Sobolev type fractional stochastic integro-differential evolution

Kerboua Mourad

doi:10.26637/mjm401/018

Abstract

In this paper, we prove the existence of $\alpha$-mild solutions for a class of fractional stochastic integrodifferential evolution equations of sobolev type with fractional sobolev stochastic nonlocal conditions in a real separable Hilbert space. To establish our main results, we use the Banach contraction mapping principle, fractional calculus, stochastic analysis and an analytic semigroup of linear operators. An example is given to illustrate the feasibility of our abstract result.

Keywords:

Fractional stochastic evolution equations, Fixed point technique, fractional stochastic nonlocal condition

Mathematics Subject Classification:

26A33, 46E39, 34K50

Authors Imprint References Funding Related Articles Downloads

Kerboua Mourad Department of Mathematics, Guelma University, 24000 Guelma, Algeria. https://orcid.org/0000-0001-8244-0269

Pages: 155-168
Date Published: 01-01-2016
Vol. 4 No. 01 (2016): Malaya Journal of Matematik (MJM)

Agarwal, R. P., Benchohra, M., & Hamani, S. "A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions." Acta Applicandae Mathematicae 109.3 (2010): 973-1033. DOI: https://doi.org/10.1007/s10440-008-9356-6

Agarwal, R. P., Lakshmikantham, V., & Nieto, J. J. "On the concept of solution for fractional differential equations with uncertainty." Nonlinear Analysis: Theory, Methods & Applications 72.6 (2010): 2859-2862. DOI: https://doi.org/10.1016/j.na.2009.11.029

Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J. J. Fractional Calculus Models and Numerical Methods. 2012. DOI: https://doi.org/10.1142/8180

Byszewski, L, and Lakshmikantham, V. "Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space." Applicable Analysis 40.1 (1991): 11-19. DOI: https://doi.org/10.1080/00036819008839989

Byszewski, L. "Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem." Journal of Mathematical Analysis and Applications 162.2 (1991): 494-505.

Byszewski, L. "Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem." Journal of Mathematical Analysis and Applications 162.2 (1991): 494-505. DOI: https://doi.org/10.1016/0022-247X(91)90164-U

Chang, Y. K., Zhao, Z. H., N'Guérékata, G. M., & Ma, R. "Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations." Nonlinear Analysis: Real World Applications 12.2 (2011): 1130-1139. DOI: https://doi.org/10.1016/j.nonrwa.2010.09.007

Cui, J., Yan, L., & Wu, X. "Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces." Journal of the Korean Statistical Society 41.3 (2012): 279-290. DOI: https://doi.org/10.1016/j.jkss.2011.10.001

Cui, J., & Yan, L. "Existence result for fractional neutral stochastic integro-differential equations with infinite delay." Journal of Physics A: Mathematical and Theoretical 44.33 (2011): 335201. DOI: https://doi.org/10.1088/1751-8113/44/33/335201

Debbouche, A., & El-Borai, M. M. "Weak almost periodic and optimal mild solutions of fractional evolution equations." Electronic Journal of Differential Equations 2009.46 (2009): 1-8.

Debbouche, A., & Nieto, J. J."Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls." Applied Mathematics and Computation 245 (2014): 74-85. DOI: https://doi.org/10.1016/j.amc.2014.07.073

Debbouche, A., & Torres, D. F. "Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions." arXiv preprint arXiv:1409.6028 (2014). DOI: https://doi.org/10.1515/fca-2015-0007

Debbouche, A., Baleanu, D., & Agarwal, R. P. "Nonlocal nonlinear integrodifferential equations of fractional orders." Boundary Value Problems 2012.1 (2012): 1-10. DOI: https://doi.org/10.1186/1687-2770-2012-78

Da Prato, G., & Zabczyk, J. Stochastic equations in infinite dimensions. Vol. 152. Cambridge university press, 2014. DOI: https://doi.org/10.1017/CBO9781107295513

El-Borai, M. M. "Some probability densities and fundamental solutions of fractional evolution equations." Chaos, Solitons & Fractals 14.3 (2002): 433-440. DOI: https://doi.org/10.1016/S0960-0779(01)00208-9

El-Borai, M. M. "On some stochastic fractional integro-differential equations." Advances in Dynamical Systems and Applications 1.1 (2006): 49-57.

Ezzinbi, K., Fu, X., & Hilal, K. "Existence and regularity in the 3b1-norm for some neutral partial differential equations with nonlocal conditions." Nonlinear Analysis: Theory, Methods & Applications 67.5 (2007): 1613-1622. DOI: https://doi.org/10.1016/j.na.2006.08.003

Grecksch, W., & Tudor, C. "Stochastic evolution equations (a Hilbert space approach)." Mathematical Research (1995).

Ichikawa, A. "Stability of semilinear stochastic evolution equations." Journal of Mathematical Analysis and Applications 90.1 (1982): 12-44. DOI: https://doi.org/10.1016/0022-247X(82)90041-5

Kerboua, M., Debbouche, A., & Baleanu, D. "Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces." Abstract and Applied Analysis. Vol. 2013 (2013), Article ID 262191, 10 pages. DOI: https://doi.org/10.1155/2013/262191

Liu, H., & Chang, J. C. "Existence for a class of partial differential equations with nonlocal conditions." Nonlinear Analysis: Theory, Methods & Applications 70.9 (2009): 3076-3083. DOI: https://doi.org/10.1016/j.na.2008.04.009

Liang, J., Liu, J., & Xiao, T. J. "Nonlocal Cauchy problems governed by compact operator families." Nonlinear Analysis: Theory, Methods & Applications 57.2 (2004): 183-189. DOI: https://doi.org/10.1016/j.na.2004.02.007

Mainardi, F. (Ed.). Fractals and fractional calculus in continuum mechanics. No. 378. Springer Verlag, 1997. DOI: https://doi.org/10.1007/978-3-7091-2664-6_7

Malinowska, A. B., & Torres, D. F. M. Fractional Calculus of Variations. Imperial College Press, Singapore, 2012. DOI: https://doi.org/10.1142/p871

Miller, K. S., & Ross, B. "An introduction to the fractional calculus and fractional differential equations." (1993).

Pazy, A. Semigroups of linear operators and applications to partial differential equations. 1983. DOI: https://doi.org/10.1007/978-1-4612-5561-1

Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198. Academic press, 1998.

Ren, Y., & Sakthivel, R. "Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps." Journal of Mathematical Physics $53.7(2012): 073517$ DOI: https://doi.org/10.1063/1.4739406

Sakthivel, R., & Luo, J. "Asymptotic stability of impulsive stochastic partial differential equations with infinite delays." Journal of Mathematical Analysis and Applications 356.1 (2009): 1-6. DOI: https://doi.org/10.1016/j.jmaa.2009.02.002

Sakthivel, R., & Ren, Y. "Exponential stability of second-order stochastic evolution equations with Poisson jumps." Communications in Nonlinear Science and Numerical Simulation 17.12 (2012): 45174523. DOI: https://doi.org/10.1016/j.cnsns.2012.04.020

Sakthivel, R., Revathi, P., & Mahmudov, N. I. "Asymptotic stability of fractional stochastic neutral differential equations with infinite delays." Abstract and Applied Analysis. Vol. 2013. Hindawi Publishing Corporation, 2013. DOI: https://doi.org/10.1155/2013/769257

Sakthivel, R., Revathi, P., & Ren, Y. "Existence of solutions for nonlinear fractional stochastic differential equations." Nonlinear Analysis: Theory, Methods & Applications 81 (2013): 70-86. DOI: https://doi.org/10.1016/j.na.2012.10.009

Wang, J., & Zhou, Y. "A class of fractional evolution equations and optimal controls." Nonlinear Analysis: Real World Applications 12.1 (2011): 262-272. DOI: https://doi.org/10.1016/j.nonrwa.2010.06.013

Wang, R. N., Xiao, T. J., & Liang, J. "A note on the fractional Cauchy problems with nonlocal initial conditions." Applied Mathematics Letters 24.8 (2011): 1435-1442. DOI: https://doi.org/10.1016/j.aml.2011.03.026

Yan, Z., & Yan, X. "Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces." Zeitschrift für angewandte Mathematik und Physik 64.3 (2013): 573-590. DOI: https://doi.org/10.1007/s00033-012-0249-1

Yan, Z., & Yan, X. "Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay." Collectanea Mathematica 64.2 (2013): 235-250. DOI: https://doi.org/10.1007/s13348-012-0063-2