Discontinuity, Nonlinearity, and Complexity
Existence of Solutions of Stochastic Fractional Integrodifferential Equations
Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 55--65 | DOI:10.5890/DNC.2018.03.005
P. Umamaheswari, K. Balachandran, N. Annapoorani
Department of Mathematics, Bharathiar University, Coimbatore 641046, India
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Abstract
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.
Acknowledgments
The authors are thankful to the referees for the improvements of the paper.
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. |