June  2022, 11(3): 925-937. doi: 10.3934/eect.2021031

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 Published  June 2022 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 and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031
References:
[1]

G. ArthiJ. 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.

[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.

[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.

[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.

[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.

[6]

A. BoudaouiT. 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.

[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.

[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.

[9]

T. CaraballoM. 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.

[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.

[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.

[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.

[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.

[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. 

[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.

[16]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore, 2000. doi: 10.1142/9789812817747.

[17]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408. 

[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.

[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. 

[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.

[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.

[22]

R. Mabel LizzyK. 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.

[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.

[24] I. Podlubny, Fractional Differential Equations, Academic press, an Diego, 1999. 
[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.

[26]

D. H. Abdel RahmanS. 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.

[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.

[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.

[29]

R. SakthivelR. GaneshY. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508. 

[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.

[31]

B. Sundara VadivooR. RamachandranJ. CaoH. 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.

[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.

[33]

J. R. WangM. 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.

[34]

X. ZhangP. AgarwalZ. LiuH. PengF. 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.

show all references

References:
[1]

G. ArthiJ. 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.

[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.

[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.

[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.

[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.

[6]

A. BoudaouiT. 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.

[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.

[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.

[9]

T. CaraballoM. 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.

[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.

[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.

[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.

[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.

[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. 

[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.

[16]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore, 2000. doi: 10.1142/9789812817747.

[17]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408. 

[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.

[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. 

[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.

[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.

[22]

R. Mabel LizzyK. 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.

[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.

[24] I. Podlubny, Fractional Differential Equations, Academic press, an Diego, 1999. 
[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.

[26]

D. H. Abdel RahmanS. 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.

[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.

[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.

[29]

R. SakthivelR. GaneshY. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508. 

[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.

[31]

B. Sundara VadivooR. RamachandranJ. CaoH. 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.

[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.

[33]

J. R. WangM. 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.

[34]

X. ZhangP. AgarwalZ. LiuH. PengF. 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.

[1]

Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040

[2]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032

[3]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

[4]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016

[5]

Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106

[6]

Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar, Chokkalingam Ravichandran. Results on controllability of non-densely characterized neutral fractional delay differential system. Evolution Equations and Control Theory, 2021, 10 (3) : 619-631. doi: 10.3934/eect.2020083

[7]

Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022028

[8]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[9]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

[10]

Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 67-84. doi: 10.3934/dcdss.2020004

[11]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[12]

Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad, Aziz Khan. S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evolution Equations and Control Theory, 2021, 10 (4) : 733-748. doi: 10.3934/eect.2020089

[13]

Xuan-Xuan Xi, Mimi Hou, Xian-Feng Zhou, Yanhua Wen. Approximate controllability of fractional neutral evolution systems of hyperbolic type. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021035

[14]

Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437

[15]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

[16]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control and Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016

[17]

Therese Mur, Hernan R. Henriquez. Relative controllability of linear systems of fractional order with delay. Mathematical Control and Related Fields, 2015, 5 (4) : 845-858. doi: 10.3934/mcrf.2015.5.845

[18]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[19]

Yajing Li, Yejuan Wang. The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027

[20]

George A. Anastassiou. Iyengar-Hilfer fractional inequalities. Mathematical Foundations of Computing, 2021, 4 (4) : 221-252. doi: 10.3934/mfc.2021004

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (500)
  • HTML views (493)
  • Cited by (0)

Other articles
by authors

[Back to Top]