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doi: 10.3934/eect.2021031
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## Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion

 1 Mathematics Department, College of Science, Qassim University, P.O.Box 6644, Buraydah 51452, Saudi Arabia 2 Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt

* Corresponding author: Hamdy M. Ahmed

Received  June 2020 Revised  May 2021 Early access July 2021

In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.

Citation: Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations & Control Theory, doi: 10.3934/eect.2021031
##### References:
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Hajji, Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion, Statistics and Probability Letters, 129 (2017), 222-229.  doi: 10.1016/j.spl.2017.06.006.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [10] J. Cui and Y. Litan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 335201, 16pp. doi: 10.1088/1751-8113/44/33/335201.  Google Scholar [11] A. Chadha and N. Pandey Dwijendra, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Analysis, 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [12] A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos, Solitons & Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023.  Google Scholar [13] M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 10 (2010), 761-783.  doi: 10.1007/s00028-010-0069-8.  Google Scholar [14] M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H > \frac{1}{2}$, Bernoulli, 12 (2006), 85-100.   Google Scholar [15] H. Gu and H. J. J. 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Luo, An averaging principle for stochastic fractional differential equations with time-delays, Applied Mathematics Letters, 105 (2020), 106290, 8pp. doi: 10.1016/j.aml.2020.106290.  Google Scholar [21] J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57.  doi: 10.1007/BF00275860.  Google Scholar [22] R. Mabel Lizzy, K. Balachandran and M. Suvinthra, Controllability of nonlinear stochastic fractional systems with distributed delays in control, Journal of Control and Decision, 4 (2017), 153-168.  doi: 10.1080/23307706.2017.1297690.  Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [24] I. Podlubny, Fractional Differential Equations, Academic press, an Diego, 1999.   Google Scholar [25] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205 (1997), 157-185.  doi: 10.1006/jmaa.1996.5177.  Google Scholar [26] D. H. Abdel Rahman, S. Lakshmanan and A. S. Alkhajeh, A time delay model of tumourimmune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Applied Mathematics and Computation, 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar [27] F. A. Rihan, C. Tunc, S. H. Saker, S. Lakshmanan and R. Rakkiyappan, Applications of delay differential equations in biological systems,, Complexity, 2018 (2018), Article ID 4584389, 3 pages. doi: 10.1155/2018/4584389.  Google Scholar [28] F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson Jumps and optimal control, Discrete Dyn. Nat. Soc., 2017(2017), Article ID 5394528, 11 pages. doi: 10.1016/j.cnsns.2013.05.015.  Google Scholar [29] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508.   Google Scholar [30] R. Sakthivel and R. Yong, Approximate controllability of fractional differential equations with state-dependent delay, Results in Mathematics, 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.  Google Scholar [31] B. Sundara Vadivoo, R. Ramachandran, J. Cao, H. Zhang and X. Li, Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects, International Journal of Control, Automation and Systems, 16 (2018), 659-669.  doi: 10.1007/s12555-017-0281-1.  Google Scholar [32] J. Wang and H. M. Ahmed, Null controllability of nonlocal Hilfer fractional stochastic differential equations, Miskolc Math. Notes, 18 (2017), 1073-1083.  doi: 10.18514/MMN.2017.2396.  Google Scholar [33] J. R. Wang, M. Feckan and Y. Zhou, A survey on impulsive fractional differential equations, Fractional Calculus and Applied Analysis, 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.  Google Scholar [34] X. Zhang, P. Agarwal, Z. Liu, H. Peng, F. You and Y. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order $q > 1$ with finite delays, Advances in Difference Equations, 2017 (2017), 1-18.  doi: 10.1186/s13662-017-1169-3.  Google Scholar

show all references

##### References:
 [1] G. Arthi, J. H. Park and H. Y. Jung, Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulation, 32 (2016), 145-157.  doi: 10.1016/j.cnsns.2015.08.014.  Google Scholar [2] G. Arthi and J. H. Park, On controllability of second-order impulsive neutral integrodifferential systems with infinite delay, IMA J. Math. Control Inf., 32 (2015), 639-657.  doi: 10.1093/imamci/dnu014.  Google Scholar [3] K. Aissani and M. Benchohra, Controllability of fractional integrodifferential equations with state-dependent delay, J. Integral Equations Applications, 28 (2016), 149-167.  doi: 10.1216/JIE-2016-28-2-149.  Google Scholar [4] H. M. Ahmed, Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion, IMA Journal of Mathematical Control and Information, 32 (2015), 781-794.  doi: 10.1093/imamci/dnu019.  Google Scholar [5] H. M. Ahmed and M. M. El-Borai, Hilfer fractional stochastic integro-differential equations, Appl. Math. Comput., 331 (2018), 182-189.  doi: 10.1016/j.amc.2018.03.009.  Google Scholar [6] A. Boudaoui, T. Caraballo and A. Ouahab, Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Applicable Analysis, 95 (2016), 2039-2062.  doi: 10.1080/00036811.2015.1086756.  Google Scholar [7] B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics and Probability Letters, 82 (2012), 1549-1558.  doi: 10.1016/j.spl.2012.04.013.  Google Scholar [8] B. Boufoussi and S. Hajji, Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion, Statistics and Probability Letters, 129 (2017), 222-229.  doi: 10.1016/j.spl.2017.06.006.  Google Scholar [9] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [10] J. Cui and Y. Litan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 335201, 16pp. doi: 10.1088/1751-8113/44/33/335201.  Google Scholar [11] A. Chadha and N. Pandey Dwijendra, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Analysis, 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [12] A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos, Solitons & Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023.  Google Scholar [13] M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 10 (2010), 761-783.  doi: 10.1007/s00028-010-0069-8.  Google Scholar [14] M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H > \frac{1}{2}$, Bernoulli, 12 (2006), 85-100.   Google Scholar [15] H. Gu and H. J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Applied Mathematics and Computation, 257 (2015), 344-354.  doi: 10.1016/j.amc.2014.10.083.  Google Scholar [16] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar [17] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408.   Google Scholar [18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006.  Google Scholar [19] J. Klamka, Stochastic controllability of linear systems with delay in control, Bulletin of the Polish Academy of Sciences, Technical Sciences, 55 (2007), 23-29.   Google Scholar [20] D. Luo, Q. Zhu and Z. Luo, An averaging principle for stochastic fractional differential equations with time-delays, Applied Mathematics Letters, 105 (2020), 106290, 8pp. doi: 10.1016/j.aml.2020.106290.  Google Scholar [21] J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57.  doi: 10.1007/BF00275860.  Google Scholar [22] R. Mabel Lizzy, K. Balachandran and M. Suvinthra, Controllability of nonlinear stochastic fractional systems with distributed delays in control, Journal of Control and Decision, 4 (2017), 153-168.  doi: 10.1080/23307706.2017.1297690.  Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [24] I. Podlubny, Fractional Differential Equations, Academic press, an Diego, 1999.   Google Scholar [25] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205 (1997), 157-185.  doi: 10.1006/jmaa.1996.5177.  Google Scholar [26] D. H. Abdel Rahman, S. Lakshmanan and A. S. Alkhajeh, A time delay model of tumourimmune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Applied Mathematics and Computation, 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar [27] F. A. Rihan, C. Tunc, S. H. Saker, S. Lakshmanan and R. Rakkiyappan, Applications of delay differential equations in biological systems,, Complexity, 2018 (2018), Article ID 4584389, 3 pages. doi: 10.1155/2018/4584389.  Google Scholar [28] F. A. Rihan, C. Rajivganthi, P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson Jumps and optimal control, Discrete Dyn. Nat. Soc., 2017(2017), Article ID 5394528, 11 pages. doi: 10.1016/j.cnsns.2013.05.015.  Google Scholar [29] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508.   Google Scholar [30] R. Sakthivel and R. Yong, Approximate controllability of fractional differential equations with state-dependent delay, Results in Mathematics, 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.  Google Scholar [31] B. Sundara Vadivoo, R. Ramachandran, J. Cao, H. Zhang and X. Li, Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects, International Journal of Control, Automation and Systems, 16 (2018), 659-669.  doi: 10.1007/s12555-017-0281-1.  Google Scholar [32] J. Wang and H. M. Ahmed, Null controllability of nonlocal Hilfer fractional stochastic differential equations, Miskolc Math. Notes, 18 (2017), 1073-1083.  doi: 10.18514/MMN.2017.2396.  Google Scholar [33] J. R. Wang, M. Feckan and Y. Zhou, A survey on impulsive fractional differential equations, Fractional Calculus and Applied Analysis, 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.  Google Scholar [34] X. Zhang, P. Agarwal, Z. Liu, H. Peng, F. You and Y. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order $q > 1$ with finite delays, Advances in Difference Equations, 2017 (2017), 1-18.  doi: 10.1186/s13662-017-1169-3.  Google Scholar
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