Existence and approximation of solution to stochastic fractional integro-differential equation with impulsive effects

• Autores: Renu Chaudhary, Dwijendra N. Pandey
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 2, 2018, págs. 181-204
• Idioma: inglés
• DOI: 10.1007/s13348-017-0199-1
• Enlaces
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• Resumen

  • In this paper, we study a stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space. Using a finite dimensional subspace, semigroup theory of linear operators and stochastic version of the well-known Banach fixed point theorem is applied to show the existence and uniqueness of an approximate solution. Next, these approximate solutions are shown to form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is then a solution of the original problem. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. In the last, we have given an example to illustrate the applications of the abstract results.

• Referencias bibliográficas
  • Bazley, N.W.: Approximation of wave equations with reproducing nonlinearities. Nonlinear Anal. 3, 539–546 (1979)
  • Bazley, N.W.: Global convergence of Faedo–Galerkin approximations to nonlinear wave equations. Nonlinear Anal. 4, 503–507 (1980)
  • Bahuguna, D., Srivastava, S.K.: Approximation of solutions to evolution integrodifferential equations. J. Appl. Math. Stoch. Anal. 9, 315–322...
  • Bahuguna, D., Shukla, R.: Approximations of solutions to nonlinear Sobolev type evolution equations. Electron. J. Differ. Equ. 31, 1–16 (2003)
  • Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications,...
  • Balasubramaniam, P., Ali, M.S., Kim, J.H.: Faedo–Galerkin approximate solutions for stochastic semilinear integrodifferential equations. Comput....
  • Balasubramaniam, P., Tamilalagan, P.: Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions...
  • Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A 44, 16...
  • Chaddha, A., Pandey, D.N.: Faedo–Galerkin approximation of solution for a nonlocal neutral fractional differential equation with deviating...
  • Chaddha, A., Pandey, D.N.: Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions. Comput....
  • Chaudhary, R., Pandey, D.N.: Approximation of solutions to a delay equation with a random forcing term and non local conditions. J. Integral...
  • Chaudhary, R., Pandey, D.N.: Approximation of solutions to stochastic fractional integro-differential equation with deviated argument. Differ....
  • Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge...
  • Dabas, J., Chauhan, A.: Existence and uniqueness of mild solutions for an impulsive neutral fractional integro-differential equation with...
  • El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440...
  • Feĉkan, M., Zhou, Y., Wang, J.-R.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear...
  • Feĉkan, M., Zhou, Y., Wang, J.-R.: Response to Comments on the concept of existence of solution for impulsive fractional differential equations....
  • Gard, T.C.: Introduction to Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 114. Dekker,...
  • Heinz, E., Von Wahl, W.: Zn einem Satz von F.W. Browder uber nichtlineare Wellengleichungen. Math. Z. 141, 33–45 (1974)
  • Hu, L., Ren, Y.: Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays. Acta Appl....
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
  • Kumar, P., Pandey, D.N., Bahuguna, D.: Existence of piecewise continuous mild solutions for impulsive functional differential equations with...
  • Kumar, P., Pandey, D.N., Bahuguna, D.: Approximation of solutions to a fractional differential equation with a deviating argument. Differ....
  • Lakshmikantham, V., Bainov, D., Simeonov, P.S.: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. World Scientic,...
  • Lin, A., Ren, Y., Xia, N.: On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators. Math....
  • Murakami, H.: On non-linear ordinary and evolution equations. Funkcial. Ekvac. 9, 151–162 (1966)
  • Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication,...
  • Miletta, P.D.: Approximation of solutions to evolution equations. Math. Methods Appl. Sci. 17, 753–763 (1994)
  • Mao, X.R.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
  • Oksendal, B.: Stochastic Differential Equations, 5th edn. Springer, Berlin (2002)
  • Pazy, A.: Semi-Groups of Linear Operator and Applications of Partial Differential Equations. Springer, New York (1983)
  • Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)
  • Park, J.Y., Jeong, J.U.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays....
  • Segal, I.: Non-linear semi-groups. Ann. Math. 2(78), 339–364 (1963)
  • Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon (1993)
  • Tamilalagan, P., Balasubramaniam, P.: Approximate controllability of fractional stochastic differential equations driven by mixed fractional...
  • Wang, J.-R., Feĉkan, M., Zhou, Y.: On the new concept of solution and existence results for impulsive fractional evolution equations. Dyn....
  • Wang, G., Ahmad, B., Zhang, L., Nieto, J.J.: Comments on the concept of existence of solution for impulsive fractional differential equations....
  • Yan, Z., Lu, F.: On approximate controllability of fractional stochastic neutral integro-differential inclusions with infiite delay. Appl....
  • Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)