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Research Article

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

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Received 12 May 2023, Accepted 17 Mar 2024, Published online: 03 Apr 2024

References

  • D. Bahuguna and M. Malik, A study of nonlocal history-valued retarded differential equations using analytic semigroups, Nonlinear Dyn. Syst. Theory 6(1) (2006), pp. 63–75.
  • D. Bahuguna and S.K. Srivastava, Approximation of solutions to evolution integro-differential equations, J. Appl. Math. Stochastic Anal. 9(3) (1996), pp. 315–322.
  • P. Balasubramaniam, M.S. Ali, and J.H. Kim, Faedo–Galerkin approximate solutions for stochastic semilinear integro-differential equations, Comput. Math. Appl. 58(1) (2009), pp. 48–57.
  • E.G. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Eindhoven, 2001.
  • N.W. Bazley, Approximation of wave equations with reproducing nonlinearities, Nonlinear Anal. TMA.3(4) (1979), pp. 539–546.
  • N.W. Bazley, Global convergence of Faedo–Galerkin approximations to nonlinear wave equations, Nonlinear Anal. TMA. 4(3) (1980), pp. 503–507.
  • R. Chaudhary, M. Malik, and D.N. Pandey, Approximation of solution to second order impulsive differential equation with finite delay, Dyn. Cont. Discr. Impulsive Syst. Series B Appl. Algo. 27 (2020), pp. 223–243.
  • R. Chaudhary, M. Malik, and D.N. Pandey, Approximation of solutions to fractional stochastic integro-differential equations of order α∈(1,2), Stochastics 92(3) (2020), pp. 397–417.
  • R. Chaudhary and D.N. Pandey, Existence and approximation of solution to stochastic fractional integro-differential equation with impulsive effects, Collect. Math. 69(2) (2018), pp. 181–204.
  • X. Chen, P. Hu, S. Shum, and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Operat. Res. 64(6) (2016), pp. 1529–1536.
  • C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum 80(1) (2010), pp. 121–142.
  • G. Cottone, M.D. Paola, and S. Butera, Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach, Probab. Eng. Mech. 26(1) (2011), pp. 101–108.
  • J. Danane, K. Allali, Z. Hammouch, and K.S. Nisar, Mathematical analysis and simulation of a stochastic COVID-19 Lévy jump model with isolation strategy, Results Phy. 23 (2021), pp. 103994.
  • G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Vol. 44, Cambridge University Press, Cambridge, 1992.
  • R. Dhayal and M. Malik, Existence and controllability of impulsive fractional stochastic differential equations driven by Rosenblatt process with Poisson jumps, J. Eng. Math. 130(1) (2021), pp. 1–8.
  • R. Dhayal, M. Malik, and S. Abbas, Solvability and optimal controls of non-instantaneous impulsive stochastic fractional differential equation of order q∈(1,2), Stochastics 93(5) (2021), pp. 780–802.
  • M. Elomari, S. Melliani, and L.S. Chadli, Conformable fractional cosine families of operators, J. Math. Sci. Model. 2(2) (2019), pp. 112–116.
  • L.C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, USA, 2012.
  • H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Elsevier, North Holland, Amsterdam, 1985.
  • M. Ferrara and A. Hadjian, Variational approach to fractional boundary value problems with two control parameters, Elect. J. Differ. Equ. 138 (2015), pp. 1–15.
  • R. Goethel, Faedo–Galerkin approximations in equations of evolution, Math. Methods Appl. Sci. 6(1) (1984), pp. 41–54.
  • Y. Guo, X.B. Shu, F. Xu, and C. Yang, HJB equation for optimal control system with random impulses, Optimization 1 (2022), pp. 1–25. doi:10.1080/02331934.2022.2154607
  • E. Heinz and W.V. Wahl, Zu einem Satz von F. W. Browder über nichtlineare Wellengleichungen, Math. Z. 141(1) (1975), pp. 33–45.
  • E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141(5) (2013), pp. 1641–1649.
  • A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl 90(1) (1982), pp. 12–44.
  • A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  • S. Kumar and P. Sharma, Faedo–Galerkin method for impulsive second-order stochastic integro-differential systems, Chaos, Solitons Fract. 158 (2022), pp. 111946. doi:10.1016/j.chaos.2022.111946
  • V. Lakshmikantham, S. Leela, and J.V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009.
  • K. Li, J. Peng, and J. Gao, Controllability of nonlocal fractional differential systems of order α∈(1,2) in Banach spaces, Reports Math. Phy. 71(1) (2013), pp. 33–43.
  • M. Malik, Existence and approximation of solutions to fractional differential equations, Math. Comput. Model. 49(5–6) (2009), pp. 1164–1172.
  • M. Malik, Faedo–Galerkin approximations to fractional integro-differential equation of order α∈(1,2) with deviated argument, Dyn. Partial Differ. Equ 13(4) (2016), pp. 351–368.
  • M. Malik and R.P. Agarwal, Approximation of solutions to impulsive functional differential equations, J. Appl. Math. Comput. 34(1–2) (2010), pp. 101–112.
  • M. Malik and A. Kumar, Controllability of fractional differential equation of order α∈(1,2) with non–instantaneous impulses, Asian J. Control 20(2) (2018), pp. 935–942.
  • X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Chichester, 1997.
  • P.D. Miletta, Approximation of solutions to evolution equations, Math. Methods Appl. Sci. 17(10) (1994), pp. 753–763.
  • K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons Inc., New York, 1993.
  • B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer Verlag, Berlin, 2003.
  • W.J. Padgett and C.P. Tsokos, A new stochastic formulation of a population growth problem, Math. Biosci. 17(1–2) (1973), pp. 105–120.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  • L. Shu, X.B. Shu, and J. Mao, Approximate controllability and existence of mild solutions for Riemann–Liouville fractional stochastic evolution equations with nonlocal conditions of order 1<α<2, Fract. Calculus Appl. Anal. 22(4) (2019), pp. 1086–1112.
  • X.B. Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1<α<2, Comput. Math. Appl. 64(6) (2012), pp. 2100–2110.
  • E. Sousa, How to approximate the fractional derivative of order 1<α≤2, Int. J. Bifurcat. Chaos. 22(4) (2012), pp. 1250075.
  • J.V. Sousa, M. Fečkan, and E.C. Oliveira, Faedo–Galerkin approximation of mild solutions of fractional functional differential equations, Nonauton. Dyn. Syst. 8(1) (2021), pp. 1–17.
  • C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Hungarica 32(1–2) (1978), pp. 75–96.
  • J. Yang, Y. Tan, and R.A. Cheke, Thresholds for extinction and proliferation in a stochastic tumour-immune model with pulsed comprehensive therapy, Commun. Nonlinear Sci. Numer. Simul.73 (2019), pp. 363–378.

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