompactness, stochastic analysis theory, solution operators and suitable fixed point theorems, we prove the existence of mild solutions for these systems. Secondly, the existence of optimal mild solutions is obtained. Thirdly, we establish the controllability of the controlled fractional impulsive stochastic functional integro-differential systems with not instantaneous impulses. Finally, an example is provided to show the application of our results.


Fractional impulsive stochastic integro-differential equations optimal mild solutions controllability measure of noncompactness fixed point 

Mathematics Subject Classification

34A37 60H15 34K50 26A33 93B05 



The authors would like to thank the editor and the reviewers for their constructive comments and suggestions. This work is supported by the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10).


  1. 1.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  2. 2.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations, Mathematics in Sciences and Engineering, 198. Academic Press, San Diego (1999)Google Scholar
  4. 4.
    El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fract. 14, 433–440 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Yan, Z.: Existence results for fractional functional integrodifferential equations with nonlocal conditions in Banach spaces. Ann. Pol. Math. 97, 285–299 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cuesta, E., Palencia, C.: A numerical method for an integro-differential equation with memory in Banach spaces: qualitative properties. SIAM J. Numer. Anal. 41, 1232–1241 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cuevas, C., Henriquez, H.R., Soto, H.: Asymptotically periodic solutions of fractional differential equations. Appl. Math. Comput. 236, 524–545 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cuevas, C., de Souza, J.C.: Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 72, 1683–1689 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hernández, E., O’Regan, D., Balachandran, K.: Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators. Indag. Math. 24, 68–82 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chalishajar, D.N., Karthikeyan, K.: Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces. Acta Math. Sci. 33, 758–772 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shu, X.-B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, 2003–2011 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chauhan, A., Dabas, J.: Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition. Commun. Nonlinear Sci. Numer. Simul. 19, 821–829 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Balachandran, K., Kiruthika, S., Trujillo, J.J.: On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput. Math. Appl. 62, 1157–1165 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Liu, Z., Li, X.: On the controllability of impulsive fractional evolution inclusions in Banach spaces. J. Optim. Theory Appl. 156, 167–182 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay. Math. Comput. Modell. 57, 754–763 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichestic (1997)zbMATHGoogle Scholar
  17. 17.
    Ren, Y., Bi, Q., Sakthivel, R.: Stochastic functional differential equations with infinite delay driven by G-Brownian motion. Math. Methods Appl. Sci. 36, 1746–1759 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sakthivel, R., Ren, Y.: Exponential stability of second-order stochastic evolution equations with Poisson jumps. Commun. Nonlinear Sci. Numer. Simul. 17, 4517–4523 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A 44, 1–18 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63, 660–668 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sakthivel, R., Luo, J.: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl. 356, 1–6 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lin, A., Ren, Y., Xia, N.: On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators. Math. Comput. Modell. 51, 413–424 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yan, Z., Yan, X.: Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay. Collect. Math. 64, 235–250 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zang, Y., Li, J.: Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions. Bound. Value Probl. 2013, 1–14 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yan, Z., Zhang, H.: Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with state-dependent delay. Electron. J. Differ. Equ. 2013, 1–29 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yan, Z., Zhang, H.: Existence of solutions to impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay. Electron. J. Differ. Equ. 2013, 1–21 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hernández, E., Pierri, M., O’Regan, D.: On abstract differential equations with non instantaneous impulses. Topol. Methods Nonlinear Anal. 46, 1067–1088 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Pierri, M., O’Regan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219, 6743–6749 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kumar, P., Pandey, D.N., Bahuguna, D.: On a new class of abstract impulsive functional differential equations of fractional order. J. Nonlinear Sci. Appl. 7, 102–114 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yu, X., Wang, J.: Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 22, 980–989 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Chalishajar, D.N., Kumar, A.: Total controllability of the second order semi-linear differential equation with infinite delay and non-instantaneous impulses. Math. Comput. Appl. 23, 1–13 (2018)Google Scholar
  34. 34.
    Yan, Z., Lu, F.: Existence results for a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay. J. Appl. Anal. Comput. 5, 329–346 (2015)MathSciNetGoogle Scholar
  35. 35.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, vol. 14. World Scientific, Singapore (1995)Google Scholar
  36. 36.
    Hernández, E., O’Regan, D.: Controllability of Volterra–Fredholm type systems in Banach spaces. J. Franklin Inst. 346, 95–101 (2009)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Chalishajar, D.N.: Controllability of mixed Volterra–Fredholm type integro-differential systems in Banach space. J. Franklin Inst. 344, 12–21 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Chalishajar, D.N.: Controllability of second order impulsive neutral functional differential inclusions with infinite delay. J. Optim. Theory Appl. 154, 672–684 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Sakthivel, R., Mahmudov, N.I., Lee, S.-G.: Controllability of nonlinear impulsive stochastic systems. Int. J. Control 82, 801–807 (2009)CrossRefGoogle Scholar
  40. 40.
    Balachandran, K., Sathya, R.: Controllability of neutral impulsive stochastic quasilinear integrodifferential systems with nonlocal conditions. Electron. J. Differ. Equ. 2011, 1–15 (2011)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Arthi, G., Park, J.H., Jung, H.Y.: Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay. Appl. Math. Comput. 248, 328–341 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ahmed, H.M.: Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion. IMA J. Math. Control Inform. 32, 781–794 (2015)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Xiong, J., Liu, G., Su, L.: Controllability of nonlinear impulsive stochastic evolution systems driven by fractional Brownian motion. Math. Probl. Eng. 2015, 1–9 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Hale, J.K., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Haase, M.: The functional calculus for sectorial operators. Operator Theory: Advances and Applications, vol. 169. Birkhauser-Verlag, Basel (2006)Google Scholar
  46. 46.
    Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Banas, J., Goebel, K.: Measure of Noncompactness in Banach Space, Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)zbMATHGoogle Scholar
  48. 48.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  49. 49.
    Mönch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Larsen, R.: Functional Analysis. Decker Inc., New York (1973)zbMATHGoogle Scholar
  51. 51.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefGoogle Scholar
  52. 52.
    Hino, Y., Murakami, S., Naito, T.: Functional-differential equations with infinite delay. In: Stahy, S. (ed.) Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsHexi UniversityZhangyePeople’s Republic of China

Personalised recommendations