Existence of Solutions of Stochastic Fractional Integrodifferential Equations
DOI:
https://doi.org/10.5890/DNC.2018.03.005Abstract
In this paper, a general class of stochastic fractional integrodifferential equations is investigated. The Picard-Lindel ¨of successive approximation scheme is used to establish the existence of solutions. The uniqueness of the solution is also studied under suitable conditions.References
[1] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
[2] Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Equations, John Wiley & Sons, New York.
[3] Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New York.
[4] Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional Integrals and Derivatives: Theory and Appliations, Gordon and Breach, Amsterdam.
[5] Itô, K. (1978), Stochastic Differential Equations,Wiley Interscience, New York.
[6] Biagini, F., Hu, Y., Øksendal, B., and Zhang, T. (2008), Stochastic calculus for fractional Brownian motion and applications, Springer, New York.
[7] Da Prato, G. and Zabczyk, J. (1992), Stochastic differential equations in infinite dimensions, Cambridge University Press, Cambridge.
[8] Pedjeu, J.C. and Sathananthan, S. (2003), Fundamental properties of stochastic integrodifferential equations-I, Existence and uniqueness results, International Journal of Pure and Applied Mathematics, 7, 337-355.
[9] Akilandeeswari, A., Balanchandran, K., Rivero, M., and Trujilo, J.J. (2017), On the solution of partial integrodifferential equations of fractional order, Tbilisi Mathematical Journal, 19, 19-29.
[10] Haseena, A., Suvinthra, M., and Annapoorani, N. (2016), On large deviations of stochastic integrodifferential equations with Brownian motion, Discontinuity, Nonlinearity and Complexity, in press.
[11] Lakshmikantham, V. and Rama Mohan Rao, M. (1995), Theory of Integrodifferential Equations, Gordon & Breach Publishers, Amsterdam.
[12] Mabel Lizzy, R., Balachandran, K., and Kim, J.K. (2016), On stochastic quasilinear evolution equations in Hilbert space, Nonlinear Functional Analysis and Applications, 21, 307-324.
[13] Mao, X. (2007), Stochastic Differential Equation and Applications, Second Edition, Horwood Publishing Limited, UK.
[14] Suvinthra, M., Balachandran, K., and Kim, J.K. (2015), Large deviations for stochastic differential equations with deviating argumentts, Nonlinear Functional Analysis and Applications, 20, 659-674.
[15] Umamaheswari, P., Balachandran, K., and Annapoorani, N. (2017), On the solution of stochastic fractional integrodifferential equations, Nonlinear Functional Analysis and Applications, 22, 35-354.
[16] Elliott, R.J. (1982), Stochastic Calculus and Applications, Springer-Verlag, New York.
[17] Friedman, A. (1975), Stochastic Differential Equations and Applications, Vol-1, Academic Press, New York.
[18] Gihman, I.I. and Skorohod, A.V. (1972), Stochastic Differential Equations, Springer, New York.
[19] Kamrani, M. (2015), Numerical solution of stochastic fractional differential equations, Numerical Algorithms, 68, 81-93.
[20] Pedjeu, J.C. and Ladde, G.S. (2012), Stochastic fractional differential equations: modeling, method and analysis, Chaos, Solitons & Fractals, 45, 279-293.
[21] Karatzas, I. and Sherve, S. (1991), Brownian Motion and Stochastic Calculus, Springer, New York.
[22] Ladde, G.S. and Lakshmikantham, V. (1980), Random differential inequalities, Academic Press, New York.
[23] Yamada, T. and Watanabe, S. (1971), On the uniqueness of solutions of stochastic differential equations, Journal of Mathematics of Kyoto University, 11, 155-167.
[24] Kloeden, P.E. and Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York.
[25] Mao, X. (2003), Numerical solutions of stochastic functional differential equations, LMS Journal of Computation and Mathematics, 6, 141-161.
[26] Øksendal, B. (2003), Stochastic Differential Equations, An Introduction with Applications, Springer-Verlag, Heidelberg.
[27] Arnold, L. (1974), Stochastic Differential Equations: Theory and Applications, JohnWiley & Sons, New York.
[28] Evans, L.C. (2014), An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence.
[29] Taniguchi, T. (1992), Successive approximations to solutions of stochastic differential equations, Journal of Differential Equations, 96 152-169.
[30] Yamada, T. (1981), On the successive approximation of solutions of stochastic differential equations, Kyoto Journal of Mathematics, 21, 501-515.
[31] Jumarie, G. (2004), Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations, Chaos, Solitons and Fractals, 22, 907-925.
[32] Jumarie, G. (2006), New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Mathematical and Computer Modelling, 44, 231-254.
[33] Allen, E.J., Novosel, S.J., and Zhang, Z. (1998), Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics and Stochastic Reports, 64, 117-142.
[34] Diethelm, K. (2010), The Analysis of Fractional Differential Equations, Springer, New York.
[35] Diethelm, K. and Ford, K. (2002), Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265, 229-248.
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